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This answers a question of Aschenbrenner, Friedl and Wilton, and provides the first examples of irreducible manifolds with $b_1=1$ that are known not to be surgery on a knot in the 3-sphere. One family consists of Seifert fibered 3-manifolds, while each member of the other family is not even homology cobordant to any Seifert fibered 3-manifold. None of our examples are homology cobordant to any manifold obtained by Dehn surgery along a knot in the 3-sphere.\\end{abstract}\n\n\\maketitle\n\n\n\\section{Introduction}\nIt is a well-known theorem of Lickorish \\cite{Lickorish:1962-1} and Wallace \\cite{Wallace:1960-1} that every closed, oriented 3-manifold is obtained by Dehn surgery on a link in the three-sphere. This leads one to wonder how the complexity of a $3$-manifold is reflected in the links which yield it through surgery, and conversely. A natural yet difficult goal in this vein is to determine the minimum number of components of a link on which one can perform surgery to produce a given 3-manifold. In particular, one can ask which 3-manifolds are obtained by Dehn surgery on a \\emph{knot} in $S^3$. If, following \\cite{Auckly:1997-1}, we define the {\\it surgery number} $DS(Y)$ of a closed 3-manifold $Y$ to be the smallest number of components of a link in $S^3$ yielding $Y$ by (Dehn) surgery, we ask for conditions under which $DS(Y) >1$.\n\n The fundamental group provides some information on this problem. Indeed, if a closed, oriented 3-manifold $Y$ has $DS(Y) = 1$, then the van Kampen theorem implies that $\\pi_1(Y)$ is normally generated by a single element (which is represented by a meridian of $K$). In particular, $\\pi_1(Y)$ has weight one and $H_1(Y;\\mathbb{Z})$ is cyclic. (Recall that the \\emph{weight} of a group $G$ is the minimum number of normal generators of $G$.)\n \nA more sophisticated topological obstruction to being surgery on a knot comes from essential 2-spheres in 3-manifolds. While Dehn surgery on a knot can produce a non-prime 3-manifold, the \\emph{cabling conjecture} \\cite[Conjecture~A]{Acuna-Short:1986} asserts that this is quite rare and occurs only in the case of $pq$-surgery on a $(p,q)$-cable knot. It would imply, in particular, that a non-prime 3-manifold obtained by surgery on a knot in $S^3$ has only two prime summands, one of which is a lens space. Deep work of Gordon-Luecke \\cite[Corollary~3.1]{Gordon-Luecke:1989-1} and Gabai \\cite[Theorem~8.3]{Gabai:1987-3} verify this in the case of homology spheres and homology $S^1\\times S^2$'s, respectively, showing more generally that if such a manifold is obtained by surgery on a non-trivial knot, then $Y$ is irreducible. \n\n\nIt is natural to ask whether these conditions are sufficient to conclude that $Y$ is obtained from $S^3$ by Dehn surgery on a knot. In the case of homology 3-spheres, Auckly \\cite{Auckly:1997-1} used Taubes' end-periodic diagonalization theorem \\cite[Theorem~1.4]{Taubes} to give examples of hyperbolic, hence irreducible, homology 3-spheres with $DS(Y)>1$. It remains unknown, however, if any of Auckly's examples have weight-one fundamental group. More recently, Hom, Karakurt and Lidman \\cite{Hom-Karakurt-Lidman:2016-1} used Heegaard Floer homology to obstruct infinitely many irreducible Seifert fibered homology 3-spheres with weight-one fundamental groups from being obtained by Dehn surgery on a knot. In \\cite{Hom-Lidman:2016-1}, Hom and Lidman gave infinitely many such hyperbolic examples, as well as infinitely many examples with arbitrary JSJ decompositions. Currently, we do not know whether the examples of \\cite{Hom-Lidman:2016-1} have weight-one fundamental groups or not.\n\nIt is interesting to note, however, that a longstanding open problem of Wiegold (\\cite[Problem 5.52]{Mazurov-Khukhro:2014-1} and \\cite[Problem 15]{Gersten:1987-1}) asks whether {\\it every} finitely presented perfect group has weight one. The question would be answered negatively if there is a homology 3-sphere whose fundamental group has weight $\\geq 2$. \n\nUsing $\\mathbb{Q}\/\\mathbb{Z}$-valued linking form and their surgery formulae for Casson invariant, Boyer and Lines \\cite[Theorem~5.6]{Boyer-Lines:1990-1} gave infinitely many irreducible homology lens spaces which have weight-one fundamental group, but are not obtained by Dehn surgery on a knot. In \\cite{Hoffman-Walsh:2015-1}, Hoffman and Walsh gave infinitely many hyperbolic examples of this sort.\n\nFor the case that $Y$ is a homology $S^1\\times S^2$, significantly less is known. Aschenbrenner, Friedl and Wilton \\cite{Aschenbrenner-Friedl-Wilton:2015-1} asked the following question. \n\n\\begin{question}[{\\cite[Question 7.1.5]{Aschenbrenner-Friedl-Wilton:2015-1}}]\\label{question:AFW}Let $M$ be a closed, orientable, irreducible $3$-manifold such that $b_1(M) = 1$ and $\\pi_1(M)$ has weight $1$. Is $M$ the result of Dehn surgery along a knot in $S^3$?\n\\end{question}\n\nNote that if $M$ as in the question does arise from surgery on a knot in $S^3$ then necessarily the surgery coefficient is zero.\n\nThe purpose of this paper is to give two families of examples that show the answer to Question \\ref{question:AFW} is negative. The first family shows that there exist homology $S^1\\times S^2$'s not smoothly homology cobordant to any Seifert manifold or to zero surgery on a knot; we recall that two closed, oriented 3-manifolds $M$ and $N$ are \\emph{homology cobordant} if there is a smooth oriented cobordism $W$ between them for which the inclusion maps $M\\hookrightarrow W \\hookleftarrow N$ induce isomorphisms on integral homology.\n\n\t\\begin{theoremalpha}\\label{theorem:A} The family of closed, oriented $3$-manifolds $\\{M_k\\}_{k\\geq1}$ described by the surgery diagram in Figure \\ref{figure:Hedden-Mark} satisfies the following.\n\t\\begin{enumerate}\t\t\n\t\t\\item $M_k$ is irreducible with first homology ${\\mathbb Z}$ and $\\pi_1(M_k)$ of weight $1$.\n\t\t\\item $M_k$ is not the result of Dehn surgery along a knot in $S^3$.\n\t\t\\item $M_k$ is not homology cobordant to Dehn surgery along a knot in $S^3$.\n\t\t\\item\\label{item:Seifert} $M_k$ is not homology cobordant to any Seifert fibered $3$-manifold.\n\n\t\t\\item $M_k$ is not homology cobordant to $M_l$ if $k\\neq l$.\n\t\\end{enumerate}\n\t\\end{theoremalpha}\n\\begin{figure}[htb!]\n \\centering\n \\includegraphics[scale=1]{figure1}\n \\caption{A surgery diagram of $M_k$ ($k\\geq 1)$.}\n \\label{figure:Hedden-Mark}\n\\end{figure}\n\nThe first property of $M_k$ is relatively elementary; in particular it follows from some general topological results about spliced manifolds. As we show in the next section, any splice of non-trivial knot complements in the 3-sphere is irreducible and has weight one fundamental group, from which our claims about $M_k$ will follow.\n\nTo show that $M_k$ is not the result of Dehn surgery along a knot in $S^3$, we use a Heegaard Floer theoretic obstruction developed by \nOzsv\\'{a}th and Szab\\'{o} in \\cite{Ozsvath-Szabo:2003-2}. They showed that certain numerical ``correction terms\" $d_{1\/2}$ and $d_{-1\/2}$ satisfy \n\\begin{equation}\\label{dconstraints}\nd_{1\/2}(M)\\leq \\tfrac{1}{2}\\quad\\mbox{and}\\quad d_{-1\/2}(M)\\geq -\\tfrac{1}{2}\n\\end{equation}\nwhenever $M$ is obtained from 0-surgery on a knot in $S^3$ (see Theorem~\\ref{thm:OS}). We will show that $d_{-1\/2}(M_k)=-\\frac{5}{2}$, and hence $M_k$ is not the result of Dehn surgery on a knot in~$S^3$. The correction terms are actually invariants of homology cobordism, from which it follows that none of the $M_k$ are even homology cobordant to surgery on a knot in~$S^3$. This feature of our examples distinguishes it from the analogous gauge and Floer theoretic results for homology spheres mentioned above. Indeed, the techniques of Auckly or Hom, Lidman, Karakurt are not invariant under homology cobordism; in the former, this is due to a condition on $\\pi_1$ in Taubes' result on end periodic manifolds, and in the latter because the reduced Floer homology is not invariant under homology cobordism (though see \\cite{Hendricks-Hom-Lidman:2018} for some results in that direction).\n\nTo show that our examples $M_k$ are not homology cobordant to any Seifert fibered 3-manifold, we prove a general result, Theorem~\\ref{theorem:dofSeifert}, about the correction terms of Seifert fibered 3-manifold $M$ with first homology $ \\mathbb{Z}$: we show that any Seifert manifold with the homology of $S^1\\times S^2$ satisfies the same constraints \\eqref{dconstraints} as the result of 0-surgery does. Part $(6)$ of our theorem immediately follows. We remark that it was only recently shown by Stoffregen (preceded by unpublished work of Fr{\\o}yshov) that there exist homology 3-spheres that are not homology cobordant to Seifert manifolds, or equivalently that not every element of the integral homology cobordism group is represented by a Seifert manifold. To be precise, Stoffregen showed in \\cite[Corollary~1.11]{Stoffregen:2015-1} that $\\Sigma(2,3,11)\\#\\Sigma(2,3,11)$ is not homology cobordant to any Seifert fibered homology 3-sphere by using homology cobordism invariants from Pin(2)-equivariant Seiberg-Witten Floer homology. \n\n\\subsubsection*{Hyperbolic examples}\n\n\tFor any closed, orientable 3-manifold $M$ with a chosen Heegaard splitting, Myers gives an explicit homology cobordism from $M$ to a hyperbolic, orientable 3-manifold \\cite{Myers:1983-1}. By using these homology cobordisms, we can obtain hyperbolic, orientable 3-manifolds $Z_k$ with first homology $\\mathbb{Z}$ which are homology cobordant to~$M_k$. Since $d_{-1\/2}$ is a homology cobordism invariant, $Z_k$ is also not the result of Dehn surgery along a knot in $S^3$ by Theorem~\\ref{thm:OS}. \n\t\n\t\\begin{corollaryalpha}There is a family of closed, orientable irreducible $3$-manifolds $\\{Z_k\\}_{k\\geq 1}$ satisfying the following.\n\t\\begin{enumerate}\n\t\t\\item $Z_k$ is hyperbolic with first homology ${\\mathbb Z}$.\n\t\t\\item $Z_k$ is not the result of Dehn surgery along a knot in $S^3$.\n\t\t\t\t\t\\item $Z_k$ is not homology cobordant to any Seifert fibered $3$-manifold.\n\n\t\t\\item $Z_k$ is not homology cobordant to $Z_l$ if $k\\neq l$.\n\t\\end{enumerate}\n\t\\end{corollaryalpha}\n\t Myers' cobordisms may not preserve the weight of the fundamental groups at hand. If $\\pi_1(Z_k)$ has weight one, then $Z_k$ would provide a negative answer to the following question.\n\t\n\t\\begin{question}Let $M$ be a closed, orientable, hyperbolic $3$-manifold with $b_1(M) = 1$ and $\\pi_1(M)$ of weight $1$. Is $M$ the result of Dehn surgery along a knot in $S^3$?\n\n\t\\end{question}\n\tWe remark that the question is also open for integral homology 3-spheres.\n\t\n\\subsubsection*{Seifert examples}\nFrom the previous remarks, it follows that the correction terms $d_{\\pm 1\/2}$ cannot show that a Seifert manifold with the homology of $S^1\\times S^2$ has $DS> 1$. Using an obstruction based on the classical Rohlin invariant instead, we prove the following.\n\n\\begin{theoremalpha}\\label{theorem:B}\nLet $\\{N_k\\}_{k\\geq 1}$ be the family of $3$-manifolds described by the surgery diagram in Figure \\ref{figure:OS}. Then\n\\begin{enumerate}\n\\item $N_k$ is irreducible with first homology ${\\mathbb Z}$ and $\\pi_1(N_k)$ of weight $1$.\n\\item $N_k$ is a Seifert manifold over $S^2$ with three exceptional fibers.\n\\item If $k$ is odd, $N_k$ is not obtained by Dehn surgery on a knot in $S^3$.\n\\item If $k$ is odd, $N_k$ is not homology cobordant to Dehn surgery along a knot in $S^3$.\n\\end{enumerate}\n\\end{theoremalpha}\n\n\\begin{figure}[htb!]\n \\centering\n \\includegraphics[scale=1]{figure2}\n \\caption{A surgery diagram of $N_k$ $(k\\geq 1)$.}\n \\label{figure:OS}\n\\end{figure}\n\nIndependent of questions involving weight or homology cobordism, our results provide the first known examples of irreducible homology $S^1\\times S^2$'s which are not homeomorphic to surgery on a knot in $S^3$. To clarify the literature, it is worth mentioning here that in \\cite[Section~10.2]{Ozsvath-Szabo:2003-2} Ozsv\\'{a}th and Szab\\'{o} argued based on the correction term obstruction that the manifold $N_1$ shown in Figure \\ref{figure:OS} is not the result of Dehn surgery on a knot in $S^3$. Unfortunately, as we mentioned above, since $N_1$ is Seifert fibered the correction terms do not actually provide obstructions to $DS = 1$. We point out in Section \\ref{section:Ozsvath-Szabo} where their calculation goes astray. \n\n\n\n\t\\begin{organization} In the next section, we establish some topological results on spliced manifolds which we will apply to our examples $M_k$. In Section~\\ref{section:background}, we briefly recall the relevant background on Heegaard Floer correction terms and the zero surgery obstruction of Ozsv\\'{a}th and Szab\\'{o}. Section~\\ref{section:dofM_k} is devoted to computation of the correction terms of $M_k$, whose values imply they are not zero surgery on knots in $S^3$ and have the stated homology cobordism properties. In Section \\ref{section:Seifert} we prove the estimates on the correction terms of Seifert manifolds and finish the proof of Theorem \\ref{theorem:A}. Section \\ref{section:rokhlin} shows how the Rohlin invariant gives a different obstruction to $DS = 1$, and in Section \\ref{section:Ozsvath-Szabo} we prove Theorem \\ref{theorem:B}.\n\t\\end{organization}\n\t\n\t\n\t\\begin{acknowledgement}The authors would like to thank Marco Golla and Jennifer Hom for their helpful comments and encouragement. Especially, Marco Golla gave several valuable comments on the correction terms of $N_1$ which are reflected in Section~\\ref{section:Ozsvath-Szabo}. Part of this work was done while Min Hoon Kim was visiting Michigan State University, and he thanks MSU for its generous hospitality and support. We also thank the University of Virginia for supporting an extended visit by Matthew Hedden in 2007, which led to the discovery of the manifolds in Theorem \\ref{theorem:A}. Matthew Hedden's work on this project was partially supported by NSF CAREER grant DMS-1150872, DMS-1709016, and an NSF postdoctoral fellowship. Min Hoon Kim was partially supported by the POSCO TJ Park Science Fellowship. Thomas Mark was supported in part by a grant from the Simons Foundation (523795, TM). Kyungbae Park was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (F2018R1C1B6008364).\n\t\t\\end{acknowledgement}\n\t\n\n\\section{Some topological preliminaries}\\label{section:topology}\t\n\nIn this section we verify the topological features---irreducibility and weight one fundamental group---of the manifolds $M_k$ in Theorem \\ref{theorem:A}. These features are consequences of the fact that the manifolds are obtained by a splicing operation. Thus we establish some general results for manifolds obtained through this construction.\n\n\t\n\tGiven two oriented $3$-manifolds with torus boundary, $X_1,X_2$, we will refer to any manifold obtained from them by identifying their boundaries by an orientation reversing diffeomorphism as a {\\em splice} of $X_1$ and $X_2$. Of course the homeomorphism type of a splice depends intimately on the choice of diffeomorphism, but this choice will be irrelevant for the topological results that follow. Note that with this definition Dehn filling is a splice with the unknot complement in $S^3$. We begin with the following observation, which indicates that the manifolds appearing in Theorem \\ref{theorem:A} are splices. \n\n\\begin{proposition}\\label{prop:splice} Let $L$ be the result of connected summing the components of the Hopf link with knots $K_1$ and $K_2$, respectively. Then any integral surgery on $L$ is a splice of the complements of $K_1$ and $K_2$.\n\\end{proposition}\n\\begin{proof} The connected sum operation can be viewed as a splicing operation. More precisely, the connected sum of a link component with a knot $K$ is obtained by removing a neighborhood of the meridian of the component and gluing the complement of $K$ to it by the diffeomorphism which interchanges longitudes and meridians. Thus the result of integral surgery on $L$ is diffeomorphic to integral surgery on the Hopf link, followed by the operation of gluing the complements of $K_1$ and $K_2$ to the complements of the meridians of the Hopf link. But the meridians of the components of the Hopf link, viewed within the surgered manifold, are isotopic to the cores of the surgery solid tori since the surgery slopes are integral. Thus, upon removing the meridians, we arrive back at the complement of the Hopf link, which is homeomorphic to $T^2\\times [0,1]$. The manifold at hand, then, is obtained by gluing the boundary tori of the complements of $K_1$ and $K_2$ to the boundary components of a thickened torus. The result follows immediately.\n\\end{proof}\n\nWe next prove that splices of knot complements in the 3-sphere have fundamental groups of weight one. This follows from a basic result about pushouts of groups.\n\n\\begin{proposition}\\label{prop:pushoutweight} Suppose that $G_1$ and $G_2$ are groups which are normally generated by elements $g_1$ and $g_2$, respectively, and that $\\phi_i:H\\rightarrow G_i$ are homomorphisms. If the image of $\\phi_1$ contains $g_1$, then the pushout $G_1\\ast_H G_2$ is normally generated by a single element\\textup{;} namely, the image of $g_2$ under the defining map $G_2\\rightarrow G_1\\ast_H G_2$.\n\\end{proposition}\n\\begin{proof} In the pushout, $g_1=\\phi_1(x)=\\phi_2(x)$. Now $\\phi_2(x)\\in G_2$, hence can be written as a product of conjugates of $g_2$. Since $g_1$ normally generates $G_1$, it follows that $g_2$ normally generates the pushout.\n\\end{proof}\n\nIt follows at once from van Kampen's theorem that that any splice of complements of knots in the 3-sphere has weight one fundamental group. Indeed, the Wirtinger presentation shows that the fundamental group of a knot complement has weight one, normally generated by a meridian. The homotopy class of the meridian is represented by a curve on the boundary, thereby verifying the hypothesis of the proposition. Of course this reasoning shows more generally that the splice of a knot complement in $S^3$ with {\\em any} manifold with torus boundary and weight one fundamental group also has fundamental group of weight one.\n\nThe discussion to this point shows that the manifolds $M_k$, being splices of knot complements, have weight one fundamental groups. We turn our attention to their irreducibility. As above, we will deduce this property from a more general result about splicing.\n\n\\vskip0.1in\nRecall that a $3$-manifold is {\\em irreducible} if any smoothly embedded $2$-sphere bounds a $3$-ball, and a surface $T$ in a 3-manifold is {\\em incompressible} if any embedded disk $D$ in the manifold for which $D\\cap T=\\partial D$ has the property that $\\partial D$ bounds a disk in $T$ as well.\n\n\\begin{proposition}\\label{prop:irreduciblesplice} Let $X_1$, $X_2$ be irreducible manifolds, each with an incompressible torus as boundary. Then any splice of $X_1$ and $X_2$ is irreducible.\n\\end{proposition}\nThe proposition applies to the complements of non-trivial knots in the $3$-sphere, which are irreducible by Alexander's characterization of the $3$-sphere \\cite{Alexander:1924} (namely, that any smooth $2$-sphere separates into two pieces, each diffeomorphic to a ball), and have incompressible boundary whenever the knot is non-trivial.\n\n\\begin{proof} The proposition follows from a standard ``innermost disk\" argument. More precisely, let $S$ be an embedded $2$-sphere in a splice of $X_1$ and $X_2$, and let $T$ denote the image of the boundary tori, identified within the splice. Then $S$ intersects $T$ in a collection of embedded circles. We claim that we can remove these circles by an isotopy of $S$. This claim would prove the proposition since, after the isotopy, the sphere lies entirely in $X_1$ or $X_2$, where it bounds a ball by hypothesis. \n\n To remove the components of $S\\cap T$, consider a disk $D\\subset S$ which intersects $T$ precisely in $\\partial D$ (a so-called ``innermost disk\", which must exist by compactness of $S\\cap T$ and the Jordan-Sch{\\\"o}nflies theorem). Since $D\\cap T=\\partial D$, the interior of $D$ must lay entirely in one of $X_1$ or $X_2$. Incompressibility of the boundary of these manifolds therefore implies $\\partial D$ bounds a disk embedded in $T$. The union of this latter disk with $D$ is an embedded sphere in either $X_1$ or $X_2$, which bounds a ball by its irreducibility. The ball can be used to isotope $S$ and remove the circle of intersection. Inducting on the number of such circles implies our claim. \n\\end{proof}\n\t\n\t\n\\section{Heegaard Floer theory and Ozsv\\'{a}th-Szab\\'{o}'s 0-surgery obstruction}\\label{section:background}\nIn this section we briefly recall the Heegaard Floer correction terms and an obstruction they yield, due to Ozsv\\'ath and Szab\\'o, to a 3-manifold being obtained by $0$-surgery on a knot in $S^3$. For more detailed exposition, we refer the reader to \\cite{Ozsvath-Szabo:2003-2}.\n\n\nLet $\\mathbb{F}$ be the field with two elements, and $\\mathbb{F}[U]$ be the polynomial ring over $\\mathbb{F}$. Let $Y$ be a closed oriented 3-manifold endowed with a spin$^c$ structure $\\mathfrak{s}$. Heegaard Floer homology associates to the pair $(Y,\\mathfrak{s})$ several relatively graded modules over $\\mathbb{F}[\nU]$, $HF^\\circ(Y,\\mathfrak{s})$, where $\\circ\\in\\{-,+,\\infty\\}$. These Heegaard Floer modules are related by a long exact sequence:\n\\begin{equation*}\n\t\\cdots\\rightarrow HF^-(Y,\\mathfrak{s})\\xrightarrow{\\iota} HF^\\infty(Y,\\mathfrak{s})\\xrightarrow{\\pi} HF^+(Y,\\mathfrak{s})\\rightarrow\\cdots.\n\\end{equation*}\nThe reduced Floer homology, denoted $HF^+_{red}(Y,\\mathfrak{s})$, can be defined either as the cokernel of $\\pi$ or the kernel of $\\iota$ with grading shifted up by one. \n\nIn the case that the spin$^c$-structure $\\mathfrak{s}$ has torsion first Chern class, the relative grading of the corresponding Floer homology modules can be lifted to an \\emph{absolute} $\\mathbb{Q}$-grading. In particular, \n $HF^\\circ(Y,\\mathfrak{s})$ is an absolutely $\\mathbb{Q}$-graded $\\mathbb{F}[U]$-module for any $\\circ\\in\\{-,+,\\infty\\}$. \n\nFor a rational homology 3-sphere $Y$, every spin$^c$ structure will have torsion Chern class, and we define the \\emph{correction term} $d(Y,\\mathfrak{s})\\in \\mathbb{Q}$ to be the minimal $\\mathbb{Q}$-grading of any element in $HF^+(Y,\\mathfrak{s})$ in the image of $\\pi$. A structure theorem \\cite[Theorem 10.1]{Ozsvath-Szabo:2004-2} for the Floer modules states that $HF^\\infty(Y,\\mathfrak{s})\\cong\\mathbb{F}[U,U^{-1}]$, from which it follows that \\[HF^+(Y,\\mathfrak{s})\\cong\\mathcal{T}^+_{d(Y,\\mathfrak{s})}\\oplus HF^+_{red}(Y,\\mathfrak{s}),\\] \nwhere $\\mathcal{T}^+_d$ denotes the $\\mathbb{Q}$-graded $\\mathbb{F}[U]$-module isomorphic to $\\mathbb{F}[U,U^{-1}]\/U\\mathbb{F}[U]$ in which the non-trivial element with lowest grading occurs in grading $d\\in \\mathbb{Q}$. Multiplication by $U$ decreases the $\\mathbb{Q}$-grading by $2$.\n \n \nA 3-manifold $Y$ with $H_1(Y;\\mathbb{Z})\\cong\\mathbb{Z}$ has a unique spin$^c$ structure with torsion (zero) Chern class; we denote this spin$^c$ structure by $\\mathfrak{s}_0$. In this setting, the structure theorem states that $HF^\\infty(Y,\\mathfrak{s})\\cong\\mathbb{F}[U,U^{-1}]\\oplus\\mathbb{F}[U,U^{-1}] $, with the two summands supported in grading $\\pm \\frac{1}{2}$ modulo 2, respectively. We define $d_{1\/2}(Y)$ and $d_{-1\/2}(Y)$ to be the minimal grading of any element in the image of $\\pi$ in $HF^+(Y,\\mathfrak{s}_0)$ supported in the grading $\\frac{1}{2}$ and $-\\frac{1}{2}$ modulo $2$, respectively. It follows that \n\\begin{equation*}\n\tHF^+(Y,\\mathfrak{s}_0)\\cong\\mathcal{T}^+_{d_{-1\/2}(Y)}\\oplus\\mathcal{T}^+_{d_{1\/2}(Y)}\\oplus HF^+_{red}(Y,\\mathfrak{s}_0).\n\\end{equation*}\n\n\nThe key features of the correction terms are certain constraints they place on negative semi-definite 4-manifolds bounded by a given 3-manifold, \\cite[Theorem~9.11]{Ozsvath-Szabo:2003-2}. Applying these constraints to the 4-manifold obtained from a homology cobordism by drilling out a neighborhood of an arc connecting the boundaries yields the following (compare \\cite[Proposition~4.5]{Levine-Ruberman:2014-1}):\n\n\\begin{Homology Cobordism Invariance} If $Y$ and $Y'$ are integral homology cobordant homology manifolds with first homology $\\mathbb{Z}$, then $d_{\\pm 1\/2}(Y)=d_{\\pm 1\/2}(Y')$.\n \\end{Homology Cobordism Invariance}\n\nThe relevance to the surgery question at hand also becomes apparent: if a 3-manifold is obtained by 0-surgery on a knot $K$ in $S^3$, then it bounds a negative semi-definite 4-manifold gotten by attaching a $0$-framed 2-handle to the 4-ball along $K$. Coupling this observation with the constraints mentioned above, and using the fact that $d_{-1\/2}(Y)=-d_{1\/2}(-Y)$ \\cite[Proposition~4.10]{Ozsvath-Szabo:2003-2}, we get the following obstruction:\n\n\n\\begin{zerosurgeryobstruction}\\cite[Corollary 9.13]{Ozsvath-Szabo:2003-2} If $Y$ bounds a homology $S^2\\times D^2$ then $d_{1\/2}(Y)\\leq\\frac{1}{2}$ and $d_{-1\/2}(Y)\\geq -\\frac{1}{2}$. \n\\end{zerosurgeryobstruction}\n\n\n\\noindent The obstruction applies, for instance, if $Y$ is homology cobordant to zero surgery on a knot in a 3-manifold that bounds a smooth contractible 4-manifold.\n\n Drawing on information from the surgery exact triangle, Ozsv\\'{a}th and Szab\\'{o} \\cite[Proposition~4.12]{Ozsvath-Szabo:2003-2} gave a refined statement of the obstruction, which determines the values of the correction terms. We rephrase their result in terms of the non-negative knot invariant $V_0(K)$ introduced by Rasmussen (under the name $h_0(K)$) in \\cite{Rasmussen:2003-1}, and used by Ni-Wu \\cite{Ni-Wu:2015-1}. To see that the following agrees with the stated reference, we recall that $d(S^3_1(K))=-2V_0(K)$, and that the $d$-invariant of $1$-surgery changes sign under orientation reversal (implying $d(S^3_{-1}(K))=2V_0(\\overline{K})$).\n\n\\begin{theorem}[{\\cite[Proposition 4.12]{Ozsvath-Szabo:2003-2}}]\\label{thm:OS}Suppose that $Y$ is obtained by $0$-surgery on a knot $K$ in~$S^3$. Then $d_{1\/2}(Y)=\\frac{1}{2}-2V_0(K)$ and $d_{-1\/2}(Y)=-\\frac{1}{2}+2V_0(\\overline{K})$ where $\\overline{K}$ is the mirror of $K$.\n\\end{theorem}\n\n\\section{Computation of $d_{\\pm 1\/2}(M_k)$}\\label{section:dofM_k}\nConsider the 3-manifold $M_k$ obtained by $(1,1)$ surgery on the link obtained from the Hopf link by connected summing one component with the right-handed trefoil $T_{2,3}$ and the other component with the $(2,4k-1)$ torus knot $T_{2,4k-1}$ as depicted in Figure~\\ref{figure:Hedden-Mark}. In this section, we compute $d_{\\pm 1\/2}(M_k)$ for any $k\\geq 1$. We assume that the reader is familiar with knot Floer homology \\cite{Ozsvath-Szabo:2004-1,Rasmussen:2003-1}.\n\n\\begin{theorem}\\label{thm:dofM_k}For any $k\\geq 1$, $d_{1\/2}(M_k)=-2k+\\frac{1}{2}$ and $d_{-1\/2}(M_k)=-\\frac{5}{2}$.\n\\end{theorem}\n\nWe briefly discuss the strategy of our computation. Consider the knot $J_k$ in $S^3_{1}(T_{2,3})$ depicted in Figure~\\ref{figure:J_k}. Since $H_1(M_k)\\cong \\mathbb{Z}$, $M_k$ is the result of surgery on $S^3_1(T_{2,3})$ along the knot $J_k$ using its Seifert framing. Note that the Seifert framing of $J_k$ is the 1-framing with respect to the blackboard framing of Figure~\\ref{figure:J_k}. Then $d_{\\pm 1\/2}(M_k)$ can be determined by the knot Floer homology $CFK^\\infty(S^3_1(T_{2,3}),J_k)$ using a surgery formula \\cite[Section~4.8]{Ozsvath-Szabo:2008-1}.\n\n\n\\begin{figure}[tb!]\n \\centering\n \\includegraphics[scale=1]{figure3}\n\t\\caption{A knot $J_{k}$ in $S^3_1(T_{2,3})$.}\n\t\\label{figure:J_k}\n\\end{figure}\n\n\nIn order to determine the aforementioned knot Floer homology complex, we first consider the meridian of $T_{2,3}$, viewed as a knot $\\mu\\subset S^3_1(T_{2,3})$. Then the relevant knot $(S^3_1(T_{2,3}),J_k)$ is simply the connected sum of two knots, $(S^3_1(T_{2,3}),\\mu)$ and $(S^3,T_{2,4k-1})$. A K\\\"{u}nneth formula for the knot Floer homology of connected sums then implies \n\\begin{equation}\\label{equation:Kunneth}CFK^\\infty(S^3_1(T_{2,3}),J_k)\\cong CFK^\\infty (S^3_1(T_{2,3}),\\mu)\\otimes CFK^\\infty (T_{2,4k-1}).\\end{equation}\n\nWe can deduce the structure of $CFK^\\infty(S^3_1(T_{2,3}),\\mu)$ using a surgery formula which, together with the K\\\"{u}nneth formula and the well-known structure of the Floer homology of torus knots, will determine the filtered chain homotopy type of $CFK^\\infty(S^3_1(T_{2,3}),J_k)$. Precisely, we prove the following:\n\n\\begin{proposition}\\label{proposition:CFKofJ_k}We have the following filtered chain homotopy equivalences.\n\\begin{enumerate}\n\\item\\label{item:CFKofmeridian} $CFK^\\infty(S^3_1(T_{2,3}),\\mu)\\cong CFK^\\infty (T_{2,-3})[-2]$.\n\\item\\label{item:CFKofJ_k} $CFK^\\infty(S^3_{1}(T_{2,3}),J_k)\\oplus A_0\\cong CFK^\\infty(T_{2,4k-3})[-2]\\oplus A_1$.\n\\end{enumerate}\nHere $[-2]$ means that the Maslov grading is shifted by $-2$, and $A_0$ and $A_1$ are acyclic chain complexes over $\\mathbb{F}[U,U^{-1}]$.\n\\end{proposition}\n\n\n\n\\begin{remark}For $N\\geq 2g(K)$, it is known that $CFK^\\infty(S^3_{-N}(K),\\mu)$ is determined by $CFK^\\infty(S^3,K)$ in \\cite[Theorem~4.2]{Hedden-Kim-Livingston:2016-1} (compare \\cite[Theorem~4.1]{Hedden:2007-1}). Since $1<2g(T_{2,3})$, we cannot apply \\cite[Theorem~4.2]{Hedden-Kim-Livingston:2016-1} to determine $CFK^\\infty(S^3_1(T_{2,3}),\\mu)$. Work in progress of Hedden and Levine on a general surgery formula for the knot Floer homology of $\\mu$ would easily yield the formula. In the case at hand, however, a surgery formula applied for $\\mu$ allows for an ad hoc argument.\\end{remark}\n\n\\begin{proof} (\\ref{item:CFKofmeridian}) The key observation is that the complement of $\\mu \\subset S^3_1(T_{2,3})$ is homeomorphic to the complement of $T_{2,3}\\subset S^3$. Indeed, this can be seen by observing that $\\mu$ is isotopic to the core of the surgery solid torus. It follows that $\\mu$ is a genus one fibered knot. Moreover, $S^3_1(T_{2,3})$ is homeomorphic to the Poincar{\\'e} sphere equipped with the opposite orientation it inherits as the boundary of the resolution of the surface singularity $z^2+w^3+r^5=0$, which is well known and easily seen to be an $L$-space homology sphere with $d$-invariant equal to $-2$. \n\nAs $\\mu$ is a genus one fibered knot in an $L$-space homology sphere, it follows readily that its knot Floer homology must have rank $5$ or $3$, and in the latter case must be isomorphic to that of one of the trefoil knots, with an overall shift in the Maslov grading by the $d$-invariant. To see this, observe that being genus one implies, by the adjunction inequality for knot Floer homology \\cite[Theorem~5.1]{Ozsvath-Szabo:2004-1}, that $\\widehat{HFK}(S^3_1(T_{2,3}),\\mu,i)=0$ for $|i|>1$. As $\\mu$ is fibered, $\\widehat{HFK}(S^3_1(T_{2,3}),\\mu,i)=\\mathbb{F}$ for $i=\\pm1$ \\cite[Theorem~5.1]{Ozsvath-Szabo:2005-1}. Moreover, the Maslov grading of the generator of $\\widehat{HFK}(S^3_1(T_{2,3}),\\mu,1)$ is two higher than that of $\\widehat{HFK}(S^3_1(T_{2,3}),\\mu,-1)$, by a symmetry of the knot Floer homology groups \\cite[Proposition~3.10]{Ozsvath-Szabo:2005-1}. Now there is a differential $\\partial$ acting on $\\widehat{HFK}(-S^3_1(T_{2,3}),\\mu)$, the homology of which is isomorphic to the Floer homology of the ambient 3-manifold. (The existence of such a ``cancelling differential\" follows from the homological method of reduction of a filtered chain complex; see \\cite[Section 2.1]{Hedden-Watson:2018-1} for details on this perspective.) This differential strictly lowers the Alexander grading, which implies that the ``middle\" group $\\widehat{HFK}(S^3_1(T_{2,3}),\\mu,0)$ is either $\\mathbb{F}^3$ or $\\mathbb{F}$. In the former case, two of the summands are supported in the same grading, which is one less than that of the top group; moreover, one of these summands is the image of $\\widehat{HFK}(S^3_1(T_{2,3}),\\mu,1)$ under~$\\partial$, and $\\partial$ maps the other summand surjectively onto $\\widehat{HFK}(S^3_1(T_{2,3}),\\mu,-1)$. The case that $\\widehat{HFK}(S^3_1(T_{2,3}),\\mu,0)=\\mathbb{F}$ divides into two sub-cases, depending on whether $\\partial$ maps the middle group surjectively onto the bottom, or the top group surjectively onto the middle. In both sub-cases the resulting knot Floer homology is thin, and hence $CFK^\\infty$ is determined by the hat groups. In the former sub-case the hat groups are isomorphic to those of the right-handed trefoil, and to those of the left-handed trefoil in the latter; in both sub-cases, their Maslov grading has an overall shift down by $2$. \n\nTo determine which of the three possibilities above arise, we recall the surgery formula for knot Floer homology. In its simplest guise, which will be sufficient for our purposes, it expresses the Floer homology of the manifold obtained by $n$-surgery, $n\\le -(2g(K)-1)$ on a null-homologous knot $(Y,K)$ as the homology of a particular sub-quotient complex of $CFK^\\infty(Y,K)$ \\cite[Theorem~4.1]{Ozsvath-Szabo:2004-1}. Its relevance to us is that $-1$-surgery on $(S^3_1(T_{2,3}),\\mu)$ is homeomorphic to $S^3$, a manifold with $\\widehat{HF}(S^3)$ of rank $1$. Since $\\mu$ is a genus one knot, we can apply the surgery formula to (re)-calculate the Floer homology of $S^3$, viewed as $-1$-surgery on $\\mu$. The surgery formula says that the homology is given as the homology of the subquotient complex of $CFK^\\infty(S^3_1(T_{2,3}),\\mu)$ generated by chains whose $\\mathbb{Z}\\oplus\\mathbb{Z}$-filtration values satisfy the constraint min$(i,j)=0$. Of the three possibilities for $\\widehat{HFK}(S^3_1(T_{2,3}),\\mu)$, all but the case of the left-handed trefoil (shifted down in grading by $2$) have the property that the relevant subquotient complex has homology of rank $3$. The stated structure of $CFK^\\infty(S^3_1(T_{2,3}),\\mu)$ follows at once.\n\n\n(\\ref{item:CFKofJ_k}) We say two chain complexes $C_0$ and $C_1$ are \\emph{stably filtered chain homotopy equivalent} if $C_0\\oplus A_0$ is filtered chain homotopy equivalent to $C_1\\oplus A_1$ for some acyclic chain complexes $A_0$ and $A_1$. By \\cite[Theorem B.1]{Hedden-Kim-Livingston:2016-1} and \\cite[Proposition 3.11]{Hom:2017-1}, it is known that the tensor product $CFK^\\infty(T_{2,4k-1})\\otimes CFK^\\infty(T_{2,-3})$ is stably filtered chain homotopy equivalent to $CFK^\\infty(T_{2,4k-3})$.\n\nBy (\\ref{item:CFKofmeridian}), the K{\\\"u}nneth formula (\\ref{equation:Kunneth}) becomes \n\\begin{equation}\\label{equation:J_k}CFK^\\infty(S^3_1(T_{2,3}),J_k)\\cong CFK^\\infty (T_{2,-3})\\otimes CFK^\\infty (T_{2,4k-1})[-2].\\end{equation}\nIt follows that the right hand side of \\eqref{equation:J_k} is stably filtered chain homotopy equivalent to $CFK^\\infty(T_{2,4k-3})[-2]$, and we obtain the desired conclusion.\n\\end{proof}\n\n\n\nNow we prove Theorem \\ref{thm:dofM_k} which states that $d_{1\/2}(M_k)=-2k+\\frac{1}{2}$ and $d_{-1\/2}(M_k)=-\\frac{5}{2}$ if $k\\geq 1$.\n\\begin{proof}[Proof of Theorem \\ref{thm:dofM_k}] Recall that $M_k$ is obtained from $S^3_1(T_{2,3})$ by surgery on $J_k$ along its Seifert framing. Since any acyclic summand does not change $d$-invariants, we have\n\\begin{align*}\n&d_{1\/2}(M_k)=d_{1\/2}(S^3_0(T_{2,4k-3}))-2=-\\tfrac{3}{2}-2V_0(T_{2,4k-3}),\\\\\n&d_{-1\/2}(M_k)=d_{-1\/2}(S^3_0(T_{2,4k-3}))-2=-\\tfrac{5}{2}+2V_0(T_{2,-4k+3}).\n\\end{align*}\nby Proposition \\ref{proposition:CFKofJ_k}(\\ref{item:CFKofJ_k}) and Theorem \\ref{thm:OS}. Strictly speaking, Theorem~\\ref{thm:OS} pertains only to surgery on knots in $S^3$, but the proof easily yields a corresponding formula for surgery on knots in an integral homology sphere $L$-space; in these cases, the correction terms inherit an overall shift by the $d$-invariant of the ambient manifold (here, $-2$). Since $k\\geq 1$, \n\\begin{align*}&V_0(T_{2,4k-3})=k-1,\\\\\n&V_0(T_{2,-4k+3})=0\\end{align*} (for example, see \\cite[Theorem~1.6]{Borodzik-Nemethi:2013-1}). This completes the proof.\n\\end{proof}\n\n\\section{Correction terms of Seifert manifolds}\\label{section:Seifert}\n\n\nIn this section we provide some general constraints on the correction terms of a Seifert fibered homology $S^1\\times S^2$. More precisely, we show $d_{-1\/2}(M)\\geq -\\frac{1}{2}$ and $d_{1\/2}(M)\\leq\\frac{1}{2}$ for any Seifert fibered homology $S^1\\times S^2$. It follows at once that none of our manifolds are homology cobordant to a Seifert fibered space. We also see that the zero surgery obstruction can say nothing about Seifert manifolds.\n\n\\vskip0.1in\n\nOur estimates hinge on the following proposition, which was pointed out to us by Marco Golla. (Compare \\cite[Theorem~5.2]{Neumann-Raymond:1978-1}.)\n\n\\begin{proposition}\\label{proposition:Seifertboundsboth}\n\tSuppose $M$ is a Seifert fibered homology $S^1\\times S^2$. Then both $M$ and $-M$ bound negative semi-definite, plumbed $4$-manifolds. \n\\end{proposition}\n\\begin{proof}\nChoose an orientation and a Seifert fibered structure of $M$. As an oriented manifold, $M$ is homeomorphic to $M(e;r_1,\\ldots,r_n)$ in Figure~\\ref{figure:Seifertmanifold} where $e$ is an integer, and each $r_i$ is a non-zero rational number. We change the Seifert invariant $(e;r_1,\\ldots,r_n)$ via the following two steps.\n\n\\begin{figure}[b]\n \\centering\n \\includegraphics[scale=1]{figure4}\n \\caption{A Seifert fibered 3-manifold $M(e;r_1,\\ldots,r_k)$.}\n \\label{figure:Seifertmanifold}\n\\end{figure}\n\n\\begin{enumerate}\n\\item If $r_i$ is an integer, remove $r_i$ from the tuple $(e;r_1,\\ldots,r_n)$ and add $r_i$ to $e$.\n\\item For each $i$, replace $r_i$ and $e$ by $r_i-\\lfloor r_i\\rfloor$ and $e+\\lfloor r_i\\rfloor$, respectively.\n\\end{enumerate}\nNote that the above procedures are realized by slam-dunk moves, so the homeomorphism type remains unchanged. For brevity, we still denote the resulting Seifert invariant of $M$ by $(e;r_1,\\ldots,r_n)$, so that each rational number $r_i$ satisfies $0 -\\frac{5}{2}$).\n\n\n\nWe conclude with an alternate proof that $d_{-1\/2}(N_k)\\geq -\\frac{1}{2}$ and $d_{1\/2}(N_k)\\leq \\frac{1}{2}$. First recall a result of Ozsv\\'{a}th and Szab\\'{o}.\n\\begin{proposition}[{\\cite[Corollary~9.14]{Ozsvath-Szabo:2003-2}}]\\label{theorem:Ozsvath-Szaboinequality} Suppose that $K$ is a knot in a homology $3$-sphere~$Y$. Let $Y_0$ be the result of Dehn surgery along $K$ via its Seifert framing. Then \n\\[d_{1\/2}(Y_0)-\\tfrac{1}{2}\\leq d(Y)\\leq d_{-1\/2}(Y_0)+\\tfrac{1}{2}.\\]\n\\end{proposition}\n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[scale=1]{figure7}\n \\caption{A knot $K$ in $S^3_{-1}(T_{2,4k-1})$.}\n \\label{figure:K_k}\n\\end{figure}\n\\begin{proposition}\\label{proposition:dofN_k}For any positive integer $k$, $d_{-1\/2}(N_k)\\geq -\\frac{1}{2}$ and $d_{1\/2}(N_k)\\leq \\frac{1}{2}$.\n\\end{proposition}\n\n\\begin{proof}Consider the knot $K\\subset S^3_{-1}(T_{2,4k-1})$ which is depicted in Figure~\\ref{figure:K_k}. By a surgery formula given in \\cite{Ni-Wu:2015-1}, $d(S^3_{-1}(T_{2,4k-1}))=2V_0(T_{2,-4k+1})=0$ since $k\\geq 1$. Then $N_k$ is the result of Dehn surgery along the knot $K\\subset S^3_{-1}(T_{2,4k-1})$ via its Seifert framing. By Theorem~\\ref{theorem:Ozsvath-Szaboinequality}, we have \n\\[d_{1\/2}(N_k)-\\tfrac{1}{2}\\leq 0\\leq d_{-1\/2}(N_k)+\\tfrac{1}{2}\\]\nfor any positive integer $k$. This completes the proof.\n\\end{proof}\n\n\n\n\\bibliographystyle{amsalpha}\n\\renewcommand{\\MR}[1]{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThin rods and bands, the latter known also as strips or ribbons, display complex geometric response under simple end loadings and clampings. \nWhile much work has been done to explore these phenomena, most of the literature pertains either to periodic boundary conditions or highly symmetric end loadings such as a wrench, in which the end-to-end vector, loading vector, and twist are coaxial. \nMost of this is also limited to the analytically tractable case of isotropic rods; here and elsewhere in this paper, the term isotropic refers to the structure rather than the material, such that the cross section has no preferred bending direction.\nHowever, the space of possible boundary conditions is much wider, and includes conditions that can interact strongly with the anisotropy of a strip or any other elastic structure with a distinguished material frame.\nIn many practical situations, the two ends of the structure may be clamped such that their material frames take any orientation with respect to each other and the end-to-end vector. As we will show, certain conditions conspire with the anisotropy to frustrate the system and couple its twist and writhe response. Clamped boundary conditions can not only create energy barriers through frustration, but may even introduce topological barriers between an undeformed ground state and excited states \\cite{baez1991topological}.\nThe present study is a preliminary exploration covering a small piece of this wider parameter space, as applied to anisotropic rods and bands.\nWe begin with symmetrically clamped buckled strips of varying width, and subject their ends to a lateral displacement parallel to the width direction.\nThe introduction of this ``shearing'' motion reveals a rich set of stable configurations and jump phenomena, including several snap-through instabilities, that to our knowledge do not appear in the literature (we encourage the reader to take a quick glance now at the supplementary video {\\texttt{widtheffect45.mp4}} \\cite{videos} to see examples of such stable states and snap-throughs).\nWe compare experimentally determined stability ranges of various configurations to results from numerical continuation of the Kirchhoff rod equations, \nand find that a perfectly anisotropic rod model captures the complicated choreography of bifurcations of narrow bands, and provides much of the backbone of the behavior of wider bands. \nWe reveal connections between various states, including higher-order unstable \\emph{elastica} modes and stable twisted states created by the rod's anisotropy.\n\nWhile the Kirchhoff equations show themselves to be a surprisingly useful tool in the analysis of strip behavior, we wish to emphasize that there is no reason to assume that such a model, which assumes that cross sections remain perpendicular to the centerline, would be appropriate for strips.\nOn the other hand, the common assumption that transverse bending of strips is governed by the constraint of developability can lead to difficulties of its own, particularly for narrow strips, issues that we will briefly touch upon in an appendix. Until such issues are resolved, it is advantageous to employ an easily implemented rod model from which the strips inherit most, or even all, of their bifurcations.\nHowever, use of such a model should not be taken to imply that a narrow strip is equivalent to a rod.\n\n \n Boundary conditions like those we explore here are potentially of interest in helping to avoid violent snap-throughs of connectors, hinges, and umbilicals in flexible and deployable systems. Geometries similar to ours appear \n as slipping folds \\cite{arya2015wrapping} in deployable space membranes, buckled elements in flexible electronics and robotics, and decorative streamers in childrens' toys \\cite{princesswand}.\nMulti-stable structures find use in compliant mechanisms \\cite{Howellbook} at all length scales.\nThe behavior of strips under our loading conditions is likely related to the phenomenon of lateral-torsional buckling, known to structural engineers \\cite{mandal2002lateral}.\n\nThere is much prior work on the configurations of naturally straight rods. Work on the general behavior and classification of solutions includes that of Antman \\cite{antman1981large,antman1975}, Maddocks \\cite{kehrbaum1997elastic}, Nizette and Goriely \\cite{nizette1999towards}, and Cognet and co-workers \\cite{ameline2017classifications}.\nNeukirch and Henderson made a detailed investigation into the connectivity of solutions for rods subject to end thrusts and coaxial twists \\cite{neukirch2002classification,henderson2004classification}. Theory, numerics, and experiment show that circular cross section rods, an integrable system, when subject to such boundary conditions will buckle, hockle into a loop, or snarl into a self-contacting twisted structure \\cite{coyne1990analysis,Yabuta1982cable,miyazaki1997analytical,goss2005experiments,van2003instability,thompson1996helix,van2000helical,goyal2005nonlinear,coleman1995theory}. Anisotropic rods, those with preferred bending directions, display even more complicated and potentially non-integrable behavior due to non-conserved twist \\cite{mielke1988spatially,champneys1996multiplicity,van1998lock,beda1992postbuckling,buzano1986secondary, goriely2001dynamics}. van der Heijden and Thompson \\cite{van1998lock} distinguish between weakly anisotropic and strongly anisotropic ``tape-like'' behavior such as that we will discuss in this paper. Integrability can also be destroyed by the addition of gravity \\cite{antman1981large,lu1995complex}, but can be preserved under addition of extensibility and shearability \\cite{stump99hockling,shi1995elastic}. \nEarly experiments by Green \\cite{green1936equilibrium, green1937elastic} showed that twist under tension makes strips unstable to the formation of multiple loops, with only a single loop forming in the absence of tension. Recent work by Chopin and Kudrolli \\cite{chopin2013helicoids} extends these findings and reveals a rich set of possible deformations and patterns under tension and twist, many of them involving stretching.\nAside from the present investigation, the only work we know of featuring lateral displacements is that of Morigaki and co-workers \\cite{morigaki2016stretching}, who begin with a slightly laterally displaced loop configuration of a strip, pull the ends, and find behavior similar to the hockling and pop-out regimes of isotropic rods.\nOther interesting boundary conditions include freely hinged conditions \\cite{perkins1990planar,lu1995complex}, and asymmetric rotation in the plane of buckling leading to snap-throughs \\cite{plaut2009vibration}. \nAnother much-studied corner of parameter space, due to its supposed relevance to DNA, is that of pre-twisted rings composed of isotropic or anisotropic rods \\cite{coleman2004theory,manning1999symmetry, hoffman2003link,tanaka1985elastic,wadati1986elastic,tobias1994dependence,starostin1996three}. The latter includes, as one particular case, the configurations of a narrow M{\\\"{o}}bius band \\cite{mahadevan1993shape,moore2015computation}.\nDichmann, Li and Maddocks \\cite{dichmann1996hamiltonian}, Li and Maddocks 1996 \\cite{li1996computation}, and Domokos and Healey \\cite{domokos2001hidden} provide insight into the connectivity of various solutions of this type.\n\nThe current paper is organized as follows. \nWe introduce the geometry of our problem and describe our experiments in Section \\ref{description}, and present the anisotropic Kirchhoff rod model in Section \\ref{rodmodel}.\nResults from experiments on narrow bands and numerical continuation of the Kirchhoff equations are compared in Section \\ref{results}, through a series of slices through parameter space. In Section \\ref{loci}, a more global view is presented through the loci of bifurcation points in parameter space, which also delineate regions of stability for different states.\nExperiments on bands of varying width are presented in Section \\ref{widtheffect} and compared with the results for narrow bands and rods.\n Many smooth and discontinuous bifurcations for both narrow and wide bands are shown in several supplementary videos \\cite{videos}, which complement the diagrams in Sections \\ref{results}-\\ref{widtheffect}. We discuss a few additional points of interest in Section \\ref{discussion}.\n In the Appendices, we show two types of configuration that exist as the bands approach the limits of developable deformation, where elastic energy focuses in conical defects at the clamped ends, give details on solving the boundary value problem, briefly discuss the minor effects of Poisson's ratio on the loci of bifurcation points, briefly contrast our results with those for isotropic (square) rods, and discuss problems that arise when employing strip models to describe the behavior of bands such as those in our experiments.\n\n\n\\section{Geometry of the experiments, methods, and errors} \\label{description}\n\nThe boundary conditions we impose are shown in Figure \\ref{fig:Intros}. We use thin (0.005 $\\pm$0.0005 inch \/ 0.127 $\\pm$ 0.013 mm) bands cut from polyester shim stock (Artus Corp., Englewood, NJ) with free length $L = 240 \\pm 0.5$ mm ($\\approx 5$ mm clamped length on either side) and various widths, the most common being $D = 3 \\pm 0.05$ mm, $30 \\pm 0.5 $ mm, and $60 \\pm 0.5$ mm, corresponding to aspect ratios $D\/L$ of $1\/80$, $1\/8$, and $1\/4$, respectively. We use a Silhouette Cameo 3 cutting machine (Silhouette America, Lindon, UT) for the narrow bands, and a paper trimmer for the wider bands. The additional accuracy was necessary for narrow bands because bifurcations of highly twisted configurations were found to be sensitive to non-uniformities in width.\n With respect to an initially flat configuration, the boundary conditions consist of a symmetric tilt angle $\\psi_0$ ($\\pm 1^{\\circ}$), a ``compression'' $\\Delta L$ ($\\pm 0.5$ mm) in the length direction, and a lateral ``shear'' $\\Delta D$ ($\\pm 0.5$ mm) parallel to the width direction at the ends. \nThe clamping is parallel to the width direction with an accuracy of $\\pm 2^{\\circ}$ for $D\/L =1\/80$, $\\pm0.2^{\\circ}$ for $D\/L = 1\/8$, and $\\pm0.1^{\\circ}$ for $D\/L=1\/4$. Typically, we fix the compression and clamping angle, and use the normalized lateral shear displacement $\\Delta D\/L$ as the primary bifurcation parameter. We align the shear direction with gravity, which mitigates its influence. A bias of $\\approx 0.5^{\\circ}$ in this alignment was introduced by the slope of the laboratory floor. We measure the range of stability\n for all observed configurations of the bands, many of which require manual manipulation to obtain. \nThe bands are never kept in any particular deformed state for longer than a few minutes, to avoid possible viscous response that could affect results. With these precautions, results are reproducible with a typical variation of about $\\pm 2$ mm (corresponding to $\\pm1\/120$ $\\Delta D\/L$) between trials, although a very few narrow band states such as the $US\\pm$ and $W$ states at $\\psi_0=60^{\\circ}$ show deviations of as much as $\\pm 5$ mm. Near bifurcations that change connectivity, multiple trials may have qualitative differences; these are discussed when they arise in Section \\ref{results}. All narrow band data come from averaging three trials; wide band data are single trials, other than some additional trials performed to estimate variation. \n\n\\begin{figure}[h!]\n\t\\centering\n\t\\begin{subfigure}[t]{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[height=1.75in]{IntroSlike.pdf}\n\t\t\\caption{}\\label{fig:SlikeRl}\n\t\\end{subfigure} \\hspace{-30pt}\n\t\\begin{subfigure}[t]{0.48\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[height=1.75in]{IntroUlike.pdf}\n\t\t\\caption{}\\label{fig:UlikeRl}\n\t\\end{subfigure} \\\\\n\t\\vspace{-0.1in}\n\t\\captionsetup[subfigure]{labelfont=normalfont,textfont=normalfont}\n\t\\caption{A thin, rectangular band of width $D$ and length $L$ is symmetrically clamped with an angle $\\psi_0$, ``compression'' $\\Delta L$, and ``shear'' $\\Delta D$. The centerline of the band carries an orthonormal director frame $(\\bm{d_1}, \\bm{d_2}, \\bm{d_3})$ corresponding to the width direction, the surface normal, and the tangent, respectively. Shown here are (\\subref{fig:SlikeRl}) $S$-like and (\\subref{fig:UlikeRl}) $U$-like configurations.\n\t}\\label{fig:Intros}\n\\end{figure}\n\nWe describe the bands in terms of the geometry of a rod, a description that is most suitable for narrow bands, though quite distinct from the description of a band as a developable surface. The mechanics of an anisotropic rod will be discussed in the following section. The description involves an orthonormal material frame $(\\bm{d_1}, \\bm{d_2}, \\bm{d_3})$ attached to the centerline of the band, with the three directors corresponding to the width direction, the surface normal, and the tangent, respectively. Figure \\ref{fig:Intros} shows such a frame superimposed on two example configurations of a wide band, which we refer to as $S$-like and $U$-like, and which we will describe in more detail in Section \\ref{results}. \n\nBands can, in theory, be approximated as developables until the shear displacement $\\Delta D\/L$ approaches a limiting value where such an isometric description is no longer possible. We discuss this further in Appendix \\ref{limitstates}. \nNote that in describing the bands as developable surfaces, the straight line generators of the surface would not coincide with the material directors of the rod description; the geometry of a developable strip differs from that of a rod with undeformed cross section. Further comment can be found in Appendices \\ref{bvp} and \\ref{stripmodel}.\n\n\n\n\t\n\t\n\\section{Anisotropic rod model}\\label{rodmodel}\n\nWe compare experimental results with a simple model of a perfectly anisotropic rod. This model assumes that the only way a band can deform is by bending around its width direction and twisting around its tangent. Bending around the surface normal is forbidden. The rod is inextensible and unshearable; its centerline is given by $\\bm{X}(s)$, where $s$ is the arc length, and the tangent can be identified with one of the directors, $\\bm{X}' = \\bm{d_3}$ (throughout this paper a prime will denote an $s$-derivative). The kinematics of the frame $(\\bm{d_1}, \\bm{d_2}, \\bm{d_3})$ are given by \n\\begin{align}\\label{rotationframe}\n\\bm{d_i}'&=\\bm{\\omega} \\times \\bm{d_i} \\, , \\\\\n\\bm{\\omega}&=\\kappa_1 \\bm{d_1} + \\tau \\bm{d_3} \\, ,\n\\end{align}\nwhere the Darboux vector $\\bm{\\omega}$ has no component normal to the strip. The generalized strains $\\kappa_1$ and $\\tau$ are the curvature in the easy (only) direction, and the twist about the tangent. For a perfectly anisotropic strip, the frame $(\\bm{d_1}, \\bm{d_2}, \\bm{d_3})$ can be identified with the Frenet-Serret frame as $(\\bm{b}, \\bm{-n}, \\bm{t})$, and the curvature $\\kappa_1$ and twist $\\tau$ with the curvature and torsion. This type of model has been used previously as an approximate model for the shape of elastic strips \\cite{mahadevan1993shape}; our present interest is primarily in bifurcations rather than shapes.\n\nLinear and angular momentum balances are provided by the Kirchhoff equations for the contact force and moment $\\bm{N}$ and $\\bm{M}$ in the absence of gravity or other distributed loads or couples,\n\\begin{equation}\\label{F&Mequilibrium}\n\\begin{aligned}\n\\bm{N} ' &=\\bf{0} \\, , \\\\\n\\bm{M} '+\\bm{d}_3 \\times \\bm{N} &=\\bf{0} \\, . \\\\\n\\end{aligned}\n\\end{equation}\n Three quantities are conserved along the centerline \\cite{van2000helical},\n\n\\begin{equation}\\label{FirstIntegral} \n\\begin{aligned}\nC_1 &=\\tfrac{1}{2}\\bm{M} \\cdot \\bm{\\omega} +\\bm{N} \\cdot \\bm{d_3} \\, ,\\\\\nC_2 &=\\bm{N} \\cdot \\bm{N} \\, ,\\\\\nC_3 &=\\bm{N} \\cdot \\bm{M} \\, .\\\\\n\\end{aligned}\n\\end{equation}\nIsotropic rods conserve the twist as a fourth quantity; our system does not. The general anisotropic rod is known to be non-integrable, but we are unaware of any published results on the presence or lack of integrability for the perfectly anisotropic case.\nWe resolve $\\bm{N}$ and $\\bm{M}$ on the moving frame as $\\bm{N} =N_i\\bm{d}_i$ and $\\bm{M} =M_i\\bm{d}_i$, and assume linear constitutive relations $M_1=EI_1 \\kappa_1$ and $M_3=GJ \\tau$, where $E$ is the Young's modulus and $G$ is the shear modulus, $I_1$ is the principal moment of inertia of the cross-section in the easy direction, and $GJ$ is the torsional rigidity. The other moment $M_2$ is a Lagrange multiplier enforcing the vanishing of curvature in the hard $\\bm{d_2}$ direction. The ratio of $E$ to $G$ involves the elastically isotropic Poisson's ratio $\\nu$, which we set to $0.25$ for the present study; this choice makes little difference to the results, as shown in Appendix \\ref{poisson}.\n There are thus six scalar balance equations. Reconstruction of the rod centerline and frame orientation is achieved through a quaternion representation leading to a set of thirteen equations.\nWe solve these using the continuation package AUTO 07P \\cite{doedel2007auto}. Details, along with the specification of boundary conditions, are discussed in \nAppendix \\ref{bvp}.\n\n\n\n\n\\section{Numerical and experimental results for narrow bands}\\label{results}\n\nIn this section, we present experimental results on narrow bands ($D\/L = 1\/80$) and compare them with numerical results from the anisotropic Kirchhoff rod model. \nWe restrict our experimental parameter space to a single compression $\\Delta L\/L = 1\/2$ and clamping angles $0 \\le \\psi_0 \\le 60^{\\circ}$.\nUsing the shear $\\Delta D\/L$ as a bifurcation parameter, we deform the bands to near the isometric limit, and find a rich and complicated landscape of stable configurations and both smooth and violent transitions.\nThese observations, which depend strongly on clamping angle, are described surprisingly well by the naive model. We keep gravity out of the model, as its effects are easily accounted for, and in practical terms it would simply break some of the symmetry of the solutions we wish to explore and create more complicated and potentially confusing figures.\n\nThe boundary conditions we impose might be easily accommodated by an isotropic rod bending in what is a forbidden direction for the perfectly anisotropic strip. Instead, our frustrated system is induced to find some combination of allowed bending and twist in order to satisfy the constraints. In short, the shear indirectly causes a non-uniform twist, and as there are multiple ways for bending and twisting energy to compete, creates a highly multi-stable system. As the shear increases, the system shifts from being compressed to being in tension. To our knowledge, these boundary conditions have not been explored before in the literature. However, Morigaki's \\cite{morigaki2016stretching} recent experiments on tensioned loops can be interpreted in terms of our boundary conditions as a clamping angle $\\psi_0$ of $180^{\\circ}$, large compression ($\\Delta L \/ L > 1$), and small shear $\\Delta D\/L$. \n\nIn general, the clamping angle will bias the strip towards two general types of configurations, with some seeming similarity to primary buckling modes of planar \\emph{elastica}. Small clamping angles favor $S$-like shapes that live both above and below the plane of clamping, and come in chiral pairs. High clamping angles favor $U$-like shapes that live mostly on one side of the plane of clamping, and are symmetric about their midpoints. Examples are shown in Figure \\ref{fig:Intros}.\nThis is consistent with what is known about the rod equations,\nnamely that all solutions are either reversibly symmetric about their midpoint or are reversibly symmetric pairs \\cite{van2003instability,neukirch2002classification,domokos2001hidden}.\nAs we increase the clamping angle, we gradually lose many of the states that exist at low angles.\n\n\n\\begin{figure}[h!]\n\t\\captionsetup[subfigure]{labelformat=empty}\n\t\\centering\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{naru.pdf}\n\t\t\\caption{$U$}\n\t\\end{subfigure}%\n\t\\hspace{0.6pt}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{narus+.pdf}\n\t\t\\caption{$US+$}\n\t\\end{subfigure} \n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{narws+.pdf}\n\t\t\\caption{$WS+$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{naruuu.pdf}\n\t\t\\caption{$uUu$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{narsw.pdf}\n\t\t\\caption{$w$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nartu+.pdf}\n\t\t\\caption{$TU+$}\n\t\\end{subfigure} \n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nartw+.pdf}\n\t\t\\caption{$TW+$}\n\t\\end{subfigure} \\\\\n\t\\vspace{-3pt}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nuconfigu.pdf}\n\t\\end{subfigure}%\n\t\\hspace{0.6pt}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nuconfigus+.pdf}\n\t\\end{subfigure} \n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nuconfigws+.pdf}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nuconfiguuu.pdf}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nuconfigsw.pdf}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nuconfigtu+.pdf}\n\t\\end{subfigure} \n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nuconfigtw+.pdf}\n\t\\end{subfigure} \\\\\n\\vspace{15pt}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{narbw.pdf}\n\t\t\\caption{$W$}\n\t\\end{subfigure}%\n\t\\hspace{0.6pt}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{narus-.pdf}\n\t\t\\caption{$US-$}\n\t\\end{subfigure} \n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{narws-.pdf}\n\t\t\\caption{$WS-$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{naruuui.pdf}\n\t\t\\caption{$uUui$}\n\t\\end{subfigure}\n\t\\hspace{0.13\\textwidth}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nartu-.pdf}\n\t\t\\caption{$TU-$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nartw-.pdf}\n\t\t\\caption{$TW-$}\n\t\\end{subfigure} \\\\\n\t\\vspace{-3pt}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nuconfigbw.pdf}\n\t\\end{subfigure}%\n\t\\hspace{0.6pt}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nuconfigus-.pdf}\n\t\\end{subfigure} \n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nuconfigws-.pdf}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nuconfiguuui.pdf}\n\t\\end{subfigure}\n\t\\hspace{0.13\\textwidth}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nuconfigtu-.pdf}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{nuconfigtw-.pdf}\n\t\\end{subfigure}\n\t\\caption{Comparison between experimental configurations of a narrow band ($D\/L=1\/80$) and renderings of the rod frame based on numerical solutions of the perfectly anisotropic rod equations, with compression $\\Delta L\/L=1\/2$, clamping angle $\\psi_0 = 15^{\\circ}$, and various values of shear $\\Delta D \/ L$.\n\tNote that the bands deform into a surface different than the rod frame rendering.\n\t There are no fitting parameters; boundary conditions and viewing angle are the same between experiments and numerics. Gravity is roughly vertical in the experimental images, and is absent in the numerical solutions; its effects are greatest on the twisted solutions $TU\\pm$ and $TW\\pm$. Thirteen states are shown, including four symmetric $\\pm$ pairs.\n}\n\t\\label{fig:ExpConfigurationwithNu}\n\\end{figure}\n\n\nFigure \\ref{fig:ExpConfigurationwithNu} shows all the types of states we observe in narrow bands, alongside renderings based on numerical solutions of the perfectly anisotropic rod equations, for a shallow clamping angle $\\psi_0 = 15^{\\circ}$ and various values of shear $\\Delta D \/ L$. We name the states in a manner that roughly describes their shapes. There are no fitting parameters; boundary conditions and viewing angle are the same between experiments and numerics. \nThe numerical solutions are rendered as strips representing the rod frame (Appendix \\ref{bvp}), with the same width as the actual bands, but we note that \\emph{the actual bands will deform into a surface that is different than the surface representing the rod frame}, so comparisons must be made carefully. Throughout this paper, we are able to identify experimental and numerical states using a combination of factors including symmetry of the shapes and the number of inflections in centerline curvature and twist, rather than from details of the shape adopted by material off of the centerline.\nFor later comparison with Figures \\ref{fig:30states} and \\ref{fig:jumpenergy} one needs to know that $s\/L \\in [0,1]$ increases from right to left in the renderings. \n\n\n Gravity is roughly vertical in the experimental images, and is absent in the numerical solutions. Its effects are generally weak, although stronger on some solutions such as the overhanging twisted solutions $TU\\pm$ and $TW\\pm$. Overall, the Kirchhoff equations reproduce the shapes of stable states quite well. Thirteen states are shown, but this includes four symmetric $\\pm$ pairs, so only nine distinct states exist.\nWe may classify them into three families. First we define a coordinate $y$ perpendicular to the clamping plane, sharing the same sign as $\\psi_0$. The $U$ family ($U$, $US\\pm$, $uUu$ and $w$) and $W$ family ($W$, $WS\\pm$ and $uUui$) tend to sit on the side of positive and negative $y$, respectively, and are mirror images when $\\psi_0 = 0^{\\circ}$. A family of twisted states ($TU \\pm$ and $TW\\pm$, mirror images when $\\psi_0 = 0^{\\circ}$) exists at low values of shear. These states, which clearly display the non-conservation of twist in anisotropic rods, can be achieved by applying a twist near the center of a $U$ or $W$ state. Alternatively, we may separate the states into reversibly symmetric $\\pm$ pairs and reversibly symmetric single solutions.\n\nWe now present a sequence of slices through parameter space for increasing values of clamping angle $0 \\le \\psi_0 \\le 60^{\\circ}$, showing the evolving solution manifolds and corresponding rod shapes for the Kirchhoff equations alongside experimentally determined ranges of stability of narrow band states. It is not difficult to link observed states with numerical solutions through qualitative comparison of the shapes and inferences about stability information from the types of bifurcations encountered. \n Changes in connectivity of the solution manifolds are also reflected in experimental data and verified in supplementary videos \\cite{videos}.\nThe slices will display the connectivity of the solution curves before and after certain transitions. These transitions are pinpointed more accurately using two-parameter continuation of bifurcation points in clamping angle-shear space, as will be shown in more detail in Figure \\ref{fig:phasediagram} in Section \\ref{loci}.\nThe shear $\\Delta D \/ L$ is the bifurcation parameter. The system is mirror-symmetric around zero shear, but we plot a small portion of the numerical negative shear results to show the loop structure of various states near the origin. For the vertical (response) axis, we choose the integrated height above the plane of clamping $\\int_{0}^{1}\\! y \\, ds$, a quantity that converges to zero for all states as the limiting shear deformation is approached, and which is identical for each $\\pm$ pair. A strip of finite width has a shear limit, discussed in Appendix \\ref{limitstates}, beyond which stretching must occur. \nFor a compression $\\Delta L \/L = 1\/2$, this limit is $\\Delta D\/L \\approx 0.854$ for our narrow bands, and $\\Delta D\/L = \\sqrt{3}\/2 \\approx 0.866$ for an ideal rod with zero width.\n We perform narrow-band experiments and continue solutions only up to $0.82$, which avoids damaging the band as well as numerical stiffness issues.\nSolution manifolds are obtained by continuation of angle and\/or displacement boundary conditions from known solutions, typically from a circle deformed through the first buckled mode of planar \\emph{elastica}. \nSome branches, such as twisted state branches, are isolated on the cross sections we present but can be reached by continuation in the full parameter space.\nCurves we wish to emphasize are plotted in black, while other closely related or connected curves are shown in grey; often, particularly at higher angles, black and grey will be used for different portions of a single continuous curve. We show more of these grey curves at lower angles on some of the plots, and replace them with dashed lines at higher angles, and often remove them entirely to overlay additional numerical or experimental results. Stability information is not shown anywhere on these plots, although it can often be inferred. An infinite number of other states exist, and are of course not shown. Branch points, and occasionally some fold bifurcations, are marked with symbols; unmarked intersections of the manifolds do not correspond to any bifurcation. Because of the $\\pm$ symmetry of the integrated height response parameter $\\int_{0}^{1}\\! y \\, ds$, most pitchfork bifurcations look like half of a pitchfork, as two lines overlap. \n Numbers on the figures identify particular bifurcations whose loci will be shown later in Figure \\ref{fig:phasediagram} in Section \\ref{loci}.\n\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{BD0states.pdf}\n\t\\caption{Some solutions (curves and renderings from the arrowed locations) and bifurcations (disks) of the perfectly anisotropic Kirchhoff rod for clamping angle $\\psi_0=0^{\\circ}$. Stability information is not shown; black and grey are used for emphasis only. The solution curves are symmetric about the zero-height axis and the zero-shear axis. Red and blue shapes are $\\pm$ pairs that share a single curve on the diagram.}\n\t\\label{fig:Config0}\n\\end{figure}\n\n\nFigure \\ref{fig:Config0} shows some of the solution manifolds for the symmetric case of zero clamping angle $\\psi_0=0^{\\circ}$. Also shown are rod frame renderings of numerical solutions at several points along the curves, many of which we identify with the configurations named in Figure \\ref{fig:ExpConfigurationwithNu}. The $\\pm$ pairs are drawn as red and blue. Several turning points (fold bifurcations) and branch points are observed, some of which are overlapping pairs. \nThe states shown include the first few stable and unstable planar modes of Euler \\emph{elastica}, which are unlabeled.\nThe loop-like curves near zero shear are highly twisted states, many of which are unstable, some of which connect with the \\emph{elastica} modes. All even-numbered modes of planar elastica and states continued from these will sit on top of one another on the horizontal axis of symmetry (zero integrated height). The connectivity along this axis is very complicated, including many (possibly an infinite number) of branch and fold bifurcations, as will be revealed when we proceed to a nonzero value of $\\psi_0$. We show only a few branch points here, and our choice of response parameter hides the presence of folds when $\\psi_0=0$.\nThis raises interesting questions. Can we assume that the entire infinite family of planar buckled modes connect through bifurcations to one or more twisted states? And how are the pitchforks distributed along the axis?\n\n\nWe are able to identify these numerical states with the stable states observed experimentally, and infer information about stability and bifurcation types. We now recognize that the $U$ and $uUu$ states lie on a single branch connected to the first-mode planar \\emph{elastica}, but are separated by two bifurcations and an unstable stretch. We will refer to this entire branch as the $U$ branch, except when it may cause confusion. There is a supercritical pitchfork at $\\Delta D\/L \\approx 0.36$ that connects the $U$ and $US\\pm$ states and causes loss of stability of the $U$ branch; stability is regained through a subcritical pitchfork at $\\Delta D\/L\\approx0.60$, with the second set of stable configurations referred to as $uUu$.\nThere are two supercritical pitchforks on the zero-height axis of symmetry at $\\Delta D\/L\\approx0.53$, one linking the $US+$ and $WS+$ states to a stable $S+$ state, the other linking the $US-$ and $WS-$ states to a stable $S-$ state. The $S\\pm$ states only exist at zero clamping angle, because the pitchforks on the horizontal axis will be broken at any non-vanishing angle. The unstable states on the low-shear side of these bifurcations connect back to the two unstable second-mode \\emph{elastica} shapes shown at zero shear.\n Note that at zero clamping angle, the $US$ and $WS$ states are equivalent. Upon symmetric change in the clamping angle, they will be distinct, and the connectivity described here will change. In this study, we don't consider asymmetric changes in clamping angle, which observation suggests will stabilize either the second-mode \\emph{elastica} or a pair of $S$-like planar shapes, depending on the value of the compression.\nBy shearing the unstable third-mode \\emph{elastica} and following the branch to high shear, we encounter three branch points, none of which appear to create any stable states. Between the second and third of these, there is a steep, but not yet folded back, section of the curve that will, upon a small change of clamping angle, become a stable section in between two folds. We plot the companion curve below the horizontal axis in grey, as it will never acquire a stable segment.\nOf the many twisted states at low shear, only two pairs of twisted states $TU\\pm$ and $TW\\pm$ (equivalent at zero clamping angle) are observed experimentally. The two loops upon which they lie are complicated pretzel-like curves, each of which provides four (pairs of) states at zero shear, of which only one is experimentally observed in narrow bands.\n \n\n\n \\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{BD0withExp.pdf}\n\t\\caption{Experimental data (red curves) from narrow bands for clamping angle $\\psi_0=0^{\\circ}$, compared with numerical solutions of the anisotropic rod equations (black and grey curves). Some solution curves branching from bifurcation points have been removed from the diagram for clarity. \n\tThe horizontal extent of the red curves is the range of stability (typical variation $\\approx \\pm 0.01\\, \\Delta D\/L$), while the vertical position of the curves is not measured data, but is made to follow near the numerical curves whenever a comparison is possible. No data was taken for $\\Delta D\/L < 0$.\nGravity causes asymmetry between $\\pm$ data. A stable $w$ state does not theoretically appear until $\\psi_0 \\ge\\, \\approx 0.135^{\\circ}$, but is observed in experiments, likely due to error in clamping or alignment.\nThere is a smooth path from the first mode of planar \\emph{elastica}, through the $U$, $US+$, and $S+$ states or the $W$, $WS-$, and $S-$ states, to approach the limiting shear.}\n\t\\label{fig:Experi0}\n\\end{figure}\n\n\n In Figure \\ref{fig:Experi0}, we compare the solutions of the anisotropic rod equations with experimental narrow band stability data for $\\psi_0=0^{\\circ}$, shown using red curves. No experimental data was obtained for negative shear, so the data is truncated at the $\\Delta D\/L = 0$ axis. Many solution curves have been removed from the figure for clarity. Only the horizontal extent of the red experimental curves has any meaning. The vertical position of these curves follows the corresponding solutions for ease of comparison, with $\\pm$ pairs separated by a small gap, or the data is plotted as a horizontal line if no corresponding solution exists. For example, the very short red line representing the $w$ state is observed experimentally, although in theory it should not appear until the clamping angle is slightly increased to $\\psi_0 \\approx 0.135^{\\circ}$. This discrepancy is likely due to some error in clamping or in vertical alignment in the presence of gravity; the system can be quite sensitive to boundary conditions close to a bifurcation. Similar comments can be made about any other asymmetries about the horizontal axis at $\\psi_0 = 0^{\\circ}$. In this and subsequent figures, gravity is responsible for observed asymmetries between $\\pm$ pairs, breaking pitchfork bifurcations like that between the $U$ and $US\\pm$ states, such that $U$ always connects with $US+$. The $US-$ and $WS+$ states are thus isolated states in between two fold bifurcations, and observed only by manual manipulation of the band, only because of the presence of gravity in a particular orientation.\nThis qualitative behavior was confirmed by augmenting the rod equations with a gravity term (see equations \\ref{Gravity} in Appendix \\ref{bvp}).\nWe observe that a first-mode \\emph{elastica} will smoothly deform through the $U$, $US+$, and $S+$ states or through the $W$, $WS-$, and $S-$ states, to approach the limiting shear, with the $\\pm$ choices being results of gravitational bias in this orientation. This process, and many other bifurcations corresponding to Figure \\ref{fig:Experi0}, are illustrated in the supplementary video {\\texttt{transition0.mp4}} \\cite{videos}.\n We will see that the numerical $W$-$WS$-$S$ path will be broken by any nonzero clamping angle, while the $U$-$US$-$S$ path will become a $U$-$US$ path at nonzero angle, and will eventually be broken at higher angles, with two merging events leading to a new smooth $U$-$w$-$uUui$ path. At $\\psi_0=0^{\\circ}$, the $uUu$ and $uUui$ states also approach the limiting shear, but are not smoothly connected to planar configurations.\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=0.85\\textwidth]{BD5states.pdf}\n \t\\caption{from narrow bands for\n\tSome solutions (curves and renderings from the arrowed locations) and bifurcations (open and closed disks, some numbered) of the perfectly anisotropic Kirchhoff rod for clamping angle $\\psi_0=5^{\\circ}$. Many bifurcations have been broken, and paths approaching the limit have been affected. Two folds 6 and 7 and the $w$ state have been created.}\n \\label{fig:Config5}\n\\end{figure}\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=0.85\\textwidth]{BD5withExp.pdf}\n \\caption{Experimental data (red curves) from narrow bands for clamping angle $\\psi_0=5^{\\circ}$, compared with numerical solutions of the anisotropic rod equations (black and grey curves). Many solution curves have been removed from the diagram for clarity. \tThe horizontal extent of the red curves is the range of stability (typical variation $\\approx \\pm 0.01\\, \\Delta D\/L$), while the vertical position of the curves is not measured data, but is made to follow near the numerical curves whenever a comparison is possible. No data was taken for $\\Delta D\/L < 0$. \t\nThere is still a smooth path from the first mode of planar \\emph{elastica}, through the $U$ and $US+$ states, to approach the limiting shear, but the corresponding path through $WS-$ has been broken; $WS\\pm$ states will now jump to $US\\pm$ states at intermediate shears.}\n \\label{fig:Experi5}\n\\end{figure}\n\n\nFigures \\ref{fig:Config5} and \\ref{fig:Experi5} show some solution manifolds, renderings, and experimental stability data for a small clamping angle, $\\psi_0=5^{\\circ}$. In Figure \\ref{fig:Config5} and some subsequent figures, some bifurcations have been numbered for convenient description and for further discussion in Section \\ref{loci} and Appendix \\ref{poisson}. The nonzero clamping angle has broken the symmetry between the $U$ and $W$ families that live primarily above and below the horizontal axis in the figures. At low shear, we can think of $U$ as the primary first-mode \\emph{elastica} state, and $W$ as the corresponding inverted state (as the clamping angle increases, its shape will more closely resemble its name, or perhaps an $M$ depending on one's orientation). All the branch bifurcations on this axis, and some off of the axis, have been broken, creating numerous folds, and revealing the complex asymmetric connectivity of the curves. \nThe breaking of the primary black pitchforks on the horizontal axis leads to the overlapping fold bifurcation pair 5 on the $WS\\pm$ branch, now separated by a jump from the $US\\pm$ branch, which branch has now merged with $S\\pm$ and smoothly approaches the limiting shear.\nFurther increases in angle will shorten the stable range of the $WS\\pm$ branch.\n Two folds 6 and 7 and an intermediate $w$ state have been created on an upper black branch. The creation occurs at $\\psi_0 \\approx 0.135^{\\circ}$ and corresponds to a cusp in $\\psi_0$-$\\Delta D\/L$ space, which will be seen later in Figure \\ref{fig:phasediagram}. Increasing the clamping angle extends the stable range of the $w$ state; the corresponding inverted grey branch becomes shallower in slope, and will never produce a stable state under our choice of clamping path.\nAll of these features of the rod equation solutions are consistent with the experimental data.\n\nIn subsequent figures, we remove many of the complicated grey solution curves, indicating their existence with small stretches of dashed lines.\n\n\nFigure \\ref{fig:Solution15} shows some solution manifolds and experimental stability data for a clamping angle $\\psi_0=15^{\\circ}$. Paths from zero to limiting shear have not changed. However, the stable extent of the $w$ state has increased, while that of the $WS\\pm$ states has decreased, with $WS+$ nearly disappearing due in part to the gravity-induced asymmetry.\n The primary twisted states $TU\\pm$ are relatively unaffected by the clamping angle change, but their inverted partners, the $TW\\pm$ states, now exist over a shorter extent. In experiments, we observe that the $WS\\pm$ states jump to the corresponding $US\\pm$ states upon increasing shear, while the $WS+$ state jumps to the $WS-$ state, and the $US-$ state jumps to the $US+$ state, upon decreasing shear, due to gravity-induced folds. The $w$ state jumps to the limiting $uUui$ state upon increasing shear, and to the $US+$ branch upon decreasing shear. \nMany of these transitions are illustrated in the supplementary video {\\texttt{transition15.mp4}} \\cite{videos}.\nWe note that much of the complexity of the solution manifolds arises from the anisotropy of the rod; Appendix \\ref{anisotropy} shows relatively simple solution manifolds for isotropic rods that may be compared with Figures \\ref{fig:Config0} and \\ref{fig:Solution15}.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{BD15degree.pdf}\n\t\\caption{Some solutions (black and grey curves) and bifurcations (open and closed disks, some numbered) of the perfectly anisotropic Kirchhoff rod equations for clamping angle $\\psi_0=15^{\\circ}$, along with experimental data (red curves). \n\t\tThe horizontal extent of the red curves is the range of stability (typical variation $\\approx \\pm 0.01\\, \\Delta D\/L$), while the vertical position of the curves is not measured data, but is made to follow near the numerical curves whenever a comparison is possible. No data was taken for $\\Delta D\/L < 0$. \n\t\tNumerically, the $w$ branch has been elongated, and the $WS\\pm$ and $TW\\pm$ branches have been shortened. Experimentally, the $WS+$ state has nearly disappeared, in part due to the action of gravity.}\n\t\\label{fig:Solution15}\n\\end{figure}\n\n\nAs we increase the clamping angle, there are many complicated changes to the solution structure. Among these, the isolated $TW\\pm$ loop partially merges with some of the complicated grey twisted curves (which we have already removed from the figures for clarity).\nThe stable $TW\\pm$ states\ndisappear at around $\\psi_0 \\approx 26.89^{\\circ}$. We don't show these, and many other, transitions here.\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{BD27p50degree.pdf}\n\t\\caption{Some solution curves and bifurcations (open and closed disks, some numbered) of the perfectly anisotropic Kirchhoff rod for clamping angle $\\psi_0=27.5^{\\circ}$. Pitchfork 4 has transformed from super- to sub-critical, and a new fold 12 has appeared. The $U$ state will now (weakly) jump to a $US$ branch, which can be followed to the limit. Fold 7 of the $w$ branch is approaching the $U$ branch. Details of this region and subsequent transitions are shown in Figure \\ref{fig:Transition1}.}\n\t\\label{fig:Solution27p5}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Transition1.pdf}\n\\caption{Details of several transitions that change the connectivity of solution curves. Closed and open disks are branch and turning points, respectively. First, the $w$ branch collides with the $U$ branch, and the high-shear portion that includes the $uUu$ branch detaches from $U$ and attaches to part of $w$. Then, folds 18 and 7 annihilate. Then, subcritical pitchforks 4 and 9 annihilate, detaching the $US\\pm$ branches from the $U$ branch, while $U$ attaches smoothly to $w$.}\n\\label{fig:Transition1}\n\\end{figure}\n\nSome solution manifolds at $\\psi_0=27.5^{\\circ}$ are shown in Figure \\ref{fig:Solution27p5}. The $WS\\pm$ branch has nearly disappeared. At $\\psi_0 \\approx 26.29^{\\circ}$, pitchfork bifurcation 4 transforms from super- to sub-critical, and a new fold bifurcation 12 appears. This means that the $U$-$US$ path is no longer smooth.\n Fold bifurcation 7 of the $w$ branch is approaching the $U$ branch, and will merge through a complicated sequence shown in detail in Figure \\ref{fig:Transition1}.\n At $\\psi_0 \\approx 27.71^{\\circ}$, the branch containing the $w$ state touches the $U$ branch, and then splits to form two new folds 13 and 18. This causes the $uUu$ branch to detach from the $U$ branch and attach to the unstable part of the $w$ branch to form an isolated branch that emerges from and loops back to the limiting shear.\n At $\\psi_0 \\approx 27.75^{\\circ}$, folds 18 and 7 annihilate each other through a cusp in the $\\Delta D\/L - \\psi_0$ plane. Subcritical pitchfork 4 and branch point 9 annihilate each other at $\\psi_0\\approx28.15^{\\circ}$; it appears that we can identify point 9 as a subcritical pitchfork at least for the small window of angles preceding this annihilation event, although stability information inferred from experiments does not allow us to make this identification in general. This annihilation process detaches the $US\\pm$ branches from the $U$ branch, while $U$ attaches smoothly to $w$ so that they are no longer distinct states, and the $U$-$US$ transition no longer occurs. Now the primary first and third modes of planar \\emph{elastica} are on the same curve, separated by a fold 6 and two branch points 8 and 0.\n \n\nFigure \\ref{fig:Solution30} shows some solution manifolds and experimental stability data for a clamping angle $\\psi_0=30^{\\circ}$. Both the $US\\pm$ and $uUu$ branches are clearly detached from what is now the $U$-$w$ branch, which is approaching and will soon collide with the $uUui$ branch. \nDue to proximity to bifurcations that change connectivity of the solutions, the experimental data is not all consistent. We draw red dotted lines to indicate that we often observe smooth transitions from $U$-$w$ to $US+$ and from $uUui$ to $U$-$w$, though at least half of the time these transitions do not occur and the data follow qualitatively with the numerical solutions. The solid red lines correspond to these ``correct'' data. Clearly, the system is sensitive to the presence of two nearby bifurcations in parameter space. \nThe jump from $U$-$w$ to $uUui$ is weak and hard to observe. This is now the main path to the limiting shear.\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=0.85\\textwidth]{BD30degree.pdf}\n \\caption{\n Some solutions (black and grey curves) and bifurcations (open and closed disks, some numbered) of the perfectly anisotropic Kirchhoff rod equations for clamping angle $\\psi_0=30^{\\circ}$, along with experimental data (red curves). \n\t\tThe horizontal extent of the red curves is the range of stability (typical variation $\\approx \\pm 0.01\\, \\Delta D\/L$), while the vertical position of the curves is not measured data, but is made to follow near the numerical curves whenever a comparison is possible. No data was taken for $\\Delta D\/L < 0$. \n\t\tNumerically, the newly formed $U$-$w$ branch, which terminates in a fold rather than reaching the limit, is approaching the $uUui$ branch. Red dotted lines indicate that sometimes smooth transitions from $U$-$w$ to $US+$ and from $uUui$ to $U$-$w$ are observed, which is inconsistent with the rest of the data (red solid lines) and the connectivity of the numerical solutions.\n}\n \\label{fig:Solution30}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{taukappan.pdf}\n\t\\caption{For five stable configurations at $\\psi_0=30^{\\circ}$, $\\Delta L\/L=1\/2$, and $\\Delta D\/L=0.625$, we plot the axial force $N_3$, the curvature $\\kappa_1$, the twist $\\tau$, and the energy density $\\varepsilon =\\frac{1+\\nu}{2} \\kappa_1 ^2 + \\tau ^2$, with $\\nu = 0.25$. The $uUui$, $w$ and $uUu$ states are reversibly symmetric about their midpoint, while the $US\\pm$ states are a reversibly symmetric pair. Increasing $s\/L$ corresponds to moving from right to left on any curve renderings in the text.}\\label{fig:30states}\n\\end{figure}\n\nAt $\\Delta D\/L=0.625$, there are five stable states. Figure \\ref{fig:30states} shows the numerically determined axial force $N_3$, the curvature $\\kappa_1$, the twist $\\tau$, and the energy density $\\varepsilon =\\frac{1+\\nu}{2} \\kappa_1 ^2 + \\tau ^2$, with $\\nu = 0.25$, for these states. At this value of shear, all states have a higher energy density near the clamps than in the middle. Interestingly, we observe that the $uUu$ state (primarily above the clamping plane) is purely tensile ($N_3 > 0$), while the corresponding $uUui$ state (primarily below the clamping plane) is compressive towards its ends and slightly tensile in the middle, although the depression at the center can be compressive at lower values of shear. In general, higher shear will lead to increased tension as the limiting states are approached.\n\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{BD37degree.pdf}\n\t\\caption{\n\tSome solutions (black and grey curves) and bifurcations (open and closed disks, some numbered) of the perfectly anisotropic Kirchhoff rod equations for clamping angle $\\psi_0=37^{\\circ}$, along with experimental data (red curves). Some of the grey curves from prior diagrams have been removed. \n\t\tThe horizontal extent of the red curves is the range of stability (typical variation $\\approx \\pm 0.01\\, \\Delta D\/L$), while the vertical position of the curves is not measured data, but is made to follow near the numerical curves whenever a comparison is possible. No data was taken for $\\Delta D\/L < 0$. \n\t\tNumerically, the $U$-$w$ branch has collided with the $uUui$ branch, creating a $U$-$w$-$uUui$ branch and a small residual $uUui$ branch. These are also observed experimentally. There is now a continuous $U$-$w$-$uUui$ path from first mode planar \\emph{elastica} to approach the limiting shear. The $WS\\pm$ states are close to disappearing numerically, and are not observed experimentally.}\n\t\\label{fig:Solution37}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{BD45degree.pdf}\n\t\\caption{\nSome solutions (black and grey curves) and bifurcations (open and closed disks, some numbered) of the perfectly anisotropic Kirchhoff rod equations for clamping angle $\\psi_0=45^{\\circ}$, along with experimental data (red curves). One grey curve has been truncated with a dashed line.\n\t\tThe horizontal extent of the red curves is the range of stability (typical variation $\\approx \\pm 0.01\\, \\Delta D\/L$), while the vertical position of the curves is not measured data, but is made to follow near the numerical curves whenever a comparison is possible. No data was taken for $\\Delta D\/L < 0$. \n\t\tThe $WS\\pm$ states have become unstable, and the residual $uUui$ branch has disappeared. }\n\t\\label{fig:Solution45}\n\\end{figure}\n\n\n\nAt $\\psi_0 \\approx 30.62^{\\circ}$, the $U$-$w$ branch collides with the $uUui$ branch. This merge-split event leads to a continuous $U$-$w$-$uUui$ path from first mode planar \\emph{elastica} to approach the limiting shear. However, part of the original $uUui$ branch remains, and can be seen between pitchfork 10 and a new fold 14 in Figure \\ref{fig:Solution37}. Figure \\ref{fig:Solution37} shows some solution manifolds and experimental stability data for a clamping angle $\\psi_0=37^{\\circ}$. \nNow the primary third mode and the inverted first mode of planar \\emph{elastica} are on the same curve, separated by several branch points 3, 10, 8, and 0, and a fold 14. \nThe $WS\\pm$ states have a very narrow extent, and will soon disappear (at $\\psi_0 \\approx 41.74^{\\circ}$) as the supercritical pitchfork 3 absorbs the fold 5 and becomes subcritical, a process we will clearly see later in Figure \\ref{fig:phasediagram}. Already we do not observe them in the experiments.\nSome of the grey curves in the lower half of previous figures, including the inverted third mode counterpart to the upper branch that contains the $w$ state, have been removed, as they have collided with other very complicated states that we have already removed. We retain a small grey hairpin curve near the limiting shear, as it will eventually link up with one of the black curves at higher angles.\nThe experiments confirm the changes in connectivity, including the presence of a residual $uUui$ branch with short extent at intermediate shear.\nInterestingly, this $uUui$ shape continues to jump to the $U$-$w$-$uUui$ branch upon increasing shear, and to the $US-$ branch upon decreasing shear, even as the range of stability shrinks and these bifurcations (fold and subcritical pitchfork) approach the same value of shear. This, along with the relatively steep slope of the branch, imply that the shapes change significantly over a narrow range of shear.\nMany transitions are illustrated in the supplementary video {\\texttt{transition37.mp4}} \\cite{videos}.\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Transition2.pdf}\n\\caption{Details of several transitions that change the connectivity of solution curves. The annihilation of branch points 8 and 10 occurs at fold point 14 and leads to the disappearance of the residual $uUui$ branch.\nClosed and open disks are branch and turning points, respectively.}\n\\label{fig:Transition2}\n\\end{figure}\n\nFigure \\ref{fig:Solution45} shows some solution manifolds and experimental stability data for a clamping angle $\\psi_0=45^{\\circ}$, after a collision between pitchfork 3 and fold 5 at $\\psi_0 \\approx 41.74^{\\circ}$ has made $WS\\pm$ states unstable, and a collision-annihilation of branch points 8 and 10 at $\\psi_0 \\approx 44.27^{\\circ}$ has eliminated the residual $uUui$ branch. Details of the latter process are shown in Figure \\ref{fig:Transition2}.\nSome complicated changes in connectivity of unstable (unobserved) states have occurred near the limiting shear, and one curve has turned back to link up with states we have already removed; we use a dashed line to truncate this curve. This connection is actually short-lived and will soon be lost again, so this dashed line will not appear again in subsequent figures.\n\n\nFigure \\ref{fig:Solution55} shows some solution manifolds and experimental stability data for a clamping angle $\\psi_0=55^{\\circ}$. At $\\psi_0 \\approx 52.09^{\\circ}$, a pair of subcritical pitchforks is born on the $uUu$ branch at $\\Delta D\/L \\approx 0.789$ . By $\\psi_0 = 55^{\\circ}$, one pitchfork 15 remains, the other has exited to the right at high shear, while a fold 16 has entered from the right. The details of this process are not known, but from the loci shown later in Figure \\ref{fig:phasediagram}, it seems that there is a curve splitting at high shear that we don't observe.\nOther curves have also developed folds that move in from the limiting shear, and additional complicated changes in connectivity of unstable (unobserved) states have occurred.\nAt this angle, the $US\\pm$ branch turns around at $\\Delta D\/L \\approx 0.856$ and connects to the loopy grey curves. We experimentally observe that both $US\\pm$ states lose stability before this value of shear, although for the $US+$ state this is at a value of shear higher than what we show in the figures.\nBifurcation 3 should now take the $W$ state to the $U$-$w$-$uUui$ branch instead of the $US\\pm$ branch. This change in path actually begins to happen at lower angles due to the destabilizing effects of gravity on the $W$ state, as can be seen in Figure \\ref{fig:Solution45}. Therefore, starting with any planar state at this high clamping angle and simply applying shear, we will approach the limit through a $U$-like state, and not an $S$-like state.\nExperimental results on the short $uUu$ branch are inconsistent, in that sometimes the state is not observed. The data shown are for ``correct'' observations. The system is sensitive due to its proximity to an event at \n$\\psi_0 \\approx 56.29^{\\circ}$, when subcritical pitchforks 11 and 15 annihilate each other, leading to the disappearance of the $uUu$ state. \n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{BD55degree.pdf}\n\t\\caption{Some solutions (black and grey curves) and bifurcations (open and closed disks, some numbered) of the perfectly anisotropic Kirchhoff rod equations for clamping angle $\\psi_0=55^{\\circ}$, along with experimental data (red curves). \n\t\tThe horizontal extent of the red curves is the range of stability (typical variation $\\approx \\pm 0.01\\, \\Delta D\/L$), while the vertical position of the curves is not measured data, but is made to follow near the numerical curves whenever a comparison is possible. No data was taken for $\\Delta D\/L < 0$. \n\t\tSeveral changes have occurred and are described in the text. The $US\\pm$ branch has a fold at higher shear than what we show here, and connects back to the loopy grey curves. The $uUu$ branch is not always observed experimentally.\t}\n\t\\label{fig:Solution55}\n\\end{figure}\n\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{BD60degree.pdf}\n\t\\caption{Some solutions (black and grey curves) and bifurcations (open and closed disks, some numbered) of the perfectly anisotropic Kirchhoff rod equations for clamping angle $\\psi_0=60^{\\circ}$, along with experimental data (red curves). \n\t\tThe horizontal extent of the red curves is the range of stability (typical variation $\\approx \\pm 0.01\\, \\Delta D\/L$), while the vertical position of the curves is not measured data, but is made to follow near the numerical curves whenever a comparison is possible. No data was taken for $\\Delta D\/L < 0$. \n\t\tThe $uUu$ branch has disappeared, and the fold 17 on the $US\\pm$ branch appears at a lower shear. Experimentally, the $W$ state is significantly destabilized by gravity.\t}\n\t\\label{fig:Solution60}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\t\\includegraphics[width=0.8\\textwidth]{jumpenergy.pdf}\n\t\t\\caption{\n\tThe bending energy density $\\varepsilon =\\frac{1+\\nu}{2} \\kappa_1 ^2$ with $\\nu = 0.25$, the twist energy density $\\tau ^2$, and the total energy density for the two states before and after the $US+$ to $uUui$ transition, the latter being the high shear portion of the $U$-$w$-$uUui$ branch.\n\tThe jump relieves a high concentration of bending energy at the $s=1$ end of the $US+$ state. The central expanse of the rod, shown in the inset, stores relatively little elastic energy, most of it in twist.\nIncreasing $s\/L$ corresponds to moving from right to left on any curve renderings in the text.}\n\t\\label{fig:jumpenergy}\n\\end{figure}\n\nFigure \\ref{fig:Solution60} shows some solution manifolds and experimental stability data for a clamping angle $\\psi_0=60^{\\circ}$.\nThe terminal fold 17 of the $US\\pm$ branch now appears at a lower shear, $\\Delta D\/L \\approx 0.771$. Experiments are qualitatively consistent with the solutions, with gravity significantly destabilizing the $W$ state at this high angle and asymmetrizing the $US\\pm$ transitions with respect to fold 17.\n\n Transitions at $\\psi_0=55^{\\circ}$ and $\\psi_0=60^{\\circ}$ are shown in the supplementary video {\\texttt{transition5560.mp4}} \\cite{videos}. It can be seen that the $US+$ to $U$-$w$-$uUui$ transition through fold 17 at $\\psi_0=60^{\\circ}$ involves a rapid rotation of one end. A similar rapid rotation is seen during local snap-through events in the tensile loading of slit sheets \\cite{marcelopersonal}. This transition is explored further in Figure \\ref{fig:jumpenergy}, which plots the bending energy density $\\varepsilon =\\frac{1+\\nu}{2} \\kappa_1 ^2$, with $\\nu = 0.25$, the twist energy density $\\tau ^2$, and the total energy density for the states just before and after this transition. It can be seen that the jump relieves a high concentration of bending energy at one end and partially relieves some twisting energy near that end while shifting its maximum to the end, and partially relieves some twisting energy at the other end. Some of the energy has moved into the central expanse of the rod, but this region stores relatively little elastic energy either before or after the transition, most of it in twist.\n\n \n\nIf we continue to increase the clamping angle above $\\psi_0 = 60^{\\circ}$, the twisted $TU\\pm$ loops will shrink and disappear through the annihilation of two folds at $\\psi_0 \\approx 74.03^{\\circ}$. The loopy structure of $US\\pm$ also shrinks and disappears after undergoing some complicated transitions which we do not investigate here.\nAdditionally, the subcritical pitchfork 3 delimiting the stability of the $W$ state approaches the zero-shear axis and annihilates with its negative-shear twin to eliminate the stable $W$ state at $\\psi_0 \\approx 76.95^{\\circ}$--- this is the classic snap-through of an inverted \\emph{elastica} arch under end rotations \\cite{plaut2009vibration}. After this, the only remaining stable configuration is $U$-$w$-$uUui$. We did not proceed past clamping angles of \n$\\psi_0=80^{\\circ}$.\n\nTo conclude this section, we remark that, despite some variable results due to sensitivity of the system near bifurcations, all of our experimental observations for narrow bands seem to be explained by the anisotropic Kirchhoff model, with allowance for the effects of gravity.\n\n\n\n\n\n\n\\section{Loci of bifurcations related to stable states}\\label{loci}\n\nThe complicated landscape of connectivity changes surveyed in the previous section can be better understood by tracing the loci of bifurcation points of the perfectly anisotropic Kirchhoff equations in a higher-dimensional parameter space. For our present study at fixed compression $\\Delta L\/L=1\/2$, this is the two-dimensional space spanned by normalized shear $\\Delta D\/L$ and clamping angle $\\psi_0$.\nFigure \\ref{fig:phasediagram} shows the paths traced in this space by many fold and branch points, numbered as on figures in Section \\ref{results}. Most of these are connected in some way with states observed in experiments, and thus delineate regions of stability for various configurations.\nThe leftmost inset shows the cusp that gives rise to folds $6$ and $7$ and the $w$ state at the small value of clamping angle $\\psi_0 \\approx 0.135^{\\circ}$.\nThe middle inset corresponds to the complicated series of transitions shown in Figure \\ref{fig:Transition1}. The upper right inset corresponds to the merge-split event between the $w$ and $uUui$ branches at $\\psi_0 \\approx 30.62^{\\circ}$.\nThe turning points connecting pitchfork 4 and branch point 9, and pitchforks 11 and 15, represent the annihilation events between these two pairs of bifurcations at $\\psi_0\\approx28.15^{\\circ}$ (Figure \\ref{fig:Transition1}) and $\\psi_0\\approx56.29^{\\circ}$, respectively.\n Continuing along the 15 curve, there is another turning point at higher shear, which implies that 15 and another pitchfork appear together on the $uUu$ branch at $\\psi_0\\approx52.09^{\\circ}$, with the other moving off to higher shears, as discussed earlier with respect to Figure \\ref{fig:Solution55}.\nBetween folds 12 and 17, there are two turning points and a cusp, indicating some complicated behavior involved in the disappearance of the $US\\pm$ states at clamping angles above $70^{\\circ}$.\nThe diagram is symmetric about zero shear; note the asymmetry between folds 1 and 2 that govern the disappearance of twisted states, such that 1 has a cusp on the zero-shear line, while 2 has a smooth turning point. The diagram also has a symmetry about zero clamping angle, but in a pairwise sense; for example, curves 1 and 2 will exchange their identities upon crossing this axis.\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{phasediagram.pdf}\n\t\\caption{Loci of various bifurcations for the perfectly anisotropic Kirchhoff rod in the plane spanned by clamping angle $\\psi_0$ and shear $\\Delta D\/L$, at fixed compression $\\Delta L\/L=1\/2$. Numbers correspond to those on figures in Section \\ref{results}. }\n\t\\label{fig:phasediagram}\n\\end{figure}\n\n\nThe loci provide some information about regions of stability. For example, with reference to the positive shear and clamping angle quadrant shown here, the $TU \\pm$ and $TW \\pm$ states are stable below curves 2 and 1, respectively, and the $W$ state is stable below curve 3. The $WS\\pm$ states are stable in the region between curves 3 and 5. The $US\\pm$ states are stable above and to the left of a curve connecting the loci of 4, 12, and 17.\n\n It is clear that clamping at large angles reduces the number of available states, and thus the occurrence of jump events. We can use a diagram like Figure \\ref{fig:phasediagram} to avoid such violent events. For example, we might wish to transform a large clamping angle, large shear $US+$ state to a large clamping angle, small shear $U$ state, without experiencing the jump that would occur upon simply reducing the shear. Instead, we can decrease the clamping angle, decrease the shear, and increase the clamping angle again. Thus we avoid crossing line 12, corresponding to a fold-induced jump, and instead cross line 4, corresponding to a supercritical pitchfork.\nFor another example, we can transform a small clamping angle, large shear $w$ state to either a small clamping angle, small shear $U$ state or a small clamping angle, large(r) shear $uUui$ state by first increasing the clamping angle, then shearing back or forward, and finally decreasing the clamping angle again. This avoids crossing lines $7$ or $6$, which are fold-induced jumps, and makes use of the continuous $U$-$w$-$uUui$ branch available at large clamping angles.\n\n\n\n\\section{Width effects: from rods to ribbons to plates}\\label{widtheffect}\n\n\\begin{figure}[h!]\n\t\\captionsetup[subfigure]{labelformat=empty}\n\t\\centering\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{widu.pdf}\n\t\t\\caption{$U1$}\n\t\\end{subfigure}%\n\t\\hspace{1pt}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{widus1+.pdf}\n\t\t\\caption{$US1+$}\n\t\\end{subfigure} \n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{widws+.pdf}\n\t\t\\caption{$WS+$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{widuuu.pdf}\n\t\t\\caption{$uUu$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{widsw.pdf}\n\t\t\\caption{$w$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\\centering\n\t\\includegraphics[height=1.056in]{U2.pdf}\n\t\\caption{$U2$}\n \\end{subfigure}\n \\hspace{-1pt}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{widus2+.pdf}\n\t\t\\caption{$US2+$}\n\t\\end{subfigure} \\\\\n\t\\vspace{10pt}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{widbw.pdf}\n\t\t\\caption{$W$}\n\t\\end{subfigure}%\n\t\\hspace{1pt}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{widus1-.pdf}\n\t\t\\caption{$US1-$}\n\t\\end{subfigure} \n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{widws-.pdf}\n\t\t\\caption{$WS-$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{widuuui.pdf}\n\t\t\\caption{$uUui$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{widswi.pdf}\n\t\t\\caption{$wi$}\n\t\\end{subfigure}\n\\hspace{0.13\\textwidth}\n\t\\begin{subfigure}[t]{0.13\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.8in]{widus2-.pdf}\n\t\t\\caption{$US2-$}\n\t\\end{subfigure}\n\t\\caption{Some experimental configurations observed in wide bands, with aspect ratio $D\/L=1\/8$, compression $\\Delta L\/L=1\/2$, clamping angle $\\psi_0 = 5^{\\circ}$, and various values of shear $\\Delta D \/ L$. Twisted states are not shown.\n\t Gravity is roughly vertical in these images.\nThe $US1\\pm$ and $US2\\pm$ states are separated by weak local jumps in the circled regions. At very low clamping angles, there is a similar separation of $WS\\pm$ into $WS1\\pm$ and $WS2\\pm$ states, but these have already merged at this clamping angle. \nThe $U1$ and $U2$ states will become connected at higher clamping angles. The red arrows on the $U2$ state point at regions of focused curvature. }\n\t\\label{fig:WidExpConfiguration}\n\\end{figure}\n\n\nIn this section, we present experimental results on wide bands ($D\/L = 1\/8$, $1\/4$) and compare them with the narrow band ($D\/L = 1\/80$) experiments and the anisotropic Kirchhoff rod model for a couple of choices of clamping angle ($\\psi_0 = 0^{\\circ}$, $15^{\\circ}$) at the same compression $\\Delta L\/L = 1\/2$.\nThe effects of gravity become less important as the width of the band increases.\nThe behavior of twisted states for intermediate width bands is quite complicated, including the appearance of new stable states and self-contact. Reserving a deeper exploration for future study, we leave this behavior out of the present discussion, other than to present later a few examples of twisted states for various intermediate width bands in Figure \\ref{fig:WidthEffect}. Also shown there are indented states, which become possible for very wide bands, and are another topic we reserve for future study. For the wider $D\/L = 1\/4$ bands, no twisted states are observed.\nAside from twisted states, several new states appear in wide bands, but these still appear to be related to the states we have already seen for narrow bands, and we can attempt to organize all the results around the Kirchhoff solutions.\nThe reversible symmetry properties of the Kirchhoff equations appear to persist in all of the experimentally observed wide band states.\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{BD0withWidExp.pdf}\n\t\\caption{Experimental data (red, brown, and blue curves) from narrow and wide bands, normalized by limiting shear, for clamping angle $\\psi_0=0^{\\circ}$, compared with numerical solutions of the anisotropic rod equations (black and grey curves). Twisted states are not included. The horizontal extent of the experimental curves is the range of stability (estimated error $\\approx \\pm 0.01\\, \\Delta D\/L$), while the vertical position of the curves is not measured data, but is made to follow near the numerical curves whenever a comparison is possible. No data was taken for $\\Delta D\/L < 0$. The $US\\pm$ states are split into two states for the intermediate-width bands. For the widest bands, the pitchforks between $U1$ and $US\\pm$ and $W$ and $WS\\pm$ are only weakly broken by gravity, and transitions to either of the $\\pm$ pair are observed.}\n\t\\label{fig:WidExperi0}\n\\end{figure}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.85\\textwidth]{BD15withWidExp.pdf}\n\t\\caption{As in Figure \\ref{fig:WidExperi0}, but with clamping angle $\\psi_0=15^{\\circ}$. The $U2$ state first appears at small nonzero clamping angles. For the widest bands, the pitchforks between $U1$ and $US\\pm$ and $W$ and $WS\\pm$ are only weakly broken by gravity, and transitions to either of the $\\pm$ pair are observed.}\n\t\\label{fig:WidExperi15}\n\\end{figure}\n\n\nFigure \\ref{fig:WidExpConfiguration} shows some of the states we observe in wide bands for a band of aspect ratio $D\/L = 1\/8$, a shallow clamping angle $\\psi_0 = 5^{\\circ}$, and various values of shear $\\Delta D \/ L$. We name the states with reference to those found in narrow bands. The $U$ state is now called $U1$, as there is a $U2$ branch stabilized at higher shears; the two will eventually connect at higher clamping angles. Increasing the shear in the $U2$ state leads to focusing of generators and bending energy, as indicated by the two red arrows. The intermediate width bands ($D\/L = 1\/8$) now feature two sets of $US\\pm$ states, which are separated by weak local jumps in the circled regions. At very low clamping angles, there is a similar separation of $WS\\pm$ into $WS1\\pm$ and $WS2\\pm$ states, but these have already merged with each other into a single $WS\\pm$ set at this small clamping angle. Twisted states are also present, but not shown in this figure. The inverted version of the $w$ state, the $wi$ state, is observed in wide bands.\n\n\n\n To compare bands of different width, we now normalize the shear $\\Delta D$ using the limiting shear for a band of given width, as discussed in Appendix \\ref{limitstates}. We have $\\Delta D_{max} \\approx 207.85$ mm for $D\/L=1\/80$, $\\Delta D_{max}=180.00$ mm for $D\/L=1\/8$ and $\\Delta D_{max} \\approx 156.33$ mm for $D\/L=1\/4$.\nFigures \\ref{fig:WidExperi0} and \\ref{fig:WidExperi15} show some solution manifolds for the perfectly anisotropic Kirchhoff rod, along with experimental data for narrow and wide bands for clamping angles $\\psi_0=0^{\\circ}$ and $\\psi_0=15^{\\circ}$, respectively. Twisted states are not included. While the $U2$ state is not present at zero clamping angle, it begins to exist at small nonzero clamping angles, and can be seen in Figure \\ref{fig:WidExperi15}. Based on its shape, it seems to correspond to a state observed in the Kirchhoff solutions but which is experimentally unstable for narrow bands. For wide bands, the $w$ state can be reached by gently poking the stable $U2$ state so that it buckles inwards near its midpoint. At both clamping angles, it is clear that increasing the width of the band stabilizes the $w$ and $wi$ states and destabilizes the $uUu$ and $uUui$ states. Recall that the $w$ state does not appear for the Kirchhoff solutions until a small nonzero clamping angle. The states all seem to follow the Kirchhoff rod backbone, but there is clearly an additional jump (probably two folds?) separating the $US1\\pm$ and $US2\\pm$ states, and another separating the $WS1\\pm$ and $WS2\\pm$ states, for intermediate width bands ($D\/L = 1\/8$) at sufficiently low clamping angles. This effect is shown in the supplementary video {\\texttt{widtheffect15.mp4}} \\cite{videos}.\n For the widest bands ($D\/L=1\/4$), the pitchforks between $U1$ and $US\\pm$ and $W$ and $WS\\pm$ are only weakly broken by gravity, and transitions to either of the $\\pm$ pair are observed, in contrast to the consistently biased choices made by narrow bands.\n\n\n\n\\begin{figure}[h!]\n\t\\captionsetup[subfigure]{labelformat=empty}\n\t\\begin{subfigure}[t]{0.18\\textwidth}\n\t\t\\includegraphics[width=0.9\\textwidth]{width3.pdf}\n\t\t\\caption{$D\/L=1\/80$}\n\t\\end{subfigure}%\n\t\\hspace{0.3pt}\n\t\\begin{subfigure}[t]{0.18\\textwidth}\n\t\t\\includegraphics[width=0.9\\textwidth]{width10.pdf}\n\t\t\\caption{$D\/L=1\/24$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.18\\textwidth}\n\t\t\\includegraphics[width=0.9\\textwidth]{width10-1.pdf}\n\t\t\\caption{$D\/L=1\/24$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.18\\textwidth}\n\t\t\\includegraphics[width=0.9\\textwidth]{width10-2.pdf}\n\t\t\\caption{$D\/L=1\/24$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.18\\textwidth}\n\t\t\\includegraphics[width=0.9\\textwidth]{width30.pdf}\n\t\t\\caption{$D\/L=1\/8$}\n\t\\end{subfigure} \\\\\n\t\t\\vspace{10pt}\n\t\\begin{subfigure}[t]{0.3\\textwidth}\n\t\t\\includegraphics[width=0.9\\textwidth]{width120-1.pdf}\n\t\t\\caption{$D\/L=1\/2$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.3\\textwidth}\n\t\t\\includegraphics[width=0.9\\textwidth]{width120-2.pdf}\n\t\t\\caption{$D\/L=1\/2$}\n\t\\end{subfigure}\n\t\\caption{Examples of states stabilized by width effects. Twisted states, different from the narrow band state shown in the upper left, are stabilized and then destabilized or eliminated by self-contact issues by increasing width. Each twisted state shown is a $-$ state, and there exist corresponding $+$ states. A pair of indented states exist for very wide bands; the elastic defects are stable only for a very narrow range of shear. All bands are compressed to $\\Delta L\/L=1\/2$; the bands with $1\/80 \\leq D\/L \\leq 1\/8$ are clamped at $\\psi_0=0^{\\circ}$ and the very wide band with $D\/L=1\/2$ is clamped at $\\psi_0=30^{\\circ}$. }\n\t\\label{fig:WidthEffect}\n\\end{figure}\n\n\n\nAt slightly higher clamping angles, the $U1$ and $U2$ states become connected for wider bands. This is in contrast to the Kirchhoff solutions and narrow band data, for which the $U1$ state (called $U$ in the prior Section \\ref{results}) merges with the $w$ state. At higher clamping angles, shearing the $U1$-$U2$ state will lead to two successive snap-throughs to the $w$ and $uUui$ states. \nThese transitions are shown in the supplementary video {\\texttt{widtheffect45.mp4}} \\cite{videos}, where it can be clearly seen that the violence of the first snap-through increases with increasing width. In general, greater width exaggerates the effect of energy focusing of the generators of a developable strip \\cite{korte2010triangular,starostin2015equilibrium,chopin2016disclinations}; the snap-through transitions release some of this stored bending energy. \nThe limiting shear is approached \\emph{via} one of four possible limiting states, either the $S$-like $US\\pm$ pair or the $U$-like $uUu$ and $uUui$ states. These contain highly focused conical singularities near their ends, as shown in Appendix \\ref{limitstates}.\n\n\nNo other new states were observed in experiments at $\\psi_0=30^{\\circ}$, $45^{\\circ}$, or $60^{\\circ}$, although interesting changes in connectivity do occur, one of which is shown in \nthe supplementary video {\\texttt{widtheffect45.mp4}} \\cite{videos}.\n\n\nWe briefly mention some other states that are stabilized by width effects, some of which are shown in Figure \\ref{fig:WidthEffect}. At a width $D\/L = 1\/24$, the narrow band $D\/L = 1\/80$ twisted state pair is replaced by three other $\\pm$ pairs. These states are sensitive to boundary conditions; one can play by hand with a band of $D\/L = 1\/40$ and see all eight twisted states by changing compression and clamping angle. Depending on the state, application of shear may lead to a snap-through, a looped structure, or self-contact. At a width of $D\/L = 1\/8$, the intermediate-width $\\pm$ pair has become unstable,\n and at higher widths the symmetric intermediate-width state requires self-contact.\nFor very fat bands, such as the $D\/L = 1\/2$ bands shown in in the figure, there is a pair of stably indented states, each of which contains a pair of ``d-cones''. These elastic defects are only stable for a very narrow range of applied shear, and upon decreasing or increasing shear will respectively annihilate or propagate through the structure to emerge through the boundaries, enabling a snap-through transition.\nBoth the twisted and indented examples suggest rich avenues of research, which we reserve for future work.\n\n\\section{Further discussion}\\label{discussion}\n\nWe have presented experimental results on the stability of thin elastic bands subject to compression, shear, and symmetric clamping. The Kirchhoff equations for perfectly anisotropic rods serve as a surprisingly good guide to the behavior of these bands, particularly when they are narrow in width. We have explored only a limited region of the parameter space of boundary conditions and band geometry, but have already stumbled on many new stable configurations and jump phenomena. Here we briefly discuss some confusing issues and avenues for future work.\n\nFirst we note again that there is a distinct difference between a Kirchhoff rod, whose cross section remains orthogonal to its centerline, and a developable strip. This difference is quite obvious in an image like Figure \\ref{fig:UlikeRl}, where the director $\\bm{d_1}$ associated with the slice of material perpendicular to the rod centerline is clearly not aligned with a straight line (generator) on the strip. This ``rod'''s cross section is actually bending in the width direction. This is why our renderings of the Kirchhoff solutions as strips representing the rod frame are not equivalent to renderings of isometrically deforming elastic strips. This point is reiterated in Appendix \\ref{bvp}. However, the assumption of developability is itself problematic, particularly when applied to narrow strips. We refer the reader to Appendix \\ref{stripmodel} for a demonstration of the limitations of strip models in the present context.\n\nFrom the behavior of wider bands, we might have expected that branch point 9 in the Kirchhoff solutions would be a pitchfork bifurcation with a stable $U2$-like state on one side. We do not observe any such stable state for narrow bands. For wider bands, the $U2$ state is somewhat shell-like, with synclastic curvature, and one can pop back and forth between $U2$ and the slightly indented $w$ state. In contrast, width appears to destabilize the highly bent and twisted $uUu$ and $uUui$ states. The stability of twisted states at low shear also shows a complicated dependence on width.\nThese and more complicated stabilization effects, such as the narrow range of boundary conditions allowing one to create defects by indenting very wide plate-like bands such as those shown in Figure \\ref{fig:WidthEffect}, raise interesting questions about the boundaries between rod-like, plate-like, and shell-like behavior in thin sheets.\nIn order to capture such width effects, one needs either a model of a two-dimensional plate or strip, or a Cosserat rod model with a more complicated structure derived from such a two-dimensional model \\cite{starostin2007shape, dias2014non, audoly2015buckling}.\nIn some such models, the width of the band can appear as a potential continuation parameter. However, strip models lead to difficulties in numerical implementation because of the singular way in which they handle inflection points, which must be added by hand and cannot arise spontaneously during continuation. These models will not admit rod-like solutions such as those shown in Figures \\ref{fig:30states} and \\ref{fig:jumpenergy}, where the twist, which for strips is identified with the torsion, does not vanish simultaneously with the curvature. More importantly, without some further modification, such models cannot be used to continue solutions such as those of the $U$-$w$-$uUui$ branch, in which inflection points smoothly appear during deformation.\n\n\nThe full space of boundary conditions includes positions and general tilts in all directions, such that the director frames at the end points may take arbitrary values in the space of rotations SO(3). In addition to energy barriers leading to multi-stability and jump phenomena, the non-simply-connected nature of SO(3) can also create topological barriers to deformation. This fact-- related to the famous ``belt trick'' of Dirac-- implies a lower bound on elastic energy for rods subject to boundary conditions in which the end tangents are parallel \\cite{baez1991topological}. It would be of great interest to expand these results to arbitrary boundary conditions, which would require some consideration of how to define an appropriate linking number for open rods \\cite{alexander1982ambiguous, van2000helical, van2007end, prior2016extended}.\n\nOur elastic system has an analogue in the geometric controls literature, in which the orientation of the frame of our perfectly anisotropic rod appears as the orientation of a vehicle that can pitch and roll, but cannot yaw, and the elastic energy appears as a quadratic cost function for the two allowed controls \\cite{Baillieul78}. However, without constraints on path length in the controls problem, the analogy is only strictly correct for an elastic setup in which the rod is not clamped, but is allowed to vary its length by sliding in or out of sleeves.\nThe ability to satisfy boundary conditions that favor a forbidden bending or steering rotation through an indirect combination of other allowed rotations reflects the fact that the commutator of two infinitesimal rotations in $\\mathbb{E}^3$ provides the third.\n\n\n\n\\section*{Acknowledgments}\nWe thank G H M van der Heijden for helpful discussions, particularly regarding the treatment of singularities in Appendix \\ref{stripmodel}. We also thank M A Dias and T J Healey for helpful comments, and P S Krishnaprasad for the reference \\cite{Baillieul78}. \n\n\n\n\n\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Availability}\nThis article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in \\cite{Stehli2020} and may be found at \\href{https:\/\/aip.scitation.org\/doi\/10.1063\/5.0023533}{https:\/\/aip.scitation.org\/doi\/10.1063\/5.0023533}. The data that support the findings of this study are available from the corresponding author upon reasonable request. \n\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn the late 1980s the study of K\\\"ahler groups, that is, fundamental groups of closed K\\\"ahler manifolds, \ntook off in spectacular fashion. While restrictions on such groups were previously known because of Hodge theory\nand because of rational homotopy theory, several deep new results were proved around 1988. I will only\nrecall two of them here. These and many other results on K\\\"ahler groups are discussed in detail in~\\cite{ABCKT}.\n\nFirstly, generalising partial results of Johnson and Rees~\\cite{JR}, Gromov proved:\n\\begin{thm}[Gromov~\\cite{G}]\\label{t:G}\nA K\\\"ahler group does not split as a nontrivial free product.\n\\end{thm}\nSecondly, building on work of Siu, Sampson and others, Carlson and Toledo proved:\n\\begin{thm}[Carlson--Toledo~\\cite{CT}]\\label{t:CT}\nNo fundamental group of a closed real hyperbolic $n$-manifold with $n\\geq 3$ is a K\\\"ahler group.\n\\end{thm}\nWhen these results were proved, several people, including Donaldson and Goldman, noticed the contrast between K\\\"ahler groups \non the one hand and three-manifold groups on the other: the latter are closed under free products, and, according to Thurston, \nmost three-manifolds with freely indecomposable fundamental group are hyperbolic. Moreover, a case by case check of the Thurston \ngeometries as explained in~\\cite{Scott} shows the following: closed three-manifolds carrying one of the geometries $S^2\\times\\mathbb{ R}$, \n$\\mathbb{ H}^2\\times\\mathbb{ R}$, $\\mathbb{ R}^3$ or $\\Sol^3$ have virtually odd first Betti number, and so their fundamental groups cannot be K\\\"ahler. \nMoreover, closed three-manifolds carrying one of the geometries $\\Nil^3$ or ${\\mathit sl}_2(\\mathbb{ R})$ have virtually positive first Betti numbers with trivial \ncup product from $H^1$ to $H^2$. Their fundamental groups cannot be K\\\"ahler by the Hard Lefschetz Theorem.\nNow, the only Thurston geometry that has not been excluded is $S^3$, where every fundamental group is finite. Since all finite \ngroups are K\\\"ahler, it was natural to expect that the intersection of three-manifold groups with the K\\\"ahler groups should consist\nexactly of the finite groups appearing as fundamental groups of three-manifolds with geometry $S^3$. The obstacle to turning \nthis expectation into a theorem, indeed a corollary of the above Theorems~\\ref{t:G} and~\\ref{t:CT}, came from three-manifolds with a \nnon-trivial JSJ decomposition along incompressible tori. While one could imagine that those manifolds containing at least some \nhyperbolic piece might yield to a generalisation of the harmonic map techniques of Carlson and Toledo~\\cite{CT}\\footnote{A first \nstep in this direction was soon taken by Hern\\'andez-Lamoneda, although his paper~\\cite{Her} was only published much later.}, \nthe case of graph manifolds seemed intractable.\n\nTwenty years ago one thought about such questions modulo Thurston's geometrisation conjecture. Since this has now \nbeen proved by Perelman~\\cite{P1,P2,KL,MT}, an unconditional result can finally be obtained. Indeed, Dimca and Suciu recently proved:\n\\begin{thm} [Dimca--Suciu~\\cite{DS}]\\label{t:main1}\nAssume that a group $\\Gamma$ is the fundamental group both of a closed K\\\"ahler manifold and of a \nclosed three-manifold. Then $\\Gamma$ is finite, and, therefore, a finite subgroup of $O(4)$.\n\\end{thm}\nOnce one proves $\\Gamma$ to be finite, it follows from Perelman's work~\\cite{P1,P2,KL,MT} that $\\Gamma$ is a finite subgroup of $O(4)$\nacting freely on $S^3$. Note that by a classical construction due to Serre, every finite group is the fundamental group of a smooth complex projective \nvariety, hence a closed K\\\"ahler manifold. By the Lefschetz hyperplane theorem one may assume this variety to be a surface.\n\nTo me, a surprising aspect of the proof given by Dimca and Suciu is that it does not follow the above outline at all, and makes little use \nof the Thurston approach to three-manifolds. In fact, their proof does not use Theorems~\\ref{t:G} and~\\ref{t:CT}. Instead, they \nconsider separately the cases of trivial and of nontrivial first Betti number. If the first Betti number of the fundamental group\nof a closed oriented three-manifold is positive, then they prove it is not K\\\"ahler using a lot of machinery of a very different sort: \ncharacteristic and resonance varieties, Catanese's \napproach to the Siu--Beauville theorem, a commutative algebra result of Buchsbaum--Eisenbud, \\ldots . Then, for the case of \nzero first Betti number, Dimca and Suciu appeal to results of Reznikov and Fujiwara pertaining to Kazhdan's property $T$.\nIt is only at this point that their proof depends on geometrisation via Fujiwara's arguments.\n\nThe present paper arose from my attempt to understand the argument of Dimca and Suciu~\\cite{DS}. From their \ntreatment of the positive Betti number case I extracted the following strategy for obtaining a contradiction: \n{\\it If $\\Gamma$ has positive first Betti\nnumber and is both the fundamental group of a closed oriented three-manifold and of a closed K\\\"ahler manifold, then\n$H^1(\\Gamma;\\mathbb{ R})$ comes from a complex curve. Therefore all cup products of classes in $H^1(\\Gamma;\\mathbb{ R})$ also \ncome from a curve, and this is incompatible with three-dimensional Poincar\\'e duality.}\n\nOne can actually implement this strategy in several different ways to prove Theorem~\\ref{t:main1}. \nHere I will give quite a different implementation from that in~\\cite{DS}, leading to a quick proof of the following:\n\\begin{thm}\\label{t:alb}\nIf $\\Gamma$ is a group with $b_1(\\Gamma)>0$ whose real cohomology algebra $H^*(\\Gamma;\\mathbb{ R})$ satisfies \n$3$-dimensional oriented Poincar\\'e duality, then $\\Gamma$ is not a K\\\"ahler group.\n\\end{thm}\nTo put this into perspective, recall that many K\\\"ahler groups are Poincar\\'e duality groups (of even dimension), cf.~\\cite{JR,Tol,Kl}.\nAlso recall that, for every $k\\geq 3$, Toledo~\\cite{Tol} constructed examples of K\\\"ahler groups of cohomological dimension $2k-1$.\nMoreover, his examples are duality (though not Poincar\\'e duality) groups.\n\nOf course, to exclude a group from being a K\\\"ahler group, it is enough that some finite index subgroup satisfy the \nassumptions of Theorem~\\ref{t:alb}. Thus Theorem~\\ref{t:alb} immediately gives:\n\\begin{cor}\\label{c:asph}\nLet $M$ be a closed aspherical three-manifold. If $M$ has a finite orientable covering that is not an $\\mathbb{ R}$-homology sphere, then\n$\\pi_1(M)$ is not a K\\\"ahler group.\n\\end{cor}\nTheorem~\\ref{t:alb} is more general than the Corollary because not every group whose real cohomology satisfies $3$-dimensional \nPoincar\\'e duality is the fundamental group of an aspherical three-manifold. This issue is related to the three-dimensional Borel\nconjecture; see Problem~3.77 on Kirby's problem list~\\cite{Kirby}.\n\nCorollary~\\ref{c:asph} proves most of Theorem~\\ref{t:main1}, since it handles not only manifolds with a nontrivial JSJ \ndecomposition, but also gives a uniform treatment of geometric cases that no longer need to be checked case by case, so \nwe obtain quite a simple proof of Theorem~\\ref{t:main1}\nfor groups with virtually positive first Betti number. Using Perelman's geometrisation theorem, the case of first Betti number \nzero can actually be reduced to Theorem~\\ref{t:CT}. In Section~\\ref{s:proofs} below we first prove Theorem~\\ref{t:alb}, and then spell out \nthe resulting straightforward proof of Theorem~\\ref{t:main1}, avoiding the difficult\narguments of Dimca--Suciu~\\cite{DS}, and the appeals to the works of Reznikov and Fujiwara. Like the original proof of~\\cite{DS},\nthe proof of Theorem~\\ref{t:main1} given here uses geometrisation only to handle the case of trivial (virtual) first Betti number.\n\nUsing the Kodaira classification of non-K\\\"ahler complex surfaces we shall also prove the following:\n\\begin{thm} \\label{t:main2}\nAssume that a group $\\Gamma$ is the fundamental group both of a closed complex surface $S$ and of a \nclosed three-manifold. Then either $\\Gamma$ is a finite subgroup of $O(4)$ and $S$ is a K\\\"ahler surface, \nor $\\Gamma$ is $\\mathbb{ Z}$ or $\\mathbb{ Z}\\oplus\\mathbb{ Z}_2$ and $S$ is a surface of class $VII$.\n\\end{thm}\nThis is interesting since in real dimension $6$ every finitely presentable group is the fundamental group of a \ncompact complex manifold, as proved by Taubes~\\cite{T}. Thus, for fundamental group questions, complex surfaces are at the \nwatershed between curves and the unrestricted case of complex three-folds, just like three-manifolds are at the watershed between \nreal surfaces and the case of four-manifolds, where all finitely presentable groups appear.\n\n\\section{Proofs}\\label{s:proofs}\n\n\\begin{proof}[Proof of Theorem~\\ref{t:alb}]\nSuppose for a contradiction that $X$ is a closed K\\\"ahler manifold with fundamental group $\\Gamma$, and let \n$\\alpha_X\\colon X\\longrightarrow T^{b_1(\\Gamma)}$\nbe its Albanese map. By the universal property of classifying maps, $\\alpha_X$ factors up to homotopy into a composition\n$$\nX\\stackrel{c_X}{\\longrightarrow} B\\Gamma \\stackrel{a}{\\longrightarrow} B\\mathbb{ Z}^{b_1(\\Gamma)} = T^{b_1(\\Gamma)} \\ ,\n$$\nwhere $c_X$ is the classifying map of the universal covering of $X$.\nOne concludes that $\\alpha_X^* = c_X^*\\circ a^*$ is trivial in real cohomology of degree $>3$ because $B\\Gamma$ has no\nsuch cohomology, and so the image of $\\alpha_X$ cannot have complex dimension $2$ or more.\nThus the image of $\\alpha_X$ is a complex curve $C$. \n\nIt is well known, and easy to see, that a one-dimensional Albanese image must be smooth, and of course it has positive genus.\nThus the Albanese map $\\alpha_X$ factors as \n$$\nX\\stackrel{c_X}{\\longrightarrow} B\\Gamma \\stackrel{\\hat a}{\\longrightarrow} C \\ .\n$$\nAll the maps above induce isomorphisms in degree one cohomology. Moreover, $\\alpha_X^*$ is nontrivial in degree $2$ \ncohomology, and so the same is true for $\\hat a^*$. However, there is no class in $H^1(\\Gamma;\\mathbb{ R})$\nthat has a nontrivial cup product with the image of $\\hat a^*$ in $H^2(\\Gamma;\\mathbb{ R})$, since this cup product comes from $C$, \nwhich has real dimension $=2$. This contradicts the assumption that $\\Gamma$ satisfies $3$-dimensional Poincar\\'e duality.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{t:main1}]\nWe need to show that an infinite three-manifold group $\\Gamma$ cannot be K\\\"ahler. Since finite coverings of \nK\\\"ahler manifolds are K\\\"ahler, we only need to exclude some finite index subgroup of $\\Gamma$, \nand so three-manifolds can be replaced by their finite coverings. In particular\nwe may assume that all three-manifolds are orientable. \n\nWe may restrict our attention to three-manifolds that are prime in the sense of being indecomposable under connected\nsums, since a nontrivial free product is never a K\\\"ahler group by Theorem~\\ref{t:G}. Such a prime\nthree-manifold is either $S^1\\times S^2$, or is aspherical, cf.~\\cite{M}. Since a K\\\"ahler group cannot be infinite cyclic,\nwe are reduced to the consideration of aspherical three-manifolds, so that, for all $3$-manifolds with positive (virtual) \nfirst Betti number, Theorem~\\ref{t:main1} follows from Corollary~\\ref{c:asph}, which in turn follows from Theorem~\\ref{t:alb}\nproved above.\n\nTo complete the proof of Theorem~\\ref{t:main1} it remains to deal with groups with vanishing first Betti number.\nThus consider a closed oriented aspherical three-manifold $M$ with infinite fundamental group $\\Gamma$ having $b_1(\\Gamma)=0$.\nIf $M$ contains an incompressible torus, then by a result of Luecke~\\cite{L}, see also~\\cite{K}, $M$ has a finite covering with \npositive first Betti number, so that Corollary~\\ref{c:asph} applied to this covering shows that\n$\\Gamma$ is not K\\\"ahler. Thus we are left with the case of an aspherical $M$ that contains no incompressible torus. \nSuch manifolds are hyperbolic by the work of Perelman~\\cite{P1,P2,KL}, and fundamental groups of hyperbolic three-manifolds are \nnever K\\\"ahler by Theorem~\\ref{t:CT}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{t:main2}]\nSuppose that $\\Gamma$ is the fundamental group of both a compact complex surface $S$ and a closed three-manifold $M$.\nAs before we may assume $M$ to be orientable.\n\nIf $S$ is K\\\"ahler, then $\\Gamma$ is finite by Theorem~\\ref{t:main1}. Conversely, if $\\Gamma$ is finite, then the first Betti number \nof $S$ vanishes, and so $S$ is K\\\"ahlerian, cf.~\\cite{Buch}. \n\nIf $S$ is not K\\\"ahlerian, then its first Betti number is odd, see again~\\cite{Buch}. We now use the Enriques--Kodaira classification\nto conclude that either $S$ is properly elliptic with $b_1(S)\\geq 3$, or $S$ is of class $VII$ with $b_1(S)=1$, cf.~\\cite{BPV,N}. In the first \ncase $\\Gamma$ is freely indecomposable and is a Poincar\\'e duality group of dimension $4$ by results of Kodaira described \nin~\\cite[Section~3 of Ch.~1]{ABCKT}. In the second case, it is known only that $\\pi_1(S)$ cannot split into $\\Gamma_1\\star\\Gamma_2$\nwith both $\\Gamma_i$ containing proper subgroups of finite index; see~\\cite[Thm.~1.35]{ABCKT}. However, since three-manifold groups\nare residually finite~\\cite{H}\\footnote{The reference~\\cite{H} treats only manifolds satisfying Thurston's geometrisation conjecture. \nBy Perelman's work~\\cite{P1,P2,KL} this is not a restriction.}, this is enough to conclude that in our case, where \n$\\pi_1(S)=\\Gamma=\\pi_1(M)$, $\\Gamma$ is indeed freely indecomposable.\n\nThus we may assume that $M$ is prime. If it is aspherical, then $\\Gamma$ is a three-dimensional Poincar\\'e duality group. \nThis means that $\\Gamma$ is not the fundamental group of a properly elliptic surface with $b_1(S)\\geq 3$ since those groups are four-dimensional \nPoincar\\'e duality groups. If $\\Gamma$ is the fundamental group of a class $VII$ surface, then we have $b_1(\\Gamma)=1$, and, by Poincar\\'e \nduality on $M$, $b_2(\\Gamma)=1$. Under the classifying map of the universal covering of $S$, $H^2(\\Gamma;\\mathbb{ R})$ injects into $H^2(S;\\mathbb{ R})$, \nwhere it becomes an isotropic subspace for the cup product for dimension reasons. (Its cup square comes from the three-dimensional $M$.) \nThus the intersection form of $S$ would have to be indefinite, which contradicts the known fact that the intersection forms of class $VII$ surfaces \nare negative definite; see~\\cite[Lemma~1.45]{ABCKT}.\n\nThus we are left to consider the case of an $M$ that is prime but not aspherical. This means that $M$ is $S^1\\times S^2$ \nif it is orientable; cf.~\\cite{M}. However, for a nonorientable $M$ we could also have the nontrivial $S^2$-bundle over $S^1$, also with \nfundamental group $\\mathbb{ Z}$, and $S^1\\times\\mathbb{ R} P^2$, with fundamental group $\\mathbb{ Z}\\oplus\\mathbb{ Z}_2$; cf.~\\cite{Scott}. Both $\\mathbb{ Z}$ and $\\mathbb{ Z}\\oplus\\mathbb{ Z}_2$\noccur as fundamental groups of Hopf surfaces. Conversely, every surface with one of these fundamental groups is of class $VII$; cf.~\\cite{BPV,N}.\nThis completes the proof of Theorem~\\ref{t:main2}.\n\\end{proof}\n\n\\section{Discussion}\n\n\\subsection{Avoiding the use of Theorem~\\ref{t:G}}\n\nIn the proof of Theorem~\\ref{t:main1} in Section~\\ref{s:proofs}, I found it most straightforward to reduce to the consideration of prime \nthree-manifolds by using Gromov's result on free products, stated as Theorem~\\ref{t:G} in the introduction. However, one can completely \nbypass the use of Theorem~\\ref{t:G}, as we now explain.\n\n\\begin{lem}\\label{l:JR}\nAssume that $\\Gamma_1$ and $\\Gamma_2$ each have a non-trivial finite quotient $f_i\\colon\\Gamma_i\\longrightarrow Q_i$. Then\ntheir free product $\\Gamma_1\\star\\Gamma_2$ has a finite index subgroup with odd first Betti number.\n\\end{lem}\n\\begin{proof}\nConsider the induced homomorphism $f\\colon\\Gamma_1\\star\\Gamma_2\\longrightarrow Q_1\\times Q_2$. By the Kurosh subgroup \ntheorem, its kernel is of the form $F_k\\star\\Gamma$, where $F_k$ is a free group of rank $k=(\\vert Q_1\\vert -1)(\\vert Q_2\\vert -1)$,\nand $\\Gamma$ is a free product of copies of the kernels of the $f_i$. For a finite quotient $g\\colon F_k\\longrightarrow Q$ of order $d$ we \nconsider the kernel $\\Delta$ of $\\bar g\\colon F_k\\star\\Gamma\\longrightarrow Q$, where $\\bar g$ restricts to $F_k$ as $g$ and is \ntrivial on $\\Gamma$. Then $\\Delta$ is isomorphic to $F_l\\star\\Gamma\\star\\ldots\\star\\Gamma$ with $d$ copies of $\\Gamma$ appearing,\nand $l=1+d(k-1)$. Thus $\\Delta\\subset \\Gamma_1\\star\\Gamma_2$ is a finite index subgroup with \n$$\nb_1(\\Delta) = l+d\\cdot b_1(\\Gamma) = 1+d\\cdot (k-1+b_1(\\Gamma)) \\ .\n$$\nChoosing $d$ to be even, we have found the desired subgroup.\n\\end{proof}\n\nSince three-manifold groups are residually finite~\\cite{H}, we have the following:\n\\begin{cor}\nIf $M$ is a non-prime three-manifold, then it has a finite covering with odd first Betti number.\n\\end{cor}\nAt the expense of appealing to residual finiteness, we can use this Corollary in place of Theorem~\\ref{t:G} to exclude non-prime manifolds\nfrom consideration in the proof of Theorem~\\ref{t:main1}. More generally, without restricting to three-manifold groups, Lemma~\\ref{l:JR}\ntells us that an arbitrary free product whose free factors admit finite quotients cannot be a K\\\"ahler group. This is exactly the special case of \nTheorem~\\ref{t:G} originally proved by Johnson and Rees~\\cite{JR}. Indeed our proof of the Lemma is a simplification of the argument \nin~\\cite{JR}.\n\n\\subsection{The necessity to discuss $\\mathbb{ R}$-homology spheres}\n\nIn the proof of Theorem~\\ref{t:main1} it was necessary to consider separately the case of groups with zero first Betti number. This step would be\nsuperfluous, if it were known that every closed three-manifold has a finite covering with positive first Betti number. If such a statement\nwere available, then one would not need Theorem~\\ref{t:CT} for the proof of Theorem~\\ref{t:main1} given here. \n\nApparently the question of whether every closed three-manifold with infinite fundamental group has virtually positive first Betti \nnumber was raised long ago by Waldhausen, Thurston, and others; see Problems~3.2 and 3.50 in Kirby's problem list~\\cite{Kirby}\nand the references given there. Curiously, those references do not include~\\cite{Hempel,L} and other papers quoted in~\\cite{Hempel},\nall of which contain a wealth of information about this problem. In any case, this problem seems to be still open.\n\n\\subsection{The second Betti number of infinite K\\\"ahler groups}\n\nCarlson and Toledo have asked whether an infinite K\\\"ahler group has virtually positive second Betti number\\footnote{The original reference\nfor their question is Section~18.16 in~\\cite{Kol}, where only a more specific version is formulated.}. If this were known to \nbe true, then, because of three-dimensional Poincar\\'e duality, we would not have to consider $\\mathbb{ R}$-homology $3$-spheres in \nthe proof of Theorem~\\ref{t:main1}. Moreover, we would not need to use geometrisation, and we would not need Theorem~\\ref{t:CT} either!\nWe refer to the paper of Klingler~\\cite{Kl} for a recent discussion of this question of Carlson and Toledo.\n\nUnfortunately, a slight misstatement occurs in~\\cite[Prop.~3.44 (i)]{ABCKT}, which implicitly asserts a positive answer to the \nquestion of Carlson and Toledo. The statement $b_2(\\pi_1(X))\\geq 1$ there should be replaced by $b_2(X)\\geq 1$ (which is trivial).\nThe Proposition in question was proved by Amor\\'os~\\cite{A}, whose paper does not contain the misstatement.\n\n\\bibliographystyle{amsplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{Omitted Calculations from Proof of \\cref{thm:fp_partial_obs}}\n\\label{sec:fp_partial_info_extras}\n\\newcommand{\\delta}{\\delta}\n\\newcommand{T}{T}\n\\newcommand{N}{N}\n\\newcommand{\\Hquants}[1]{v_{{#1}}}\n\\newcommand{\\estHquants}[1]{\\hat{v}_{{#1}}}\n\\newcommand{\\Hiquants}[2]{w_{{#2}}}\n\\newcommand{\\estHiquants}[2]{\\hat{w}_{{#2}}}\n\\newcommand{\\yquants}[1]{u_{{#1}}}\n\\newcommand{\\beta}{\\beta}\n\\newcommand{\\epsilon_1}{\\epsilon_1}\n\\newcommand{\\epsilon_1\/2}{\\epsilon_1\/2}\n\\newcommand{\\epsilon_1\/2}{\\epsilon_1\/2}\nWe further condition on the above and define an estimate of $G_i(x_j) \n= \\int_{x_j}^1 \\frac{1}{H(z)}\\, dH_i(z)$,\n\\[\n \\hat{G}_i(x_j) = \\sum_{s=j}^{|X|-1} \\lr{\\hat{H}_i(x_{s+1}) - \\hat{H}_i(x_s)}\/{\\hat{H}(x_s)}.\n\\]\nUsing the mean value theorem, there exists a set of points $\\{\\zeta_j\\}$ with\n$\\zeta_j \\in (x_{j}, x_{j+1}]$ and \n\\[\n G_i(x_j) \n = \\int_{x_j}^1 \\frac{1}{H(z)}\\, dH_i(z)\n = \\sum_{s=j}^{|X|-1} \\frac{H_i(x_{s+1}) - H_i(x_{s})}{H(\\zeta_s)}.\n\\]\nWe first bound the difference between our piecewise estimate and the true $G_i$\non the set $X$: \n\\begin{align*}\n \\abs{G_i(\\hat{v}_t) - \\hat{G}_i(\\hat{v}_t)} \n &= \n \\abs*{\n \\sum_{s=j}^{|X|-1} \\frac{H_i(x_{s+1}) - H_i(x_{s})}{H(\\zeta_s)} - \n \\sum_{s=j}^{|X|-1} \\frac{\\hat{H}_i(x_{s+1}) - \\hat{H}_i(x_s)}{\\hat{H}(x_s)}\n } \\\\\n &\\leq \n \\abs*{\n \\sum_{s=j}^{|X|-1} \\frac{H_i(x_{s+1}) - H_i(x_{s})}{H(\\zeta_s)} - \n \\sum_{s=j}^{|X|-1} \\frac{\\hat{H}_i(x_{s+1}) - \\hat{H}_i(x_s)}{H(\\zeta_s)}\n } \\\\\n &\\qquad +\n \\abs*{\n \\sum_{s=j}^{|X|-1} \\frac{\\hat{H}_i(x_{s+1}) - \\hat{H}_i(x_s)}{H(\\zeta_s)}\n - \\sum_{s=j}^{|X|-1} \\frac{\\hat{H}_i(x_{s+1}) - \\hat{H}_i(x_s)}{\\hat{H}(x_s)}\n } \\\\\n &\\leq \n \\frac{2}{\\gamma} \\cdot \\sum_{s=1}^{|X|} \\abs{H_i(x_{s}) - \\hat{H}_i(x_{s})}\n + \\max_{s \\in [|X|]}\\ \\ \\abs*{\n \\frac{1}{H(x_{s+1})} - \\frac{1}{\\hat{H}(x_s)}\n } \\\\\n &\\leq \n \\frac{2|X|\\beta}{\\gamma} + \\frac{\\delta + 2 \\cdot \\epsilon_1 + \\beta}{\\gamma^2}\n \\\\\n &\\leq \n \\frac{2|X|\\beta}{\\gamma} + \\frac{1}{\\gamma^2} \\max_{s \\in [|X|]}\\ \\ \\abs*{\n {H(\\zeta_s)} - \\hat{H}(x_s)\n } \n\\end{align*}\n\\subsection{Proof of \\cref{thm:sp_bid_insert}}\n\\label{app:sp_partial_info_proof}\n\nWe combine this result with the same binary search from\n\\cref{sec:1stPrice:additionalBidder} to identify {\\em quantiles} of the\nfunctions $\\widehat{F}_i$. In particular, fix $\\epsilon > 0$ and let\n\\[\n W = \\{w_a \\coloneqq \\gamma + a \\cdot \\frac{\\epsilon}{2} \n \\ |\\ a \\in \\mathbb{N} \\text{ and } \\gamma + a \\cdot \\frac{\\epsilon}{2} \\leq 1 \\} \n \\cup \\{1\\}.\n\\]\nAn identical argument from the one from the proof of \\cref{thm:fp_partial_obs}\nshows that for any $\\epsilon > 0$, we can use binary search to \nfind $\\hat{z}_{j,a}$ such that \n\\begin{align}\n \\label{eq:sp_quantile_recovery}\n \\abs{F_j(\\hat{z}_{j,a}) - w_a} \\leq \\frac{\\epsilon}{2} \n \\text{ for all $j \\in [k]$ and $a \\in [|W|]$}\n \\\\ \n \\nonumber\n \\text{ w.p. } \\qquad \n 1 - \\frac{4k}{\\epsilon}\\log\\lr{\\frac{4L}{\\epsilon}}\\exp\\lbrb{-\\epsilon^2 \\gamma n_1 \/ 192}\n\\end{align}\nusing $C \\cdot n_1\\cdot k \\cdot \\log(4L\/\\epsilon) \/ \\epsilon$ samples \nfor a universal constant $C$ \n(in particular, see \nthe proof of \\cref{thm:fp_partial_obs}, set $\\delta = \\epsilon_1 =\n\\epsilon\/2$, and use \\cref{lem:sp_reserve_conc} for the pointwise guarantee in\nplace of \\eqref{eq:fp_pointwise_conc}).\n\nConditioning on this event, we can thus define the piecewise-constant functions\n$\\widehat{F}_j$ for $j \\in [k]$ as \\[\n\\widehat{F}_j(x) = \\sum_{a \\in [|W|]} \\bm{1}\\lbrb{x \\in [z_{j,a}, z_{j, a+1})} \\cdot \\lr{\\gamma + a\\cdot \\frac{\\epsilon}{2}}\n\\]\nNow, consider any $x \\in [p, 1]$ and define $a \\in \\mathbb{N}$ such that \n$x \\in [\\hat{z}_{j, a}, \\hat{z}_{j,a+1}]$, so that by construction \n$\\widehat{F}_j(x) = w_{j,a}$. Then:\n\\begin{align*}\n F_j(x) &\\geq F_j(\\hat{z}_{j,a}) &\\text{(monotonicity of CDF)}\\\\\n &= w_{j,a+1}\n + (F_j(\\hat{z}_{j,a}) - w_{j,a}) \n + (w_{j,a} - w_{j,a+1}) \\\\\n &\\geq w_{j,a+1} - \\epsilon\/2 - \\epsilon\/2 &\\text{(definition of $W$ and \\eqref{eq:sp_quantile_recovery})} \\\\\n &\\geq \\widehat{F}_j(x) - \\epsilon &\\text{(definition of $\\widehat{F}_j$)}\n\\end{align*}\nSimilarly, $F_j(x) \\leq \\widehat{F}_j(x) + \\epsilon$. \n\nIt remains to handle $x \\in [p, \\hat{z}_{j,0}]$: note that by (strict)\nmonotonicity of the $\\widehat{F}_j$, we must have $\\widehat{F}_j^{-1}(\\gamma)\n\\leq p$. Thus, if $p \\leq x \\leq \\hat{z}_{j,0}]$,\n\\begin{align*}\n F_i(x) \\geq F_i(p) \\geq \\gamma = \\widehat{F_i}(x),\n\\qquad \\text{ and } \\qquad \n F_i(x) \\leq F_i(\\hat{z}_{j,0}) \\leq \\gamma + \\frac{\\epsilon}{2} \\leq \\widehat{F}_i(x) + \\epsilon.\n\\end{align*}\n\nThus, \n$\\abs{F_j(x) - \\widehat{F}_j(x)} \\leq \\epsilon$ over the entire interval $[p,\n1]$ with probability at least $1 - \\delta$, as long as\n\\[\n n \\geq \\frac{C k \\log(k\/\\epsilon) \\log(L\/\\epsilon)^2}{\\epsilon^3\\gamma},\n\\]\nfor a universal constant $C$, completing the proof.\n\\qed\n\\subsection{Miscellaneous Results}\n\\label{ssec:misc_res_sec_price}\n\nHere, we present miscellaneous results used in various parts of our proof. The first lemma shows that the functions, $U^*_i$, for different $i$ are within a constant factor of each other.\n\n\\uiujcomp*\n\\begin{proof}\n We have:\n \\begin{equation*}\n U^*_i (x) - U^*_i (y) = \\int_{y}^x \\sum_{l \\neq i} f_l (z) \\prod_{m \\neq i, l} F_m (z) dz \\leq \\lprp{\\frac{\\eta}{\\alpha}}^3 (U^*_j (x) - U^*_j (y))\n \\end{equation*}\n where the second inequality follows from \\cref{as:second_price_bdd}:\n \\begin{equation*}\n \\forall l, l', z \\in [0, 1]: f_l(z) \\prod_{m \\neq i, l} F_m (z) \\leq \\lprp{\\frac{\\eta}{\\alpha}}^3 \\cdot f_{l'}(z) \\prod_{m \\neq j, l'} F_{m} (z).\n \\end{equation*}\n\\end{proof}\n\n\n\n\\begin{lemma}\n \\label{lem:uh_disc_apx}\n We have, for all $i \\in [k]$ and all $\\tau \\in [T]$,\n \\begin{align*}\n \\forall l \\in \\ell^{(\\tau)}: \\lprp{\\hat{U}_i (x_{\\tau, l}) - \\hat{U}_i (x_{\\tau, l - 1})} \\leq \\frac{1}{1 - x_{\\tau, l}} \\cdot \\Delta^{(\\tau)}_{i, l} \\leq 2 \\lprp{\\hat{U}_i (x_{\\tau, l}) - \\hat{U}_i (x_{\\tau, l - 1})} \\\\\n \\forall \\hat{U}_i (x_\\tau) - \\hat{U}_i (x_{\\tau - 1}) \\leq \\sum_{l = 1}^{\\ell^{(\\tau)}} \\frac{1}{1 - x_{\\tau, l}} \\cdot \\Delta^{(\\tau)}_{i, l} \\leq 2 \\cdot \\lprp{\\hat{U}_i (x_\\tau) - \\hat{U}_i (x_{\\tau - 1})}\n \\end{align*}\n\\end{lemma}\n\\begin{proof}\n For the lower bound, we have $\\forall l \\in \\ell^{(\\tau)}$:\n \\begin{align*}\n \\frac{1}{1 - x_{\\tau, l}} \\cdot \\Delta^{(\\tau)}_{i, l} &= \\frac{1}{n} \\cdot \\sum_{j = 1}^n \\frac{1}{(1 - x_{\\tau, l})} \\cdot \\bm{1} \\lbrb{Z_j = i, x_{\\tau, l - 1} < Y_j \\leq x_{\\tau, l}} \\\\\n &\\geq \\frac{1}{n} \\cdot \\sum_{j = 1}^n \\frac{1}{(1 - Y_j)} \\cdot \\bm{1} \\lbrb{Z_j = i, x_{\\tau, l - 1} < Y_j \\leq x_{\\tau, l}} = \\hat{U}_i (x_{\\tau, l}) - \\hat{U}_i (x_{\\tau, l - 1}).\n \\end{align*}\n Summing the above inequality over $l$ concludes the proof of the lower bound. Similarly, for the upper bound, we get $\\forall l \\in \\ell^{(\\tau)}$:\n \\begin{align*}\n &\\frac{1}{1 - x_{\\tau, l}} \\cdot \\Delta^{(\\tau)}_{i, l} - (\\hat{U}_i (x_{\\tau, l}) - \\hat{U}_i (x_{\\tau, l - 1})) \\\\\n &= \\frac{1}{n} \\cdot \\sum_{j = 1}^n \\lprp{\\frac{1}{(1 - x_{\\tau, l})} - \\frac{1}{(1 - Y_j)}} \\cdot \\bm{1} \\lbrb{Z_j = i, x_{\\tau, l - 1} < Y_j \\leq x_{\\tau, l}} \\\\\n &\\leq \\frac{1}{n} \\cdot \\sum_{j = 1}^n \\lprp{\\frac{\\delta}{(1 - x_{\\tau, l})(1 - Y_j)}} \\cdot \\bm{1} \\lbrb{Z_j = i, x_{\\tau, l - 1} < Y_j \\leq x_{\\tau, l}} \\\\\n &\\leq \\frac{1}{n} \\cdot \\sum_{j = 1}^n \\lprp{\\frac{1}{2 \\cdot (1 - Y_j)}} \\cdot \\bm{1} \\lbrb{Z_j = i, x_{\\tau, l - 1} < Y_j \\leq x_{\\tau, l}} = \\frac{1}{2} \\lprp{\\hat{U}_i (x_{\\tau, l}) - \\hat{U}_i (x_{\\tau, l - 1})}.\n \\end{align*}\n Again, re-arranging and summing over $l$ concludes the proof. \n\\end{proof}\n\n \n \n \n\n\\subsection{Proof of \\cref{lem:qi_lb}}\nWe will proceed similarly to the proof of \\citep{lebrun2006uniqueness}, who\nuse a similar technique to prove strict monotonicity (i.e., a lower bound of\nzero). In particular, proving a quantitative lower bound requires carefully \ncontrolling additional terms that cancel in the original proof.\n\nFor $1 \\leq i \\leq n$, we define \n\\[\n b'_i = \\inf \\left\\{\n b' \\in [0, 1]:\\ \n \\frac{d}{db}\\log\\lr{G_i(v_i(b))} > L(b)\\ \\text{ for all } b \\in (b', 1]\n \\right\\},\n\\]\nand let $i$ be such that $b'_i = \\max_{1 \\leq k \\leq n} b'_k$. \nOur goal is to prove that $b_i' < \\rho$, since (by construction) this would\nimply that our desired property is true on the entire range.\n\nBy continuity of $(d\/db) \\log\\, G_i(v_i(b))$ and of $L(b)$, at the point\n$b_i'$ we must have that\n\\[\n \\frac{d}{db} \\log\\, G_i(\\vi{b}) = L(b).\n\\] \nSuppose that $b_i' \\geq \\rho$---by our definition of effective\nsupport, $\\Gi{b_i'} \\geq \\gamma$. Re-arranging the characterization of the\nBayes-Nash equilibrium \\eqref{eq:equilibrium_characterization} (cf.\n\\cref{lemma:characterization_equilibrium}), \n\\begin{align*}\n (v_i(b) - b) \\cdot \\frac{d}{db} \\log\\lr{G_i(v_i(b))} = \\frac{1}{n-1}\\lr{-(n-2) + \\sum_{j \\neq i} \\frac{v_i(b) - b}{v_j(b) - b}}.\n\\end{align*}\nTaking the derivative with respect to $b$ yields\n\\begin{align}\n D(b) \n \\label{eq:lebrun_derivative_1}\n &= \\sum_{j \\neq i} \\frac{v_i'(b)}{v_j(b) - b} - \\sum_{j \\neq i} \\frac{(v_i(b) - b)v_j'(b)}{(v_j(b) - b)^2} + \\sum_{j \\neq i} \\frac{v_i(b) - v_j(b)}{(v_j(b) - b)^2}. \n\\end{align}\n\n\\noindent Our next goal is to upper-bound the value of\n\\eqref{eq:lebrun_derivative_1} at $b_i'$. \nFirst, note that for all $j \\neq i$, our construction of $b_i'$ implies that\n$b_i' \\in [b_j', 1]$ (since $i = \\arg\\max_k b_k'$), and so\n\\begin{equation*}\n \\frac{1}{v_i(b_i') - b_i'} - \\frac{1}{v_j(b_i') - b_i'} = \\frac{d}{db} \\log\\lr{G_j(v_j(b_i'))} - \\frac{d}{db} \\log\\lr{G_i(v_i(b_i'))} \\geq 0,\n\\end{equation*}\nmeaning that \n\\begin{equation}\n \\vi{b_i'} \\leq \\vj{b_i'} \\implies \n \\frac{v_i(b) - v_j(b)}{(v_j(b) - b)^2} \\leq 0.\n \\label{eq:value_monotone_1}\n\\end{equation}\nThus, we can safely ignore the (negative) final term when upper bounding\n\\eqref{eq:lebrun_derivative_1}. \nTurning our attention to the second term, \\eqref{eq:log_deriv_diff}\nimplies the existence of at least one $j \\neq i$ such that \n\\begin{align}\n \\nonumber\n \\frac{d}{db} \\log\\, G_j(\\vj{b}) &\\geq \\frac{1}{n-1} \\frac{1}{v_i(b) - b}, \\text{ and rearranging, } \\\\\n \\label{eq:value_monotone_2}\n (v_i(b) - b) v_j'(b_i') &\\geq \\frac{G_j(v_j(b_i'))}{(n-1)\\cdot g_j(v_j(b_i'))} \\geq \\frac{\\gamma}{(n-1)\\cdot \\eta}.\n\\end{align}\nSince $v_j(b), b \\in [0, 1]$ and $v_j(b_i') \\geq v_i(b_i')$, we have $0 \\leq v_j(b_i') - b_i' \\leq 1$,\nand in turn \n\\begin{equation}\n \\frac{(v_i(b) - b) v_j'(b_i')}{(v_j(b_i') - b_i')^2} \\geq \\frac{\\gamma}{(n-1)\\cdot \\eta}\n\\end{equation}\nfor at least one $j \\neq i$, and thus \n\\begin{equation}\n -\\sum_{j \\neq i} \\frac{(v_i(b) - b) v_j'(b_i')}{(v_j(b_i') - b_i')^2} \\leq -\\frac{\\gamma}{(n-1)\\cdot \\eta}.\n\\end{equation}\nFinally, recall that \n\\[\n L(b_i') = \\frac{v_i'(b_i') \\cdot g_i(\\vi{b_i'})}{G_i(\\vi{b_i'})}\n \\implies \n v_i'(b_i') = \\frac{L(b_i') \\cdot G_i(\\vi{b_i'})}{g_i(\\vi{b_i'})}.\n\\]\nCombining this with \\eqref{eq:value_monotone_1} and\n\\eqref{eq:value_monotone_2}, and using that $\\vj{b_i'} \\geq \\vi{b_i'}$ yields\n\\begin{align*}\n D(b_i') &\\leq \\frac{-\\gamma}{(n-1)\\cdot \\eta} + \\sum_{j \\neq i} \\frac{L(b_i')}{v_j(b_i') - b_i'} \n \\leq \\frac{-\\gamma}{(n-1)\\cdot \\eta} + \\frac{(n - 1) \\cdot L(b_i')}{\\vi{b_i'} - b_i'}\n = 0,\n\\end{align*}\nwhere the last equality is by definition of $L$. Meanwhile, by definition\n\\begin{align}\n \\nonumber\n D(b) &= \\frac{d}{db} \\lbrb{ (v_i(b) - b) \\cdot \\frac{d}{db} \\log\\lr{G_i(v_i(b))} } \\\\ \n \\nonumber\n &= \\lr{v'_i(b) - 1} \\cdot \\frac{d}{db} \\log\\, G_i(\\vi{b}) \n + (v_i(b) - b) \\cdot \\frac{d^2}{db^2} \\log\\, G_i(v_i(b)) \\\\\n \\label{eq:Db_lhs}\n D(b_i') &\\geq (0 - 1)\\cdot L(b_i') + (v_i(b_i') - b_i') \\cdot \\frac{d^2}{db^2} \\log\\, G_i(v_i(b)).\n\\end{align}\nCombining the above results yields\n\\(\n (v_i(b_i') - b_i') \\cdot \\frac{d^2}{db^2} \\log\\, G_i(v_i(b))\n \\leq -L(b_i') < 0,\n\\)\nand since $v_i(b_i') - b_i' > 0$,\nthis must mean $\\frac{d}{db} \\log\\, G_i(\\vi{b}) < 0$. However, this in turn implies that there exists an\n$\\epsilon > 0$ such that $\\log\\, \\Gi{b_i'} < x$, which is a contradiction of\nour definition of $\\bi'$. Thus, our initial assumption ($b_i' > \\delta$) is\nimpossible, which proves the desired result.\n\\section{Introduction}\n\\input{sections\/intro.tex}\n\n\\subsection{Preliminaries} \n\\label{sec:prelims}\n\\input{sections\/preliminaries.tex}\n\n\\section{Estimation from First-price Auction Data} \n\\label{sec:max_sel_alg} \n\\input{sections\/first_price\/overview}\n\n\\subsection{Estimation of Bid Distributions} \n\\label{sec:1stPrice:bids}\n\\input{sections\/first_price\/bid_estimation}\n\n\\subsection{Estimation of Value Distributions}\n\\label{sec:value_estimation}\n\\input{value_estimation}\n\n\n\\section{Estimation from Second-price Auction Data}\n\\label{sec:sp_sel_alg}\n\\input{sections\/second_price\/overview.tex}\n\n\\subsection{Approach}\n\\label{ssec:sec_price_approach}\n\\input{sections\/second_price\/approach}\n\n\\input{sections\/second_price\/finite_sample}\n\\input{sections\/second_price\/partial_info}\n\n\\section{Conclusion}\nIn this work, we presented efficient methods for estimating first- and\nsecond-price auctions under independent (asymmetric) private values and partial\nobservability. \nOur methods come with convergence guarantees that are uniform in that\ntheir error rates do not depend on the bid\/value distributions being estimated.\nThese methods and the corresponding finite-sample guarantees build on a long\nline of work in Econometrics that establishes either identification\nresults, or estimation results under restrictive assumptions such as\nsymmetry or full bid observability.\n\n\\section{Acknowledgements}\nThis work is supported by NSF Awards CCF-1901292, DMS-2022448 and\nDMS2134108, a Simons Investigator Award, the Simons Collaboration on the Theory\nof Algorithmic Fairness, a DSTA grant, the DOE PhILMs project\n(DE-AC05-76RL01830), an Open Philanthropy AI Fellowship and a Microsoft Research-BAIR Open Research Commons grant.\n\n\\printbibliography\n\n\n\n\\subsubsection{Proof of \\cref{thm:1stPrice:likely}} \n\\label{sec:proof:1stPrice:likely}\nWe start by considering the effective-support setting.\nOur first result will be an\ninformation theoretic result enabling identification of $F_i$ with access to the\nfunction $H$ and the measure $H_i$ (without requiring a density function).\n\n\\begin{lemma} \\label{lem:it_ident}\n For all $i \\in [k]$ and all $x \\in (0, 1)$ such that $F_i(x) > 0$ and \n $H(x) > 0$,\n \\begin{equation*}\n F_i (x) = \\exp \\left(- \\int_{x}^1 \\frac{1}{H(y)} \\ d H_i\\right).\n \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n Using Lemma 3.1 from \\cite{matematyczny2002chain} we have that\n \\[\n \\log(F_i(1)) - \\log(F_i(x)) = \\int_x^1 \\frac{1}{F_i(y)} \\ d F_i \n = \\int_x^1 \\frac{\\prod_{j \\neq i} F_j(y)}{\\prod_{j \\in [k]} F_j(y)} \\ d F_i \n = \\int_x^1 \\frac{1}{H(y)} d H_i,\n \\]\n where the last equality follows from the continuity of $F_i$'s and the properties\n of Riemann-Stieltjes integration. The lemma follows by observing that \n $F_i(1) = 1$.\n\\end{proof}\n\nWe now focus our attention on obtaining good estimates of the quantity within the\nexponential on the right hand side in \\cref{lem:it_ident}. We introduce the\nfollowing notation: \n\\begin{gather*}\n \\hat{H}(x) \\triangleq \\frac{1}{n} \\sum_{j = 1}^n \\bm{1} \\lbrb{Y_j \\leq x} \n \\qquad \n \\hat{G}_i (x) \\triangleq \\frac{1}{n} \\sum_{j = 1}^n \\frac{1}{\\hat{H} (Y_j)} \\bm{1} \\lbrb{Y_j \\geq x \\text{ and } Z_j = i}\n\\end{gather*}\n\n\\noindent Based on the above definitions we can define our estimate for $F_i$ as\n\\( \\hat{F}_i (x) = \\exp \\left(-\\hat{G}_i(x)\\right). \\)\n\n\\noindent Our next goal is to prove that $\\hat{F}_i$ is close to $F_i$ for every\nvalue $y \\in [0, 1]$ such that $H(y) \\ge \\gamma$.\n\nNow we establish \nconcentration of $\\hat{H}$. \nBy the DKW inequality \\cite{dkw}:\n\\begin{equation*}\n \\max_{x \\in [0, 1]} \\abs*{\\hat{H}(x) - H (x)} \\leq \\frac{1}{20} \\cdot \\gamma^2 \\eps\n\\end{equation*}\nwith probability at least $1 - \\delta \/ 2$ for our setting of $n$. Conditioning on\nthe above event and observing that $H(x) \\geq \\gamma$ for all $x \\geq p$, \n\\begin{equation} \\label{eq:ih_err}\n \\max_{x \\in [p, 1]} \\abs*{{1\/H(x)} - {1\/\\hat{H}(x)}} \\leq \\frac{\\eps}{10}.\n\\end{equation}\nTo establish concentration of $\\hat{G}_i(x)$, we introduce another quantity\n$\\wt{G}_i$, defined as follows:\n\\begin{equation*}\n \\wt{G}_i(x) \\triangleq \\frac{1}{n} \\cdot \\sum_{j = 1}^n \\frac{1}{H(Y_i)} \\bm{1} \\lbrb{Y_j \\geq x \\text{ and } Z_j = i}.\n\\end{equation*}\nWe have from \\eqref{eq:ih_err} that $\\abs{\\hat{G}_i (x) - \\wt{G}_i (x)} \\leq\n\\frac{\\eps}{10}$ for all $x \\in [p, 1]$.\nThus, it suffices to establish concentration of $\\wt{G}$ around $G$. We first\nprove concentration on a discrete set of points and interpolate to the rest of \nthe interval. Define $U_i$ and $V_i$ as:\n\\begin{equation*}\n U_i = \\left\\{\\gamma + i \\cdot \\frac{\\eps}{10}: i \\in [N] \\cup \\{0\\} \\text{ and } \\gamma + i \\cdot \\frac{\\eps}{10} \\leq 1\\right\\} \\cup \\{1\\} \\text{ and } V_i = F_i^{-1} (U_i).\n\\end{equation*}\nand let $V = \\cup_{i \\in [k]} V_i$. We have for all $x \\in V, i \\in [k]$, by\nHoeffding's inequality that:\n \\begin{equation}\n \\label{eq:gt_bound}\n \\abs{\\wt{G}_i (x) - G_i (x)} \\leq \\frac{\\eps}{10}\n \\quad \n \\text{with probability at least $1 - \\delta \/ 2$.}\n \\end{equation}\n We now condition on the above event as well. By combining \\cref{eq:ih_err,eq:gt_bound}, we get:\n \\begin{align*}\n \\forall x \\in V, i \\in [k]: \\abs{\\hat{G}_i(x) - G_i(x)} \n &\\leq \\eps\/5, \\text{and so for all $x \\in V, i \\in [k]$, we have:} \\\\\n \\exp \\lbrb{-{\\eps\/5}}\\cdot F_i(x) \\leq \\hat{F}_i(x) \n &\\leq \\exp \\lbrb{{\\eps\/5}}\\cdot F_i(x).\n \\end{align*}\n We now extend from $V$ to the rest of $[p,1]$. Note that $\\hat{G}_i(x)$ is a decreasing function of $x$. Hence, $\\hat{F}_i$ is an increasing function of $x$. Now, let $x \\in (p, 1) \\setminus V$ and $i \\in [k]$. We must have $x_l, x_h \\in V$ with $x_l < x \\leq x_h$ satisfying $F_i(x_h) - F_i(x_l) \\leq \\eps\/10$. We now get:\n \\begin{align*}\n &\\hat{F}_i(x) \\leq \\hat{F}_i(x_h) \\leq \\exp\\lbrb{{\\eps\/5}} \n \\cdot F_i(x_h) \\leq \\exp\\lbrb{{\\eps\/5}} F_i(x_l) + \n {\\eps}\/{8} \\leq \\exp\\lbrb{{\\eps\/5}} F_i(x) + \\eps\/8,\n \\\\\n &\\hat{F}_i(x) \\geq \n \\exp\\lbrb{-{\\eps\/5}} \n F_i(x_l) \\geq \\exp\\lbrb{-{\\eps\/5}} F_i(x_h) + \n {\\eps\/10} \\geq \\exp\\lbrb{- {\\eps\/5}} F_i(x) + {\\eps\/10}.\n \\end{align*}\n The above two inequalities and our condition on $\\eps$ conclude the proof. \n \\hfill \\qed\n\n\\subsubsection{Proof of \\cref{thm:1stPrice:fullSupport}} \n\\label{sec:proof:1stPrice:fullSupport}\n\\noindent We now leverage our effective-support recovery result to recover bid\ndistributions on their full support (in Wasserstein distance). \nUnder the ``lower bound on density'' assumption,\n\\begin{align} \\label{eq:lowerBoundH}\n H(\\eta) = \\prod_{j \\in [k]} F_j(\\eta) \\ge (\\lambda \\cdot \\eta)^k.\n\\end{align}\nNow, setting $\\gamma = (\\lambda \\cdot \\eta)^k$ and using\n\\cref{thm:1stPrice:likely} we have that \n$\\tilde{\\Theta}\\left(\\frac{\\log(k\/\\delta)}{\\lambda^k \\cdot \\eta^k \\cdot \\eta^2}\\right)$ \nsamples suffice to find estimates $\\hat{F}_i$ such that the additive error\nbetween $\\hat{F}_i$ and $F_i$ is at most $\\eta$ in the interval $[\\eta, 1]$. For\nevery $i$, the maximum possible mass in the interval $[0, \\eta]$ with respect to\nthe measure $F_i$ is $1$. Therefore, any two measures with support $[0, \\eta]$ \nmass at most $1$ have a Wasserstein distance of at most $\\eta$. Also, in the subset\n$[\\eta, 1]$ of the support we have that since the longest distance in the support \nis at most $1$ and $\\max_{x \\in [\\eta, 1]} \\abs*{\\hat{F}_i(x) - F_i(x)} \\le \\eta$ \nwe have that the Wasserstein distance of the measures $\\hat{F}_i$ and $F_i$\nconditioned on the support $[\\eta, 1]$ is at most $\\eps \\cdot 1$. Thus,\n\\[ \\calW(\\hat{F}_i, F_i) \\le 2 \\cdot \\eta. \\]\nSetting $\\eta = \\eps\/2$ the theorem follows. \\hfill \\qed\n\n\\subsubsection{Proof of \\cref{thm:1stPrice:density}} \\label{sec:proof:1stPrice:density}\n\n We are going to use the estimation $\\hat{F}_i$ from \\cref{thm:1stPrice:likely}\ntogether with the Lipschitzness of $f_i$ to prove this theorem. Let $h > 0$ and\n$\\epsilon_0 > 0$ be parameters that we will determine later. We define, for\nevery $x \\in [p, 1]$, an density estimate\n\\[ \\hat{f}_i(x) \\triangleq \\frac{1}{h} (\\hat{F}_i(x + h) - \\hat{F}_i(x)), \\]\nwhere due to \\cref{thm:1stPrice:likely} we have \n$\\abs{\\hat{F}_i(x + h) - F_i(x + h)} \\le \\eps_0$ and \n$\\abs{\\hat{F}_i(x) - F_i(x)} \\le \\eps_0$ for $n =\n\\tilde{\\Theta}\\left(\\frac{\\log(k\/\\delta)}{\\gamma^4 \\eps^2}\\right)$ samples.\nThen,\n\\begin{align*}\n \\int_p^1 \\abs*{\\hat{f}_i(x) - f_i(x)} \\ d x \n &= \\int_p^1 \\abs*{\\frac{1}{h} (\\hat{F}_i(x + h) - \\hat{F}_i(x)) - f_i(x)} \\ d x \\\\\n & \\le \\int_p^1 \\abs*{\\frac{1}{h} (F_i(x + h) - F_i(x)) - f_i(x)} \\ d x + 2 \\eps \\\\\n & = \\int_p^1 \\abs*{\\frac{1}{h} \\lprp{\\int_{x}^{x + h} f_i(z) \\ d z} - f_i(x)} \\ d x + \\frac{2 \\eps}{h} \\\\\n &\\le \\int_p^1 \\frac{1}{h} \\lprp{\\int_{x}^{x + h} \\abs*{f_i(z) - f_i(x)} \\ d z} \\ d x + \\frac{2 \\eps}{h} \\\\\n \\intertext{now due to the Lipschitzness of $f_i$ we have that}\n \\int_p^1 \\abs*{\\hat{f}_i(x) - f_i(x)} \\ d x \n &\\le \\int_p^1 \\frac{1}{h} \\lprp{\\int_{x}^{x + h} L \\cdot \\abs*{z - x} \\ d z} \\ d x + \\frac{2 \\eps}{h} \\\\\n & = \\int_p^1 \\frac{L}{h} \\lprp{\\frac{(x + h)^2}{2} - \\frac{x^2}{2} - h \\cdot x} \\ d x + \\frac{2 \\eps}{h} \\\\ \n &= \\int_p^1 \\frac{L}{h} \\cdot h^2 \\ d x + 2 \\eps \\le L \\cdot h + \\frac{2 \\eps}{h}.\n\\end{align*}\nTherefore, if we choose $h = \\sqrt{\\eps_0\/L}$ and we also set $\\eps = \\eps_0^2\/(9 L)$ \nthe theorem follows.\n\n\\subsection{Lower Bound for Full-Support Estimation} \\label{sec:1stPrice:lowerBound}\n\n\\newcommand{\\mrm{Unif}}{\\mrm{Unif}}\n\n\\noindent Here, we establish lower bounds proving the optimality of \\cref{thm:1stPrice:fullSupport}. We prove:\n\\begin{enumerate}\n \\item the exponential dependence on $k$ incurred in \\cref{thm:1stPrice:fullSupport} is necessary and\n \\item the distributions cannot be recovered in Kolmogorov distance in their whole support.\n\\end{enumerate}\nIn both these cases, we will construct a pair of distributions $\\{f_i\\}_{i = 1}^k$ and $\\{f^\\prime_i\\}_{i = 1}^k$ satisfying the bounded density condition of \\cref{thm:1stPrice:fullSupport} such that:\n\\begin{enumerate}\n \\item $f_1$ and $f^\\prime_1$ have $\\calW (f_1, f^\\prime_1) \\geq \\Omega(\\eps)$ and $\\mathrm{d}_{\\mathrm{K}} (f_1, f^\\prime_1) \\geq 1 \/ 2$ and \n \\item Fewer than $\\Omega((\\lambda \\eps)^{-(k - 1)})$ fail to distinguish them with large probability.\n\\end{enumerate}\nThe main intuition behind our construction \nis that learning the\nbehavior of any of the densities below $\\eps$ requires observing \n$Y \\leq \\eps$ and this only happens with probability $\\eps^{-k}$. \n\n\\begin{theorem}\n \\label{thm:main_lb}\n Let $k \\in \\mb{N}$, and let $\\eps, \\lambda \\in (0, 1\/2)$. \n Then, there exist two tuples of distributions $\\mc{D} = \\{f_i\\}_{i = 1}^k$ and \n $\\mc{D}^\\prime = \\{f^\\prime_i\\}_{i = 1}^k$ \n with the first price auction model (\\cref{def:max_sel_obs}) \n on $\\mc{D}, \\mc{D}^\\prime$ satisfies the density bound condition from\n \\cref{thm:1stPrice:fullSupport} such that for any estimator $\\hat{\\mu}$, we\n have: \n \\begin{equation*}\n \\max \\lprp{\n \\mb{P} \\lbrb{\\calW \\lprp{\\hat{\\mu} (\\lbrb{(Y_i, Z_i)}_{i = 1}^n), \\mc{D}} \\geq \\frac{\\eps}{8}}, \n \\mb{P} \\lbrb{\\calW \\lprp{\\hat{\\mu} (\\lbrb{(Y_i^\\prime, Z_i^\\prime)}_{i = 1}^n), \\mc{D}^\\prime} \\geq \\frac{\\eps}{8}}} \\geq \\frac{1}{3}\n \\end{equation*}\n where $(Y_i, Z_i)$, $(Y^\\prime_i, Z^\\prime_i)$ are drawn i.i.d from\n $\\mc{D}$ and $\\mc{D}^\\prime$ respectively if $n \\leq\n \\frac{1}{10} \\cdot (\\lambda \\eps)^{-(k - 1)}$. \n\\end{theorem}\n\\begin{proof}\n Let $\\mc{D}_1 = \\{f_1, \\dots, f_k\\}$ and $\\mc{D}_2 = \\{f^\\prime_1, \\dots,\n f^\\prime_k\\}$ denote the two sets of distributions characterizing our\n first price auction model (\\cref{def:max_sel_obs}). \n We will have $f_i = f^\\prime_i$ for all $i > 1$.\n \\begin{equation*}\n f^\\prime_i = f_i = \\lambda \\cdot \\mrm{Unif} ([0, 1]) + (1 - \\lambda) \\cdot \\mrm{Unif}([3\/4, 1])\n \\quad \n \\text{ for all } i > 1.\n \\end{equation*}\n However, $f_1$ and $f_1^\\prime$ will have large Wasserstein and Kolmogorov\n distance:\n \\begin{align*}\n f_1 &= \\lambda \\cdot \\mrm{Unif} ([0, 1]) + \n (1 - \\lambda) \\cdot \\mrm{Unif} ([0,\\eps \/ 4]) \\\\\n f^\\prime_1 &= \\lambda \\cdot \\mrm{Unif} ([0, 1]) + (1 - \\lambda) \\cdot \\mrm{Unif} ([3\\eps \/ 4, \\eps]).\n \\end{align*}\n Let $(Y, Z)$ and $(Y^\\prime, Z^\\prime)$ be distributed according to the\n first price auction model with respect to $\\mc{D}_1$ and $\\mc{D}_2$. \n We now define the events $E$ and $E'$ on $(Y, Z)$ and $(Y', Z')$ as follows:\n \\begin{equation}\n \\label{eq:pe_bnd}\n E = \\{Y \\in (\\eps, 1]\\} \\text{ and } E^\\prime = \\{Y^\\prime \\in (\\eps, 1]\\} \\implies \\P \\lbrb{E} = \\P \\lbrb{E^\\prime} = 1 - (\\lambda \\eps)^{k - 1}\n \\end{equation}\n By construction, $(Y,Z)$ and $(Y^\\prime,Z^\\prime)$ have the same\n distribution conditioned on $E$ and $E'$.\n \n Now, let $\\bm{W} = \\{(Y_i, Z_i)\\}_{i \\in [n]}$ and $\\bm{W}^\\prime = \\{(Y_i,\n Z_i)\\}_{i = 1}^n$ be collections of $n$ i.i.d samples from\n $\\mc{D}$ and $\\mc{D}^\\prime$ respectively, and let $\\hat{\\mu}$ denote any\n estimator of the first price auction model. We show that $\\hat{\\mu}$ has\n large error on at least one of $\\mc{D}$ or $\\mc{D}^\\prime$. Letting $F$\n (respectively, $F^\\prime$) denote the event that $E$ (respectively,\n $E^\\prime$) holds for all of the $(Y_i, Z_i)$ (respectively, $(Y^\\prime_i,\n Z^\\prime_i)$), we have: \n \\begin{align*}\n \\mb{P} \\lr{\\calW(\\hat{\\mu} (\\bm{W}), \\mc{D}) \\leq \\nicefrac{\\eps}{8}} \n = \\mb{P}(F) \\cdot \\mb{P}\\lr{\\calW(\\hat{\\mu} (\\bm{W}), \\mc{D}) \\leq\n \\nicefrac{\\eps}{8} \\vert F} \n + \\mb{P} (\\bar{F}) \\cdot \\mb{P} \\lr{\\calW(\\hat{\\mu} (\\bm{W}), \\mc{D}) \\leq \\nicefrac{\\eps}{8} | \\bar{F}}.\n \\end{align*}\n Now, if $n \\leq \\frac{1}{10} (\\lambda\n \\eps)^{-(k-1)}$, we have from \\cref{eq:pe_bnd} and a union bound that\n $\\mb{P}(F) \\geq 9\/10$. \n Furthermore, note that conditioned on $F$ and $F^\\prime$,\n $\\bm{W}$ and $\\bm{W}^\\prime$ have the same distribution and $\\calW(f_1,\n f_1^\\prime) \\geq \\eps \/ 4$. Assuming the probability in the above equation\n is greater than $2 \/ 3$, we may re-arrange the above equation as follows:\n \\begin{align*}\n \\frac{2}{3} \\leq \\mb{P} (F) \\mb{P} \\lbrb{\\calW(\\hat{\\mu} (\\bm{W}), \\mc{D}) \\leq \\frac{\\eps}{8} \\biggr| F} + \\frac{1}{10} \\leq \\mb{P} (F^\\prime) \\mb{P} \\lbrb{\\calW(\\hat{\\mu} (\\bm{W}^\\prime), \\mc{D}^\\prime) \\geq \\frac{\\eps}{8} \\biggr| F^\\prime} + \\frac{1}{10}.\n \\end{align*}\n By re-arranging the above equation, we have that either:\n \\begin{align*}\n \\mb{P} \\lbrb{\\calW(\\hat{\\mu} (\\bm{W}), \\mc{D}) \\leq \\frac{\\eps}{8}} \\leq \\frac{2}{3},\n \\text{ or }\n \\mb{P} \\lbrb{\\calW(\\hat{\\mu} (\\bm{W}^\\prime), \\mc{D}^\\prime) \\leq \\frac{\\eps}{8}} \\leq \\frac{2}{3}\n \\end{align*}\n concluding the proof of the theorem.\n\\end{proof}\nNote that the probabilities $1 \/ 3$ chosen in the above theorem is not a\nsubstantial restriction as any algorithm successfully distinguishing between\n$\\mc{D}$ and $\\mc{D}^\\prime$ with probability bounded away from\n$\\nicefrac{1}{2}$ can be boosted to arbitrarily high probability by simple\nrepetition. As a simple consequence of this construction, we can \nrule out estimation in Kolmogorov distance:\n\n\\begin{theorem}\n \\label{thm:lb_kol}\n Let $n \\in \\mb{N}$ and $\\hat{\\mu}$ be an estimator for the First-Price-Auction model. Then, for all $\\delta > 0$, there exists a First-Price-Auction model characterized by $\\mc{D} = \\{f_i\\}_{i = 1}^k$ satisfying the bounded density condition of \\cref{thm:1stPrice:fullSupport} satisfying:\n \\begin{equation*}\n \\mb{P} \\lprp{\\mathrm{d}_{\\mathrm{K}} (\\hat{\\mu} (\\bm{W}), \\mc{D}) \\leq \\frac{1}{4}} \\leq \\frac{1}{2} + \\delta.\n \\end{equation*}\n where $\\bm{W} = \\{(Y_i, Z_i)\\}_{i = 1}^n$ are drawn i.i.d from the first\n price auction model on $\\mc{D}$.\n\\end{theorem}\n\\begin{proof}\n We will prove the lemma via contradiction. Let $n, \\hat{\\mu}$ be such that the there exists $\\delta > 0$ such that for all First-Price-Auction models, $\\mc{D}$, satisfying the bounded density condition:\n \\begin{equation*}\n \\mb{P} \\lprp{\\mathrm{d}_{\\mathrm{K}} (\\hat{\\mu} (\\bm{W}), \\mc{D}) \\leq \\frac{1}{4}} \\geq \\frac{1}{2} + \\delta.\n \\end{equation*}\n Note that by repeating the experiment $\\Omega(1 \/ \\delta^2)$ times, we may boost the success probability to $9 \/ 10$ by taking the pointwise median of the resulting estimates. However, from our construction in the proof of \\cref{thm:main_lb}, we have by picking $\\eps$ small enough in the construction that there exists a distribution, $\\mc{D}$ such that:\n \\begin{equation*}\n \\mb{P} \\lbrb{\\mathrm{d}_{\\mathrm{K}} \\lprp{\\hat{\\mu} (\\bm{W}), \\mc{D}} \\geq \\frac{1}{4}} \\geq \\frac{1}{3}\n \\end{equation*}\n as all the distributions we construct have Kolmogorov distance greater than $1\/2$ between them. This yields the contradiction, proving the theorem.\n\\end{proof}\n\n\\input{sections\/first_price\/partial_info}\n\n\\subsection{Estimation from Partial Observations} \n\\label{sec:1stPrice:additionalBidder}\n\n In this section we show how our results in the previous sections can be \ntranslated to the partial observation model introduced by \\cite{blum2015learning} \ndefined below.\n\n\\begin{definition}[Partial Observation Data] \\label{def:max_partial_obs}\n Let $\\{F_i\\}_{i = 1}^k$ be $k$ cumulative distribution functions with support \n $[0, 1]$, i.e. $F_i(x) = 0\\ \\forall x < 0$ and $F_i(1) = 1$. A sample $(r, Y, Z)$ from a\n first-price auction with bid distributions $\\{F_i\\}_{i = 1}^k$ is generated as\n follows:\n \\begin{enumerate}\n \\item we, the observer, pick a price $r \\in [0, 1]$, and let $X_{k + 1} = r$\n \\item generate $X_i \\ts F_i$ independently for all $i \\in [k]$,\n \\item observe a winner $Z = \\argmax_{i \\in [k+1]} X_i$.\n \\end{enumerate}\n\\end{definition}\n\nAt first glance, it seems like the access to partial observation data is more\nrestrictive than the access to the first-price auction data that we defined in \n\\cref{def:max_sel_obs}. Nevertheless, we show that partial observations \nsuffice to run the same \nestimation used in \\cref{sec:1stPrice:bids}. \n\n\\begin{theorem}[First-Price Auctions -- Partial Observations]\n \\label{thm:fp_partial_obs}\n Let $\\{Z_i\\}_{i = 1}^n$ be $n$ i.i.d. partially observed samples from the same \n first-price auction as per \\cref{def:max_partial_obs} and assume that the cumulative \n distribution functions $F_i$ are continuous and admit Lipschitz-continuous\n densities $f_i$ with constant $L$. Then, given $p, \\gamma \\in [0, 1]$ such\n that $\\Pr(X_{i \\in [k]} \\leq p) \\geq \\gamma$, \n there exists a polynomial-time\n estimation algorithm, that computes the cumulative distribution functions $\\hat{F}_i$\n for $i \\in [k]$, so that for every $\\eps \\in (0, \\gamma\/2]$ it holds that \n \\[ \\Pr\\left(\\max_{x \\in [p, 1]} \\abs*{\\hat{F}_i(x) - F_i(x)} \\le \\eps\\right) \\ge 1 - \\delta \\] \n for all $i \\in [k]$ assuming that \n $n = \\Theta\\lr{\n \\frac{k}{\\gamma^6\\epsilon^5}\\log\\lr{\\frac{k}{\\gamma^2\\epsilon\\alpha}}\\log\\lr{\\frac{L}{\\gamma^2 \\epsilon}}\n }$\n\\end{theorem}\n\\begin{proof}\nNote that under the partial observation model, we can estimate\n$\\Pr(Y \\geq x)$ for any fixed $x$ by setting the reserve price to $x$ (i.e., bidding\n$x$) and counting the number of times that the planted bid wins the auction.\nMore precisely, we can define %\n\\[\n \\hat{H}(x) = 1 - \\frac{1}{n_1} \\sum_{i=1}^{n_1} \\mathbf{1} \\lbrb{Z_i = k+1}.\n\\]\n\nWe can similarly define, for any given agent $i$, an estimator for the\nprobability that the agent wins with price less than or equal to $x$:\n\\[\n \\hat{H}_i (x) \\triangleq \n \\frac{1}{n_2} \\sum_{j=1}^{n_2} \\bm{1}\n \\lbrb{Z_j = i \\text{ at reserve price } 0}\n - \n \\frac{1}{n_2} \\sum_{j = 1}^{n_2} \\bm{1} \n \\lbrb{Z_j = i \\text{ at reserve price $x$}}.\n\\]\n\n\\newcommand{\\delta}{\\delta}\n\nBy construction $\\hat{H}_i \\rightarrow H_i$ and $\\hat{H} \\rightarrow H$ where\n$H_i$ and $H$ are as defined earlier. Similarly to our strategy in the proof of\n\\cref{thm:1stPrice:likely}, let\n\\begin{equation*}\n U = \\left\\{\\gamma + i \\cdot \\delta : i \\in \\mathbb{N} \\cup \\{0\\} \\text{ and } \\gamma + i \\cdot \\delta \\leq 1\\right\\} \\cup \\{1\\} \n \\text{ and } V = H^{-1} (U)\n \\text{ and } W = H_i^{-1} (U)\n\\end{equation*}\n\n\\newcommand{T}{T}\n\\newcommand{N}{N}\n\\newcommand{\\Hquants}[1]{v_{{#1}}}\n\\newcommand{\\estHquants}[1]{\\hat{v}_{{#1}}}\n\\newcommand{\\Hiquants}[2]{w_{{#2}}}\n\\newcommand{\\estHiquants}[2]{\\hat{w}_{{#2}}}\n\\newcommand{\\yquants}[1]{u_{{#1}}}\n\\newcommand{\\beta}{\\beta}\n\\newcommand{\\epsilon_1}{\\epsilon_1}\n\\newcommand{\\epsilon_1\/2}{\\epsilon_1\/2}\n\\newcommand{\\epsilon_1\/2}{\\epsilon_1\/2}\n\nFor convenience, define $N = |U| \\leq \\delta^{-1}$, and recall that\n$k$ is the number of agents in the auction.\nOur first goal is to obtain a set of estimates $\\estHquants{j} \\approx\n\\Hquants{j}$ for the quantiles of $H$ and another set $\\estHiquants{i}{j}\n\\approx \\Hiquants{i}{j}$ for the quantiles of each $H_i$.\nTo accomplish this, we will run, for each $\\yquants{j} \\in U$, \n$T$ iterations of binary search between $0$ and $1$. In particular, \nwe initialize $\\estHquants{j}^{(0)} = 1$ then, for each successive iteration\n$t$, a Hoeffding bound shows that \n\\begin{align}\n\\label{eq:fp_pointwise_conc}\n\\Pr\\lr{ \\abs*{\\hat{H}(\\estHquants{j}^{(t)}) - H(\\estHquants{j}^{(t)})} \\geq \\epsilon_1\/2} \n< 2\\exp\\lbrb{-2 \\lr{\\epsilon_1\/2}^2 n}. \\\\\n\\Pr\\lr{ \\abs*{\\hat{H}_i(\\estHiquants{i}{j}^{(t)}) - H_i(\\estHiquants{i}{j}^{(t)})} \\geq \\epsilon_1\/2} \n< 2\\exp\\lbrb{-\\lr{\\epsilon_1\/2}^2 n\/2}.\n\\end{align}\nWe condition on the above events by taking a union bound over all agents, all\nsearch steps, and all points $u_i \\in U$.\nNow, for each iteration $t$ of the binary search:\n\\begin{enumerate}\n \\item If $|\\hat{H}(\\hat{v}_i^{(t)}) - u_t| \\leq \\epsilon_1\/2$,\n then $|H(\\hat{v}_i^{(t)}) - u_t| \\leq \\epsilon_1$, so we terminate\n and set $\\hat{v}_i = \\hat{v}_i^{(t)}$,\n \\item otherwise if $\\hat{H}(\\hat{v}_i^{(t)}) - u_t > \\epsilon_1\/2$ then\n $H(\\hat{v}_i^{(t)}) > u_t$ and we search the upper interval,\n \\item otherwise $\\hat{H}(\\hat{v}_i^{(t)}) - u_t < -\\epsilon_1\/2$ and so\n $H(\\hat{v}_i^{(t)}) < u_t$ and we search the lower interval.\n\\end{enumerate}\nWe perform an analogous process to find the $\\estHiquants{i}{j}$.\nThis ensures the correctness of the binary search, and\nsetting $T = \\log(2L\/\\epsilon_1)$, where $L$ is a Lipschitz\nconstant of $H$ and all $H_i$, guarantees that after performing\nthis search for each $u_i$, we will find $\\hat{V}$ and $\\hat{W}$ such that\n\\begin{align}\n \\label{eq:cond1_partial_info}\n |H(\\hat{v}_j) - u_j| &\\leq \\epsilon_1 \\qquad \n \\text{for all}\\qquad j \\in [|U|], \\text{ and} \\\\\n |H_i(\\estHiquants{i}{j}) - u_j| &\\leq \\epsilon_1 \\qquad \n \\text{for all}\\qquad j \\in [|U|] \\text{ and } i \\in [k] \\\\\n \\nonumber\n \\text{w.p.} \\qquad & 1 - 2 T N \\exp \\lbrb{-2(\\epsilon_1\/2)^2} - 2k T N \\exp \\lbrb{-(\\epsilon_1\/2)^2\/2} \n\\end{align}\n\n\\noindent In order to define our approximation of $G_i$, we will\nconsider the list of indices $X = V \\cup W_i$, i.e., the union of the\nestimated quantiles of $H$ and $H_i$. Using Hoeffding's inequality,\n\\begin{align*}\n |H(x_j) - \\hat{H}(x_j)| &\\leq \\beta \\qquad \\text{for all}\\qquad j \\in [|X|], \\text{ and} \\\\\n |H_i(x_j) - \\hat{H}_i(x_j)| &\\leq \\beta \\qquad \\text{for all}\\qquad j \\in [|X|], \\text{ and} \\\\\n \\nonumber\n \\text{w.p.} \\qquad & 1 - 4kN \\exp\\lbrb{2n \\beta^2} \\geq 1 - \\frac{4k}{\\delta} \\exp\\lbrb{2n \\beta^2}\n\\end{align*}\n\nWe further condition on the above and define an estimate of $G_i(x_j) \n= \\int_{x_j}^1 \\frac{1}{H(z)}\\, dH_i(z)$,\n\\[\n \\hat{G}_i(x_j) = \\sum_{s=j}^{|X|-1} \\lr{\\hat{H}_i(x_{s+1}) - \\hat{H}_i(x_s)}\/{\\hat{H}(x_s)}.\n\\]\nUsing the mean value theorem (see Appendix \\ref{sec:fp_partial_info_extras} for more detail), \n\\begin{align*}\n \\abs{G_i(\\hat{v}_t) - \\hat{G}_i(\\hat{v}_t)} \n &\\leq \n \\frac{2}{\\gamma} \\cdot \\sum_{s=1}^{|X|} \\abs{H_i(x_{s}) - \\hat{H}_i(x_{s})}\n + \\max_{s \\in [|X|]}\\ \\ \\abs*{\n \\frac{1}{H(x_{s+1})} - \\frac{1}{\\hat{H}(x_s)}\n } \\\\\n &\\leq \n \\frac{2|X|\\beta}{\\gamma} + \\frac{\\delta + 2 \\cdot \\epsilon_1 + \\beta}{\\gamma^2}\n\\end{align*}\n\nWe now extend our approximation from the set of points $\\{x_i\\}$ to the entire\ninterval $[\\rho, 1]$. Note that for any $x$ in this interval, there exists an\n$x_h, x_{h+1} \\in X$ such that $x \\in (x_h, x_{h+1}]$. Furthermore, both $G_i$\nand $\\hat{G}_i$ are monotonic in $x$ by construction, and \n\\[\n \\abs*{G_i(x_{h+1}) - G_i(x_h)} = \\abs*{\\int_{x_h}^{x_{h+1}} \\frac{1}{H(z)}\\, dH_i} \\leq \\frac{1}{\\gamma} \\cdot (\\delta + 2\\epsilon_1),\n\\]\nsince the $x_i$ are at least as close as the quantiles of $H_i$, while\n$\\frac{1}{H(z)} \\leq \\frac{1}{\\gamma}$ by assumption. Using this inequality and\nthe monotonicity of $G_i$ yields: \n\\begin{align*}\n G_i(x) \\geq G_i(x_h) \n &\\geq \\hat{G}_i(x_{h+1}) - \\frac{1}{\\gamma} (\\delta + 2\\epsilon_1) - \\frac{2|X|\\beta \\gamma + \\delta + 2 \\epsilon_1 + \\beta}{\\gamma^2} \\\\\n &\\geq \\hat{G}_i(x_{h+1}) - \\frac{(2|X| \\gamma + 1)\\beta + (\\gamma + 1)(\\delta + 2\\epsilon_1)}{\\gamma^2} \\\\\n &\\geq \\hat{G}_i(x) - \\frac{4(k+1) \\gamma \\beta\/\\delta + 2\\delta + 4\\epsilon_1}{\\gamma^2} \\\\\n \\text{Analogously, } G_i(x) &\\leq \\hat{G}_i(x) + \\frac{4(k+1) \\gamma \\beta \/ \\delta + 2\\delta + 4\\epsilon_1}{\\gamma^2}\n\\end{align*}\nNow, set:\n\\(\n \\delta = \\frac{\\gamma^2\\epsilon}{6},\n \\epsilon_1 = \\frac{\\gamma^2 \\epsilon}{24},\n\\) and \\(\n \\beta = \\frac{\\gamma\\epsilon^2}{24(k+1)},\n\\)\nso that\n\\(\n |\\hat{G}_i(x) - G_i(x)| \\leq \\frac{\\epsilon}{2}\n\\)\nwith probability \n\\begin{align*}\n &1 - \\frac{12}{\\gamma^2\\epsilon}\\log\\lr{\\frac{48L}{\\gamma^2 \\epsilon}}\\exp \\lbrb{- \\frac{\\gamma^4\\epsilon^2}{1152} n}\n - \\frac{12k}{\\gamma^2\\epsilon}\\log\\lr{\\frac{48L}{\\gamma^2 \\epsilon}}\\exp \\lbrb{- \\frac{\\gamma^4\\epsilon^2}{4608} n}\n - \\frac{24k}{\\gamma^2\\epsilon} \\exp\\lbrb{-\\frac{\\gamma^2 \\epsilon^4}{288(k+1)^2} n}.\n\\end{align*}\nThus, setting\n$$n = \\frac{4608(k+1)^2}{\\gamma^4\\epsilon^4}\\log\\lr{\\frac{3}{\\alpha}\n\\frac{24k}{\\gamma^2\\epsilon} \\log\\lr{\\frac{48L}{\\gamma^2 \\epsilon}}}$$\nmakes this probability $1 - \\alpha$. Now, the total number of samples required\nfor this approach is $O(N \\cdot k \\cdot T \\cdot n)$, concluding the proof.\n\\end{proof}\n\\begin{remark}[Inserting bids may change equilibria]\n As we highlighted above, using access to the partial observation data we can\n estimate the distributions $H_i$ to within $\\epsilon$ error and thus apply a\n similar algorithm to the ordinary first-price setting. \n Observe, however, that by inserting arbitrary bids to get good\n estimates of the functions $F_i$, the econometrician can affect the bidding\n strategy of the agents and thus interfere with the equilibrium point \n of the first-price auction. \n This is not true for our model in\n \\cref{def:max_sel_obs}, where the econometrician is a passive observer (in\n particular, observations do not interfere with the \n equilibrium of the agents) and hence the bid distributions can lead\n to an estimation of the value distributions as well (as we show in\n \\cref{sec:value_estimation}). \n\\end{remark}\n\n\n\n\\subsection{Fixed Point Definition}\n\\label{ssec:fp_def}\n\nHere, we formally define the version of the fixed point iteration used in our algorithm. \nRecall that the CDFs of the bid generating distributions, $F_i$, satisfy:\n\\begin{equation*}\n \\forall i \\in [k]: U^*_i (x) \\coloneqq \\prod_{j \\neq i} F_j (x) = \\int_{0}^x \\frac{1}{1 - F_i (z)} \\cdot g_i (z) dz. \n\\end{equation*}\nRecasting the above equation in terms of the functions, $U^*_i$, we obtain:\n\\begin{equation}\n \\forall i \\in [k]: U^*_i (x) \\coloneqq \\prod_{j \\neq i} F_j (x) = \\int_{0}^x \\frac{g_i (z)}{1 - H_i (z)} dz \\text{ with } H_i (z) = \\frac{\\prod_{j \\neq i} (U^*_j (z))^{1 \/ (k - 1)}}{(U^*_i (z))^{(k - 2) \/ (k - 1)}}.\n \\tag{FP-CONT}\n \\label{eq:fixed_point}\n\\end{equation}\nIn our algorithm, we approximate the functions, $U^*_i$, by piecewise constant functions on intervals of width $\\delta$ (a parameter which we set later, in \\ref{eq:par_set}) and approximate solutions to the above fixed-point equation. We will prove that the approximation errors as well as errors due to computational and statistical constraints remain small despite these choices.\n\nIn the finite sample setting, we do not have access to the precise population functions, $G_i$ and hence, approximate them with their empirical counterparts:\n\\begin{equation*}\n \\forall x \\in [0, 1]: \\hat{G}_i (x) \\coloneqq \\frac{1}{n} \\cdot \\sum_{j = 1}^n \\bm{1} \\lbrb{W_j = i, Y_j \\leq x}\n\\end{equation*}\n\nBefore running our main algorithm, we first compute (very) coarse approximations\n$\\smash{\\hat{U}_i(\\cdot)}$ to the functions $\\smash{U_i^*(\\cdot)}$; it turns out\n(see \\cref{alg:est_u} and \\cref{lem:u_appx})\nthat we can compute such loose approximations efficiently as a simple\nconsequence of the DKW inequality \\citep{dkw}.\n\nOur algorithm then operates in stages: we divide the interval $[\\nu, 1 - \\theta]$\n(we set $\\nu$ and $\\theta$ later, in \\eqref{eq:par_set}) into \na finite number of ``macro-intervals'', each of which\ncontains a number of micro-intervals of width $\\delta$. The macro-intervals are\ndefined by endpoints $\\nu \\coloneqq x_{0} < x_{1} \\dots < x_{T} \\leq 1 - \\theta \/ 2$, where $T$ and the width of each macro-interval are chosen dynamically based on observed data. Within each macro-interval $[x_{\\tau - 1}, x_{\\tau}]$ are $\\ell^{(\\tau)}$ micro-intervals of width $\\delta$, defined by endpoints $x_{\\tau - 1} \\coloneqq x_{\\tau,0} < x_{\\tau,1} < \\ldots < x_{\\tau,\\ell^{(\\tau)}} \\eqqcolon x_{\\tau}$. Picking the endpoints $\\{x_{\\tau}\\}_{\\tau \\in [T]}$ requires balancing contractivity of the fixed-point iteration with the overall number of macro-intervals which determine the accuracy of the final solution. We formally describe our algorithm for computing the endpoints in \\cref{alg:macro_ints}. To ease notation in the remainder of the section, we use $\\Delta^{(\\tau)}_{i, m}$ to denote differentials of $\\hat{G}_i$ across the grid points and extend the definition of $x_{\\tau, \\ell}$:\n\\begin{gather*}\n \\forall i \\in [k], \\tau \\in [T], l \\in [\\ell^{(\\tau)}]: \\Delta^{(\\tau)}_{i, l} \\coloneqq \\hat{G}_i (x_{\\tau, l}) - \\hat{G}_i (x_{\\tau, l - 1}) \\\\\n \\forall \\tau \\in [T], \\ell \\in \\mb{N} \\cup \\{0\\}: x_{\\tau, \\ell} \\coloneqq x_{\\tau - 1} + \\ell \\cdot \\delta.\n\\end{gather*}\n\nThe resulting intervals define a discretization of the fixed-point equation\n\\eqref{eq:fixed_point}. As we previously discussed, our algorithm operates in\nstages: at stage $\\tau$, we assume access to an estimate $\\smash{V_i^{(\\tau)}\n\\approx U^*_i(x_{\\tau,0})}$ from the previous stage \n(recall that $\\smash{x_{\\tau,0} = x_{\\tau-1,\\ell^{(\\tau-1)}}}$ by definition).\nWe instantiate $\\smash{U^{(\\tau)} \\in \\mathbb{R}^{k \\times \\ell^{(\\tau)}}}$ \nwhere $\\smash{U^{(\\tau)}_{i, l}}$ is designed to approximate $U^*_i (x_{\\tau,\nl})$.\nWe estimate the elements of $\\smash{U^{(\\tau)}}$ via the following discretized\nfixed-point iteration: \n\\begin{align*}\n \\phi^{(\\tau)}_{i, l} (U^{(\\tau)}) \n &= \\mathrm{clip} \\lprp{\n \\sum_{m = 1}^l \\frac{1}{1 - H^{(\\tau)}_{i, m}(U^{(\\tau)})} \\cdot \\Delta^{(\\tau)}_{i, m} + V^{(\\tau)}_{i}\n ,\\frac{1}{2\\eta} \\cdot \\hat{U}_i (x_{\\tau, l}), \\frac{2}{\\alpha} \\cdot \\hat{U}_i (x_{\\tau, l})} \\\\\n H^{(\\tau)}_{i, m} (U^{(\\tau)}) &= \\mathrm{clip} \\lprp{\n \\frac{\\prod_{j \\neq i} (U^{(\\tau)}_{j, m})^{\\frac{1}{(k - 1)}}}{(U^{(\\tau)}_{i, m})^{\\frac{(k - 2)}{(k - 1)}}}\n ,\\alpha x_{\\tau, m}, \\min \\lprp{1 - \\alpha (1 - x_{\\tau, m}), \\eta x_{\\tau, m}}}. \\tag{FP} \\label{eq:fp_fin_def}\n\\end{align*}\nIn \\eqref{eq:fp_fin_def}, $\\hat{U}_i(\\cdot)$ are the aforementioned coarse\napproximations to $U_i^*(\\cdot)$, and $V_i^{(\\tau)}$ is the estimate of $U^*_i\n(x_{\\tau - 1})$ obtained by running $L$ iterations of the fixed point\niteration $\\phi^{(\\tau - 1)}$ on the previous macro-interval (where $L$ is set in\n\\eqref{eq:par_set}). We initialize the algorithm with\n\\[\n V_i^{(0)} \\coloneqq \\hat{G}_i (\\nu).\n\\]\n\nWe present a summary of the entire algorithm corresponding to\n\\cref{thm:sec_price_fin_samp} in \\cref{alg:sp_estimation}, and spend the\nremainder of this section proving that this algorithm yields an\n$\\eps$-approximation to $U^*_i(\\cdot)$ for all $i \\in [k]$: \nThe precise choices of our parameters are provided below:\n\\begin{gather*}\n \\theta \\coloneqq \\frac{\\eps}{16 \\eta},\n \\quad \\delta \\coloneqq \\lprp{\\frac{\\alpha}{8\\eta} \\nu}^{32k}, \n \\quad \\eps_g \\coloneqq \\delta \\cdot \\lprp{\\frac{\\alpha \\nu}{8 \\eta}}^{32k}, \n \\quad L \\coloneqq \\log (4 \/ \\eps_g) \\\\\n \\nu \\coloneqq \\min \n \\lbrb{\\lprp{\\frac{\\alpha}{2\\eta}}^{256}, \\exp \\lprp{- 2^{32} k \\lprp{\\frac{\\eta}{\\alpha}} \\log \\lprp{\\frac{2\\eta}{\\alpha}}}, \\lprp{\\frac{\\theta}{2}}^{\\lprp{\\frac{4\\eta}{\\alpha}}^{16}}, \\lprp{\\frac{\\alpha \\eps}{32 \\eta}}^{24}\n } \\tag{PAR} \\label{eq:par_set}\n\\end{gather*}\nThe parameter $\\theta$ is chosen to be suitable small such that $F_i (x)$ is well approximated by $0$ when $x \\leq \\theta$ and $1$ when $x \\geq 1 - \\theta$. Hence, we only need to obtain good estimates in the range $[\\theta, 1 - \\theta]$. However, as shown in \\cref{sssec:err_prop}, our estimation errors grow exponentially in the number of macro-intervals, $T$ but drop rapidly as our start point decreases (\\cref{lem:err_prop}). Consequently, $\\nu$ is picked to be significantly smaller than $\\theta$ to ensure our starting error is small. The parameters $\\delta$ and $\\eps_g$ are then chosen small with respect to $\\nu$ to ensure the approximation errors caused by the discretization of our fixed point iteration and statistical errors from approximating the functions $G_i(\\cdot)$ with $\\hat{G}_i$ remain small. Finally, the parameter which determines how many times we iterate our fixed point equation, $L$, is picked to control errors due to computational limits.\n\n\\begin{algorithm}[h]\n \\caption{Our algorithm for estimating value distributions in second-price auctions.}\n \\label{alg:sp_estimation}\n \\begin{algorithmic}[1]\n \\Procedure{Estimate-Second-Price}{$\\{(Y_j, Z_j)\\}_{j=1}^n, \\delta, \\alpha, \\eta, \\nu, \\theta$}\n \\State $\\{x_{\\tau, m}: \\tau \\in [T], m \\in [\\ell^{(\\tau)}]\\} \\gets $\n \\Call{Construct-Macro-Intervals}{$\\{(Y_j, Z_j)\\}_{j=1}^n, \\delta, \\alpha,\n \\eta, \\nu, \\theta$}\n \\For{$i \\in \\{1,\\ldots,k\\}$}\n \\State $V_i^{(0)} \\coloneqq \\hat{G}_i (\\nu)$\n \\State $\\hat{U}_i(\\cdot) \\gets $ \\Call{Coarse-Approximate-$U$}{\n $\\{(Y_j, Z_j)\\}_{j=1}^n$, $i$} \\Comment{See \\cref{lem:u_appx}}\n \\State $U_i(x) \\gets \\hat{G}_i (\\nu) \\cdot \\bm{1} \\lbrb{x \\leq \\nu}$ \\Comment{Estimate of $U^*_i$}\n \\EndFor\n \\For{$\\tau \\in \\{1,\\ldots T\\}$}\n \\State Initialize $U^{(\\tau)} \\in \\mathbb{R}^{k \\times \\ell^{(\\tau)}}$\n such that $U^{(\\tau)}_{i,m} \\gets \\hat{U}_i(x_{\\tau,m})$\n \\Comment{Initialize with coarse approx.}\n \\State Define $\\phi^{(\\tau)}: \\mathbb{R}^{k \\times \\ell^{(\\tau)}} \\to\n \\mathbb{R}^{k \\times \\ell^{(\\tau)}}$ as in \\eqref{eq:fp_fin_def}\n \\State \\textbf{repeat $L$ times}: $U^{(\\tau)} \\gets \\phi^{(\\tau)}(U^{(\\tau)})$\n \\State $U_i(x) \\gets U_i(x) + \\sum_{m=1}^{\\ell^{(\\tau)}} U^{(\\tau)}_{i,m} \\cdot\n \\bm{1}\\lbrb{x \\in [x_{\\tau,m-1}, x_{\\tau,m})}$\n \\Comment{Extend our estimate}\n \\State $V^{(\\tau+1)}_{i} \\gets U^{(\\tau)}_{i,\\ell^{(\\tau)}}$\n \\EndFor\n \\State \\Return $\\{U_i\\}_{i=1}^k$\n \\EndProcedure\n \\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsection{Proof of \\cref{thm:sec_price_fin_samp}}\n\\label{ssec:sec_price_proof}\nIn this subsection, we prove \\cref{thm:sec_price_fin_samp}. The sole\nprobabilistic condition we require is the empirical concentration of the\nfunctions $\\hat{G}_i$ around $G_i$ (defined in \\eqref{eq:sp_gi_def}), \nwhich we establish below:\n\\begin{lemma}\n \\label{lem:epsg_lem}\n We have, for $\\eps_g$ as defined in \\eqref{eq:par_set},\n as long as $n \\geq \\log (2 k \/ \\rho) \/ (2\\eps_g^2)$,\n \\begin{equation*}\n \\mb{P} \\lbrb{\\forall i \\in [k]: \\norm{\\hat{G}_i - G_i}_\\infty \\leq \\eps_g} \\geq 1 - \\rho.\n \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n For each $i \\in [k]$ and $j \\in [n]$, define the random variables, \n $\\smash{A^i_j \\coloneqq Y_j \\cdot \\bm{1} \\{Z_j = i\\} + \\bm{1} \\{Z_j \\neq i\\}}$. \n The CDFs of the $A^i_j$ variables correspond to $G_i$, while their empirical\n CDFs correspond to $\\hat{G}_i$. The lemma thus follows from the DKW\n inequality \\citep{dkw} and a union bound over $i$. \n\\end{proof}\n\n\\noindent For the the rest of the proof, we condition on the result of\n\\cref{lem:epsg_lem} and proceed as follows:\n\\begin{itemize}\n \\item In \\cref{sssec:init_est_sp}, we show the functions $\\hat{U}_i$ in the\n definition of $\\phi^{(\\tau)}$ \\eqref{eq:fp_fin_def} (i.e., the coarse\n approximations of $U^*_i$) may be efficiently estimated from data.\n \\item In \\cref{ssec:macro_ints}, we show how to construct the\n aforementioned data-driven micro- and macro-intervals using the functions\n $\\hat{U}_i$.\n \\item Subsequently, in \\cref{sssec:contr}, we analyze the contractivity\n properties of $\\phi^{(\\tau)}$ over each macro-interval, allowing application\n of \\cref{thm:banach}. \n \\item In \\cref{sssec:err_prop}, we show how errors incurred in early stages\n of the procedure are exponentially compounded for each new\n \\emph{macro-interval} requiring careful control over the number of such\n intervals, $T$. \n \\item Finally, we bound $T$ and prove \\cref{thm:sec_price_fin_samp} in\n \\cref{sssec:iter_count}. \n\\end{itemize}\n\n\\subsubsection{Approximate Estimation}\n\\label{sssec:init_est_sp}\n\\begin{algorithm}[h]\n \\caption{Constructing macro- and micro-intervals for the fixed-point iteration.}\n \\label{alg:est_u}\n \\begin{algorithmic}[1]\n \\Procedure{Coarse-Approximate-$U$}{$\\{(Y_j, Z_j)\\}_{j=1}^n, i$}\n \\State \\Return $x \\mapsto \\frac{1}{n} \\sum_{j = 1}^n \\frac{1}{1 - Y_j} \\cdot \\bm{1} \\lbrb{Z_j = i, Y_j \\leq x}$\n \\EndProcedure\n \\end{algorithmic}\n\\end{algorithm}\n\nHere we describe the construction of $\\hat{U}_i$ used in \\eqref{eq:fp_fin_def}, ensuring the truncation range contains the true parameter values; i.e. we establish the following lemma:\n\\begin{lemma}\n \\label{lem:u_appx}\n Let $\\hat{U}_i: [0, 1 - \\theta \/ 4] \\to \\mb{R}$ be monotonic functions defined as follows:\n \\begin{equation*}\n \\hat{U}_i (x) \\coloneqq \\frac{1}{n} \\sum_{j = 1}^n \\frac{1}{1 - Y_j} \\cdot \\bm{1} \\lbrb{W_j = i, Y_j \\leq x}.\n \\end{equation*}\n Then, for all $x, y \\in [0, 1 - (\\theta \/ 4)]$ such that $\\max \\lbrb{U^*_i(x) - U^*_i (y), \\hat{U}_i (x) - \\hat{U}_i (y)} \\geq \\lprp{\\frac{\\alpha \\nu}{8\\eta}}^{32 k}$, we have:\n \\[\n \\frac{\\alpha}{2} (U^*_i (x) - U^*_i (y)) \\leq \\hat{U}_i (x) - \\hat{U}_i (y) \\leq 2 \\eta (U^*_i (x) - U^*_i (y)).\n \\]\n\\end{lemma}\n\\begin{proof}\n Fix $x, y$ satisfying the required constraints.\n We have:\n \\begin{align*}\n \\E [\\hat{U}^i(x) - \\hat{U}^i(y)] \n &= \\E \\left[\\frac{1}{1 - Y} \\cdot \\bm{1} \\lbrb{W = i, y < Y \\leq x}\\right] \\\\\n &= \\int_{x}^y \\frac{1}{(1 - z)} \\cdot (1 - F_i (z)) \\sum_{j \\neq i} f_j (z) \\prod_{k \\neq j, i} F_k (z) dz,\n \\end{align*}\n and hence (since $\\alpha z \\leq F_i(z) \\leq \\eta z$), we get:\n \\begin{equation*}\n \\alpha (U^*_i (x) - U^*_i (y)) \\leq \n \\E [\\hat{U}^i(x) - \\hat{U}^i(y)] \n \\leq \\eta (U^*_i (x) - U^*_i (y)).\n \\end{equation*}\n Now, we will show that the estimate $U^i$ can be uniformly estimated for all $i, x, y$. For empirical analysis, we have by the integration by parts formula:\n \\begin{align*}\n \\hat{U}_i(x) - \\hat{U}_i(y) &\\coloneqq \\frac{1}{n} \\cdot \\sum_{j = 1}^n \\frac{1}{1 - Y_j} \\cdot \\bm{1} \\lbrb{W_j = i, y < Y_j \\leq x} \n \\\\\n &= \\lprp{\\frac{\\hat{G}_i (y)}{1 - y} - \\frac{\\hat{G}_i (x)}{1 - x}} - \\int_{y}^x \\frac{1}{(1 - z)^2} \\hat{G}_i (z) dz.\n \\end{align*}\n Similarly, we have for the population counterparts:\n \\begin{equation*}\n \\E [\\hat{U}^i(x) - \\hat{U}^i(y)] \n = \\lprp{\\frac{G_i (y)}{1 - y} - \\frac{G_i (x)}{1 - x}} - \\int_{y}^x \\frac{1}{(1 - z)^2} \\cdot G_i (z) dz.\n \\end{equation*}\n From the previous two displays (and conditioning on \\cref{lem:epsg_lem}), we get:\n \\begin{align*}\n \\abs{\\hat{U}_i (x) - \\hat{U}_i (y) - \\E [\\hat{U}^i(x) - \\hat{U}_i(y)]} \n &\\leq 2 \\cdot \\frac{\\eps_g}{\\theta\/4} + \n \\int_{y}^x \\frac{\\eps_g}{(\\theta\/4)^2} dz \\\\\n &\\leq (\\theta\/2) \\cdot \\frac{16\\eps_g}{\\theta^2} + \n (x - y) \\frac{16 \\eps_g}{\\theta^2} \\leq \\frac{32\\eps_g}{\\theta^2}.\n \\end{align*}\n Hence, we get:\n \\begin{equation*}\n \\alpha (U^*_i (x) - U^*_i (y)) - \\frac{32 \\eps_g}{\\theta^2} \\leq \\hat{U}_i (x) - \\hat{U}_i (y) \\leq \\eta (U^*_i (x) - U^*_i (y)) + \\frac{32 \\eps_g}{\\theta^2}\n \\end{equation*}\n By \\ref{eq:par_set}, this concludes the proof when $U^*_i (x) - U^*_i (y) \\geq \\lprp{\\nicefrac{\\alpha \\nu}{8 \\eta}}^{32k}$. In the alternative case (i.e when $\\hat{U}_i (x) - \\hat{U}_i (y) \\geq \\lprp{\\nicefrac{\\alpha \\nu}{8 \\eta}}^{32k}$), we again get from the above equation:\n \\begin{gather*}\n \\alpha (U^*_i (x) - U^*_i (y)) \\leq \\hat{U}_i (x) - \\hat{U}_i (y) + \\frac{32 \\eps_g}{\\theta^2} \\leq 2 \\cdot \\lprp{\\hat{U}_i (x) - \\hat{U}_i (y)} \\\\\n \\eta (U^*_i (x) - U^*_i (y)) \\geq \\hat{U}_i (x) - \\hat{U}_i (y) - \\frac{32 \\eps_g}{\\theta^2} \\geq \\frac{1}{2} \\cdot \\lprp{\\hat{U}_i (x) - \\hat{U}_i (y)}\n \\end{gather*}\n completing the proof in this case as well.\n\\end{proof}\n\n\\subsubsection{Defining Macro- and Micro-Intervals}\n\\label{ssec:macro_ints}\nWe now describe our procedure for splitting up our estimation interval $[\\nu,\n1-\\theta]$ into macro- and micro-intervals. Recall that the micro-intervals (of\nfixed length $\\delta$) define the intervals on which our approximation will be\npiecewise-constant, while the macro-intervals (consisting of collections of\nmicro-intervals whose size is dynamically determined by data) define which\ngroups of micro-intervals are estimated simultaneously in each stage (i.e., by\nthe same fixed-point iteration \\eqref{eq:fp_fin_def}).\n\nWe construct these macro- and micro-intervals according to a few different\nconsiderations: \n\\begin{itemize}\n \\item The length of the micro-intervals $\\delta$ must be small enough to\n bound the approximation error of our estimates (we set $\\delta$ in\n \\eqref{eq:par_set});\n \\item We set the length of the macro-intervals (dynamically) to ensure that\n the discretized version of fixed-point iteration remains contractive\n \\item At the same time, we must ensure that the total number of macro-intervals,\n $T$, does not grow too rapidly, as estimation errors incurred in earlier\n stages of the algorithm are exponentially amplified in later stages. \n\\end{itemize}\n\nWe construct our micro- and macro-intervals in a recursive fashion, detailed in \\cref{alg:macro_ints}. The algorithm assumes access to the coarse approximations $\\hat{U}_i$ to $U^*_i$ for any $i \\in [k]$---we established access to such approximations in the previous section (\\cref{lem:u_appx}). In particular, starting from $\\nu$ as the first endpoint, we set the macro-interval length based on a worst-case estimate of the resulting fixed-point contractivity, obtained using the coarse approximations. This quantity which plays a crucial role in our algorithm design and subsequent analysis is defined below:\n\n\\begin{equation*}\n \\forall \\tau \\in [T], \\ell \\in \\mathbb{N} \\text{ s.t } x_{\\tau, \\ell} \\leq 1 - \\frac{\\theta}{2}: \\gamma^{(\\tau)}_\\ell \\coloneqq \\max_{i \\in [k]} 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\sum_{m = 1}^{\\ell} \\frac{x_{\\tau, m}}{(1 - x_{\\tau, m})^2} \\cdot \\Delta^{(\\tau)}_{i, m}.\n\\end{equation*}\n\n\\begin{algorithm}[H]\n \\caption{Constructing macro- and micro-intervals for the fixed-point iteration.}\n \\label{alg:macro_ints}\n \\begin{algorithmic}[1]\n \\Procedure{Construct-Macro-Intervals}{$\\{(Y_j, Z_j)\\}_{j=1}^n, \\delta,\n \\alpha, \\eta, \\nu, \\theta$}\n \\State $x_{0} \\gets \\nu,\\ \\tau \\gets 1$\n \\State Let $\\hat{U}_i(\\cdot) \\gets $ \\Call{Coarse-Approximate-$U$}{\n $\\{(Y_j, Z_j)\\}_{j=1}^n$, $i$} for all $i \\in [k]$ \\Comment{See \\cref{lem:u_appx}}\n \\While{$x_{\\tau - 1} < 1 - \\theta$}\n \\State Number of micro-intervals in this macro-interval:\n \\State $\\ell^{(\\tau)} \\gets \\max \\lbrb{\\ell \\in \\mb{N}: x_{\\tau,\n \\ell} \\leq \\min(2 \\cdot x_{\\tau - 1}, 1 - \\theta \/ 2) \\text{ and\n } \\gamma^{(\\tau)}_\\ell \\leq 1\/4}.$\n \\State $x_{\\tau} \\gets x_{\\tau, \\ell^{(\\tau)}}, \\tau \\gets \\tau + 1$\n \\EndWhile\n \\State $T \\gets \\tau - 1$\n \\State \\Return $\\{x_{t}\\}_{t \\in [T]}$\n \\EndProcedure\n \\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection{Contractivity Analysis}\n\\label{sssec:contr}\n\nWe now establish contractivity of the mappings, $\\phi^{(\\tau)}$ in\n\\cref{lem:contr}, allowing us to apply Banach's fixed point theorem\n(\\cref{thm:banach}). In particular, we establish contractivity in the\ninfinity-norm; i.e, for some $\\rho < 1$: \n\\begin{equation*}\n \\norm*{\\phi^{(\\tau)} (U) - \\phi^{(\\tau)} (U')}_\\infty \\leq \\rho \\norm{U - U'}_\\infty \\text{ where } \\norm{M}_\\infty = \\max_{i, j} \\abs{M_{i, j}}.\n\\end{equation*}\nDenoting the Jacobian of $\\phi^{(\\tau)}$ by $J_{\\phi^{(\\tau)}} (\\cdot)$, we consequently bound its $1$-norm defined below:\n\\begin{equation*}\n \\norm*{J_{\\phi^{(\\tau)}} (U^{(\\tau)})}_1 \\coloneqq \\max_{i, l} \\sum_{\\substack{j \\in [k] \\\\ m \\in [\\ell^{(\\tau)}]}} \\lprp{J_{\\phi^{(\\tau)}} (U^{(\\tau)})}_{(i, l), (j, m)} = \\max_{i, l} \\sum_{\\substack{j \\in [k] \\\\ m \\in [\\ell^{(\\tau)}]}} \\abs*{\\frac{\\partial \\phi^{(\\tau)}_{i, l} (U^{(\\tau)})}{\\partial U^{(\\tau)}_{j, m}}}\n\\end{equation*}\n\nBefore proceeding further, we first establish that $\\phi^{(\\tau)}$ is a valid\nfixed-point iteration, in the sense that it maps a compact set $S$ to itself:\n\\begin{lemma}\n\\label{lem:phi_monotone}\nThe mapping $\\phi^{(\\tau)}$ maps the set $S^{(\\tau)}$ defined as follows\nonto itself:\n\\begin{equation*}\n S^{(\\tau)} \\coloneqq \\lbrb{U^{(\\tau)} \\in \\mb{R}^{k \\times \\ell^{(\\tau)}}: \n \\frac{1}{2\\eta} \\cdot \\hat{U}_i (x_{\\tau, l}) \\leq U^{(\\tau)}_{i, l} \n \\leq \\frac{2}{\\alpha} \\cdot \\hat{U}_i (x_{\\tau, l}) \\text{ and } U^{(\\tau)}_{i, l} \\leq U^{(\\tau)}_{i, l + 1}}.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof}\n The first constraint follows from \\ref{eq:fp_fin_def} while the second\n follows from the fact that $\\phi^{\\tau}$ is a clipping of a monotonic\n function onto a monotonically growing range (\\cref{lem:u_appx}). \n\\end{proof}\n\nHaving established the existence of a set $S^{(\\tau)}$ such that $\\phi^{(\\tau)}:\nS^{(\\tau)} \\to S^{(\\tau)}$, we can proceed to bounding the contractivity of the\nfixed-point iteration itself:\n\\begin{lemma}\n \\label{lem:contr}\n We have, for all $U^{(\\tau)} \\in S^{(\\tau)}$,\n \\begin{equation*}\n \\norm*{J_{\\phi^{(\\tau)}} (U^{(\\tau)})}_1 \\leq \\max_{i \\in [k]} 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\sum_{m = 1}^{\\ell^{(\\tau)}} \\frac{x_{\\tau, m}}{(1 - x_{\\tau, m})^2} \\cdot \\Delta^{(\\tau)}_{i, m}.\n \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n First, define the function $\\Gamma^{(\\tau)}_{i,l}$ as \n \\begin{align*}\n \\Gamma_{i,l}^{(\\tau)}(U^{(\\tau)}) \\coloneqq \n \\sum_{m = 0}^l \\frac{1}{1 - H^{(\\tau)}_{i, m}(U^{(\\tau)})} \\cdot \\Delta^{(\\tau)}_{i, m} + V^{(\\tau)}_{i},\n \\end{align*}\n so that $\\phi^{(\\tau)}_{i,l}(U^{(\\tau)}) = \\text{clip}(\n \\Gamma_{i,l}^{(\\tau)},\n \\frac{1}{2\\eta} \\cdot \\hat{U}_i (x_{\\tau, l}), \n \\frac{2}{\\alpha} \\cdot \\hat{U}_i (x_{\\tau, l}) )$.\n Direct calculation shows for any $i, j, l, m$,\n \\begin{align}\n \\label{eq:gamma_deriv_sp}\n \\frac{\\partial}{\\partial U^{(\\tau)}_{j, m}} \\Gamma^{(\\tau)}_{i,l}(U^{(\\tau)}) \n &= \\frac{1}{\\lr{1 - H^{(\\tau)}_{i, m}(U^{(\\tau)})}^2} \\cdot \\Delta^{(\\tau)}_{i, m}\n \\cdot\n \\frac{\\partial H^{(\\tau)}_{i,m}(U^{(\\tau)})}{\\partial U^{(\\tau)}_{j, m}}.\n \\end{align}\n It remains to compute the partial derivative ${\\partial\n H^{(\\tau)}_{i,m}(U^{(\\tau)})} \/ {\\partial U^{(\\tau)}_{j, m}}$, which (from\n \\eqref{eq:fp_fin_def}) corresponds to\n \\begin{align}\n \\label{eq:h_deriv_sp}\n \\frac{\\partial H^{(\\tau)}_{i,m}(U^{(\\tau)})}{\\partial U^{(\\tau)}_{j, m}} \n &= \\lr{\\frac{1}{k-1} - \\bm{1}\\lbrb{i = j}} \\frac{1}{U^{(\\tau)}_{j,m}} H^{(\\tau)}_{i,m}(U^{(\\tau)}).\n \\end{align}\n Now, combining \\eqref{eq:gamma_deriv_sp} and \\eqref{eq:h_deriv_sp} allows us\n to consider a single term in the Jacobian of $\\phi^{(\\tau)}$:\n \\begin{align*}\n \\abs*{\\frac{\\partial \\phi^{(\\tau)}_{i, l} (U^{(\\tau)})}{\\partial U^{(\\tau)}_{j, m}}}\n \\leq \n \\abs*{\\frac{\\partial \\Gamma^{(\\tau)}_{i, l} (U^{(\\tau)})}{\\partial U^{(\\tau)}_{j, m}}}\n \\leq \n \\abs*{\\frac{1}{k-1} - \\bm{1}\\lbrb{i = j}}\n \\cdot\n \\abs*{\\frac{H^{(\\tau)}_{i,m}(U^{(\\tau)})}{\\lr{1 - H^{(\\tau)}_{i, m}(U^{(\\tau)})}^2} \\cdot\n \\Delta^{(\\tau)}_{i, m}\n \\cdot\n \\frac{1}{U^{(\\tau)}_{j,m}} \n }.\n \\end{align*}\n By construction (i.e., by the clipping in \\eqref{eq:fp_fin_def}), \n $H^{(\\tau)}_{i, m}(U^{(\\tau)}) \\leq \\min(\\eta \\cdot x_{\\tau, m}, 1 - \\alpha(1 - x_{\\tau,m}))$, \n and so:\n \\begin{align*}\n \\abs*{\\frac{\\partial \\phi^{(\\tau)}_{i, l} (U^{(\\tau)})}{\\partial U^{(\\tau)}_{j, m}}}\n &\\leq \\frac{\\eta}{\\alpha^2} \\cdot \\abs*{\\frac{1}{k - 1} - \\bm{1} \\lbrb{i = j}} \\cdot \\abs*{\\frac{x_{\\tau, m}}{(1 - x_{\\tau, m})^2} \\cdot \\frac{1}{U^{(\\tau)}_{j, m}} \\cdot \\Delta^{(\\tau)}_{i, m}}.\n \\end{align*}\n Now, note that $\\smash{U^{(\\tau)}_{j, m}}$ is either equal to\n $\\smash{\\hat{U}_j(x_{\\tau, m})}$ (at initialization), or it is the output of\n $\\phi^{(\\tau)}_{j, m}$. As a result (again, by the\n clipping construction in \\eqref{eq:fp_fin_def}), we must have (also using\n our approximation result from \\cref{lem:u_appx})\n \\[\n U^{(\\tau)}_{j, m} \\geq \\frac{1}{2\\eta} \\hat{U}_j(x_{\\tau, m})\n \\geq \\frac{\\alpha}{4\\eta} U^*_j(x_{\\tau, m}).\n \\]\n We will also need the following Lemma relating the functions $U_i^*(\\cdot)$\n across agents:\n \\begin{restatable}{lemma}{uiujcomp}\n \\label{lem:ui_uj_comp}\n Let $U^*_i(\\cdot)$ be as defined in \\eqref{eq:fixed_point}. Then, for\n all $i, j \\in [k]$ and for any $x, y \\in [0, 1]$ with $x > y$, we have\n that:\n \\begin{equation*}\n \\lprp{\\frac{\\alpha}{\\eta}}^3 (U^*_j (x) - U^*_j (y)) \\leq U^*_i (x) - U^*_i (y) \\leq \\lprp{\\frac{\\eta}{\\alpha}}^3 (U^*_j (x) - U^*_j (y)).\n \\end{equation*}\n \\end{restatable}\n \\begin{proof}\n Algebraic manipulation of \\eqref{eq:fixed_point} and\n \\cref{as:second_price_bdd}---full proof in \\cref{ssec:misc_res_sec_price}.\n \\end{proof}\n \\noindent Applying both of these results to the Jacobian entry and applying\n \\cref{lem:u_appx} again,\n \\begin{align*}\n \\abs*{\\frac{\\partial \\phi^{(\\tau)}_{i, l} (U^{(\\tau)})}{\\partial U^{(\\tau)}_{j, m}}}\n &\\leq 4 \\cdot \\frac{\\eta}{\\alpha^2} \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^4 \\cdot \\abs*{\\frac{1}{k - 1} - \\bm{1} \\lbrb{i = j}} \\cdot \\abs*{\\frac{x_{\\tau, m}}{(1 - x_{\\tau, m})^2} \\cdot \\frac{1}{U^*_i (x_{\\tau, m})} \\cdot \\Delta^{(\\tau)}_{i, m}} \\\\\n &\\leq 8 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\abs*{\\frac{1}{k - 1} - \\bm{1} \\lbrb{i = j}} \\cdot \\abs*{\\frac{x_{\\tau, m}}{(1 - x_{\\tau, m})^2} \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\cdot \\Delta^{(\\tau)}_{i, m}}.\n \\end{align*}\n \\noindent Finally, summing the previous equation over all $j, m$ yields:\n \\begin{align*}\n \\norm*{J_{\\phi^{(\\tau)}}}_{1} \\leq \\max_{i \\in [k]} 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\sum_{m = 1}^{\\ell^{(\\tau)}} \\frac{x_{\\tau, m}}{(1 - x_{\\tau, m})^2} \\cdot \\Delta^{(\\tau)}_{i, m}.\n \\end{align*}\n\\end{proof}\nAs a direct consequence of this Lemma, we can show that $\\phi^{(\\tau)}(\\cdot)$\nis $\\nicefrac{1}{4}$-contractive for our particular setting of $\\delta$:\n\\begin{corollary}\n \\label{cor:contractive}\n For any $\\tau \\in [T]$, let $\\phi^{(\\tau)}$ be as defined in\n \\eqref{eq:fp_fin_def} and $\\delta$ as set in \\eqref{eq:par_set}, \n \\[\n \\norm*{J_{\\phi^{(\\tau)}}}_{1} \\leq \\frac{1}{4}.\n \\]\n\\end{corollary} \n\n\\subsubsection{Error Propagation}\n\\label{sssec:err_prop}\nOne consequence of our stagewise estimation procedure is that error incurred\nduring the early stages {\\em compounds} into the later stages. \nIn this subsection, we establish a bound on the rate at which this error\ngrows---in particular, we show that the total error incurred over the course of\nour entire estimation procedure is at most exponential in the number of\n\\emph{macro-intervals}, $T$.\n\nWe first introduce some notation. Let $\\smash{\\wt{U}^{(\\tau)}}$ \nbe estimates resulting from running the fixed-point iteration $\\phi^{(\\tau)}$\n\\eqref{eq:fp_fin_def} $L$ times---the results of the last subsection (in\nparticular, \\cref{cor:contractive}) imply the existence of a unique fixed-point \n$\\smash{\\wt{U}^{(\\tau)}}_{\\text{fixed}}$ for which \n\\begin{equation}\n \\label{eq:fp_error}\n \\norm*{\n \\wt{U}^{(\\tau)} - \\wt{U}^{(\\tau)}_{\\text{fixed}}\n }_{\\infty} \\leq \\frac{1}{4^L}.\n\\end{equation}\nWe will compare the estimates $\\wt{U}^{(\\tau)}$ to the ``ground-truth'' \nvariables $\\overline{U}^{(\\tau)} \\in \\mb{R}^{k \\times \\ell^{(\\tau)}}$, defined \nas:\n\\begin{equation*}\n \\forall \\tau \\in [T], i \\in [k], l \\in \\ell^{(\\tau)}: \\bar{U}^{(\\tau)}_{i, l} \\coloneqq U^*_i (x_{\\tau, l}).\n\\end{equation*}\nWith this notation in hand, we can bound the total error incurred from our\nstage-wise estimation algorithm, which we accomplish via induction:\n\\begin{lemma}\n \\label{lem:err_prop}\n Let $T \\in \\mathbb{N}$ be the number of macro-intervals in our estimation\n algorithm. Then,\n \\begin{equation*}\n \\max_{\\tau \\in [T]}\\ \\norm*{\\wt{U}^{(\\tau)} - \\bar{U}^{(\\tau)}}_\\infty \\leq 2^{T} (2\\eta \\nu)^{k}.\n \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n We will prove the lemma by induction on the number of macro intervals. Concretely, we will prove the following claim via induction on $\\tau$:\n \\begin{equation*}\n \\norm*{\\wt{U}^{(\\tau)} - \\bar{U}^{(\\tau)}} \\leq 2^{\\tau} (2\\eta \\nu)^{k} \\tag{IND} \\label{eq:ind_err}.\n \\end{equation*}\n We first establish the base case:\n \\begin{align*}\n \\abs{\\hat{G}_i (\\nu) - U^*_i (\\nu)} &\\leq \\eps_g + \\abs{G_i (\\nu) - U^*_i (\\nu)} \\\\\n &= \\eps_g + \\int_{0}^\\nu (1 - (1 - F_i(z))) \\cdot \\sum_{j \\neq i} f_j (z) \\prod_{m \\neq i, j} F_m (z) dz \\\\\n &\\leq \\eps_g + \\eta \\nu \\int_{0}^\\nu \\sum_{j \\neq i} f_j (z) \\prod_{m \\neq i, j} F_m (z) dz \\\\ \n &\\leq \\eps_g + \\eta \\nu U^*_i (\\nu) \\leq (2\\eta \\nu)^{k}\n \\end{align*}\n \n \\noindent For the induction step, suppose \\eqref{eq:ind_err} is true for\n all intervals up to $\\tau - 1$. From \\eqref{eq:fp_error}, \n it suffices to bound the error between $U^{(\\tau)}_{\\mrm{fixed}}$ and $\\bar{U}^{(\\tau)}$:\n \\begin{align*}\n \\norm{\\bar{U}^{(\\tau)} - U^{(\\tau)}_{\\mrm{fixed}}}_\\infty &\\leq \\norm{\\bar{U}^{(\\tau)} - \\phi^{(\\tau)}(\\bar{U}^{(\\tau)})}_\\infty + \\norm{\\phi^{(\\tau)}(\\bar{U}^{(\\tau)}) - U^{(\\tau)}_{\\mrm{fixed}}}_\\infty \\\\\n &= \\norm{\\bar{U}^{(\\tau)} - \\phi^{(\\tau)}(\\bar{U}^{(\\tau)})}_\\infty + \\norm{\\phi^{(\\tau)}(\\bar{U}^{(\\tau)}) - \\phi^{(\\tau)} (U^{(\\tau)}_{\\mrm{fixed}})}_\\infty \\\\\n &\\leq \\norm{\\bar{U}^{(\\tau)} - \\phi^{(\\tau)}(\\bar{U}^{(\\tau)})}_\\infty + {\\norm{\\bar{U}^{(\\tau)} - U^{(\\tau)}_{\\mrm{fixed}}}_\\infty}\/{4} \\\\\n \\norm{\\bar{U}^{(\\tau)} - U^{(\\tau)}_{\\mrm{fixed}}}_\\infty \n &\\leq \\frac{4}{3}\\norm{\\bar{U}^{(\\tau)} - \\phi^{(\\tau)}(\\bar{U}^{(\\tau)})}_\\infty \\numberthis \\label{eq:ust_baru_contr_bnd}. \\\\\n \\end{align*}\n For the RHS, we have for fixed $i \\in [k], \\ell \\in [\\ell^{(\\tau)}]$:\n \\begin{align*}\n &\\abs*{\\bar{U}^{(\\tau)}_{i, \\ell} - (\\phi^{(\\tau)} (\\bar{U}^{(\\tau)}))_{i, \\ell}} \n = \\abs*{\\bar{U}_{i, \\ell} - \\sum_{l = 1}^\\ell \\frac{1}{1 - H^{(\\tau)}_{i, l} (\\bar{U}^{(\\tau)})} \\Delta^{(\\tau)}_{i, l} - V_i^{(\\tau)}} \\\\\n &= \\abs*{\\int_{x_{\\tau - 1}}^{x_{\\tau, \\ell}} \\frac{1}{1 - F_i (z)} \\cdot g_i (z) dz + U^*_i (x_{\\tau - 1}) - \\sum_{l = 1}^\\ell \\frac{1}{1 - F_i(x_{\\tau, l})} \\Delta^{(\\tau)}_{i, l} - V_i^{(\\tau)}} \\\\\n &\\leq \\abs*{\\int_{x_{\\tau - 1}}^{x_{\\tau, \\ell}} \\frac{1}{1 - F_i (z)} \\cdot g_i (z) dz - \\sum_{l = 1}^\\ell \\frac{1}{1 - F_i(x_{\\tau, l})} \\Delta^{(\\tau)}_{i, l}} + \\abs*{U^*_i (x_{\\tau - 1}) - V_i^{(\\tau)}} \\\\\n &\\leq \\abs*{\\int_{x_{\\tau - 1}}^{x_{\\tau, \\ell}} \\frac{1}{1 - F_i (z)} \\cdot g_i (z) dz - \\sum_{l = 1}^\\ell \\frac{1}{1 - F_i(x_{\\tau, l})} \\Delta^{(\\tau)}_{i, l}} + \\norm*{\\wt{U}^{(\\tau - 1)} - \\bar{U}^{(\\tau - 1)}}_\\infty. \n \\numberthis \\label{eq:u_phiu_bnd}\n \\end{align*}\n \n \\noindent The second term in the above expression is bounded by our induction hypothesis---for the first term, we have:\n \\begin{align*}\n &\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n \\abs*{\\int_{x_{\\tau - 1}}^{x_{\\tau, \\ell}} \\frac{1}{1 - F_i (z)} \\cdot g_i (z) dz - \\sum_{l = 1}^\\ell \\frac{1}{1 - F_i(x_{\\tau, l})} \\Delta^{(\\tau)}_{i, l}} \\\\\n &= \\abs*{\\sum_{l = 1}^\\ell \\int_{x_{\\tau, l - 1}}^{x_{\\tau, l}} \\frac{1}{1 - F_i (z)} \\cdot g_i (z) dz - \\frac{1}{1 - F_i(x_{\\tau, l})} (\\hat{G}_i (x_{\\tau, l}) - \\hat{G}_i (x_{\\tau, l - 1}))} \\\\\n &\\leq \\abs*{\\sum_{l = 1}^\\ell \\int_{x_{\\tau, l - 1}}^{x_{\\tau, l}} \\lprp{\\frac{1}{1 - F_i (z)} - \\frac{1}{1 - F_i (x_{\\tau, l})}} \\cdot g_i (z) dz} + \\\\\n &\\qquad \\qquad \\abs*{\\sum_{l = 1}^\\ell \\frac{1}{1 - F_i(x_{\\tau, l})} (\\hat{G}_i (x_{\\tau, l}) - \\hat{G}_i (x_{\\tau, l - 1}) - (G_i (x_{\\tau, l}) - G_i (x_{\\tau, l - 1})))} \\\\\n &\\leq \\sum_{l = 1}^\\ell \\int_{x_{\\tau, l - 1}}^{x_{\\tau, l}} \\frac{(F_i (x_{\\tau, l}) - F_i (z))}{(1 - F_i (z))(1 - F_i (x_{\\tau, l}))} \\cdot g_i (z) dz + \n \\frac{8 \\eps_g \\ell^{(\\tau)}}{\\alpha \\theta} \\\\\n &\\leq \\frac{4 \\eta \\delta}{(\\alpha \\theta)^2} \\int_{x_{\\tau, 0}}^{x_{\\tau, \\ell}} g_i (z) dz + \\frac{8\\eps_g \\ell^{(\\tau)}}{\\alpha \\theta} \\\\\n &\\leq \\frac{4 \\eta \\delta}{(\\alpha \\theta)^2} + \\frac{8\\eps_g \\ell^{(\\tau)}}{\\alpha \\theta}.\n \\numberthis \\label{eq:u_phiu_int_bnd}\n \\end{align*}\n \\cref{eq:fp_error,eq:ust_baru_contr_bnd,eq:u_phiu_bnd,eq:u_phiu_int_bnd} conclude the induction step with \\eqref{eq:par_set} and \\eqref{eq:ind_err}.\n \n\\end{proof}\n\n\\subsubsection{Bounding the Number of Macro Intervals}\n\\label{sssec:iter_count}\nWe now turn to the most technical component of our proof of\n\\cref{thm:sec_price_fin_samp}, namely bounding $T$ (the number of\nmacro-intervals in our algorithm). \nIn \\cref{lem:T_bnd}, we employ a potential function argument and track the\ngrowth of $U^*_i$ at the end points of the macro-intervals $\\{ x_{\\tau} \\}_{\\tau\n\\in [T]}$ to argue that the total number of such intervals must be bounded.\n\n\\paragraph{Preliminaries.} \nBefore proving \\cref{lem:T_bnd}, we establish a few\npreliminary results. Our first proves establishes a threshold beyond which the functions $U^*_i$ are lower bounded by a constant.\n\\begin{lemma}\n \\label{lem:u_ub_lem}\n We have:\n \\begin{equation*}\n \\forall i \\in [k], x \\in \\lsrs{1 - \\frac{1}{4 \\eta k}, 1}: U^*_i (x) \\geq \\frac{3}{4}.\n \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n Let $X_i \\overset{iid}{\\thicksim} F_i$. Then, by the union bound:\n \\(\n \\mb{P} \\lbrb{\\exists i: X_i \\geq 1 - \\frac{1}{4\\eta k}} \\leq \\frac{1}{4},\n \\)\n concluding the proof.\n\\end{proof}\n\nOur second result is an upper bound on the densities of the ``densities'' $g_i (\\cdot)$.\n\\begin{lemma}[Bounding the density of $g_i(\\cdot)$]\n \\label{lem:g_bnd}\n Under \\cref{as:second_price_bdd}, we have that for all $i \\in [k]$ and all\n $x \\in [0, 1]$, the density $g_i$ satisfies:\n \\begin{equation*}\n g_i(x) \\leq \\frac{\\eta^2}{\\alpha}.\n \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n Fix $x \\in [0, 1]$ and let $i^* = \\argmax_{i \\in [k]} F_i (x)$. Then, for any $i\n \\in [k]$, we get:\n \\begin{align*}\n g_i(x) &= (1 - F_i (x)) \\sum_{j \\neq i} f_j(x) \\prod_{\\ell \\neq i, j} F_\\ell (x) \\leq \\eta (1 - F_i (x)) \\sum_{j \\neq i} \\prod_{\\ell \\neq i, j} F_\\ell (x) \\\\\n &\\leq \\lprp{\\frac{\\eta^2}{\\alpha}} \\cdot (1 - F_{i^*} (x)) \\sum_{j \\neq i} \\prod_{\\ell \\neq i, j} F_\\ell (x) \\leq \\lprp{\\frac{\\eta^2}{\\alpha}} \\cdot (1 - F_{i^*} (x)) \\sum_{j \\neq i} \\prod_{\\ell \\neq i, j} F_{i^*} (x) \\\\\n &= (k - 1) \\lprp{\\frac{\\eta^2}{\\alpha}} (1 - F_{i^*} (x)) (F_{i^*} (x))^{k - 2} \\leq \\lprp{\\frac{\\eta^2}{\\alpha}} \\cdot \\lprp{1 - \\frac{1}{k - 1}}^{k - 2} \\leq \\frac{\\eta^2}{\\alpha}\n \\end{align*}\n where the first inequality is \\cref{as:second_price_bdd}, and the second is $(k-1)(1-x)x^{k-2} \\leq 1\\ \\forall\\ k \\geq 2$. \n\\end{proof}\n\nOur next three results prove coarse niceness properties on the macro-intervals constructed by \\cref{alg:macro_ints}. The first proves that \\cref{alg:macro_ints} constructs a finite number of intervals and hence, terminates in polynomial time from our settings of $\\delta$ \\eqref{eq:par_set}. \n\\begin{lemma}[Macro-intervals are non-empty]\n Running \\cref{alg:macro_ints} with our\n particular parameters \\eqref{eq:par_set} yields a set of non-empty\n macro-intervals that cover the interval $[\\nu, 1-\\theta]$. That is,\n \\begin{equation*}\n \\ell^{(\\tau)} > 0\\ \\forall\\ \\tau \\in [T] \\qquad \n \\text{ and } \n \\qquad \n x_T \\geq 1 - \\theta.\n \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n First, from \\cref{alg:macro_ints}, we have that $x_{\\tau, 0} < 1 - \\theta$\n (otherwise the algorithm terminates). \n Furthermore, we have $x_{\\tau, 0} + \\delta < 2 x_{\\tau, 0}$, since $x_{\\tau,\n 0} > \\nu > \\delta$ from our definition of $\\delta$ and $\\nu$\n \\eqref{eq:par_set}. Similarly, $x_{\\tau, 0} + \\delta < 1 - \\theta\/2$ since\n $x_{\\tau,0} < 1 - \\theta$ and $\\delta < \\theta\/2$ (again, by\n \\eqref{eq:par_set}). \n \n It only remains to show that $\\gamma^{(\\tau)}(1)$ as\n defined in Line 8 of \\cref{alg:macro_ints} is less than $1\/4$.\n From \\cref{lem:u_appx,lem:g_bnd}, our bounds on $\\eps_g, \\theta, \\delta$\n (\\ref{eq:par_set}) and the fact that $U^*_i (\\nu) \\geq (\\alpha \\nu)^{k - 1}$:\n \\begin{align*}\n \\gamma^{(\\tau)}_1 &\\coloneqq \\max_{i \\in [k]} 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\cdot \\frac{x_{\\tau - 1} + \\delta}{(1 - x_{\\tau - 1} - \\delta)^2} \\cdot \\Delta^{(\\tau)}_{i, m} \\\\\n &\\leq \\max_{i \\in [k]} 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\cdot \\frac{x_{\\tau - 1} + \\delta}{(1 - x_{\\tau - 1} - \\delta)^2} \\cdot (\\hat{G}_i (x_{\\tau - 1} + \\delta) - \\hat{G}_i (x_{\\tau - 1})) \\\\\n &\\leq \\max_{i \\in [k]} 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\cdot \\frac{x_{\\tau - 1} + \\delta}{(1 - x_{\\tau - 1} - \\delta)^2} \\cdot \\lprp{\\frac{\\eta^2 \\delta}{\\alpha} + 2 \\eps_g} \\\\\n &\\leq \\max_{i \\in [k]} 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{2}{\\alpha} \\cdot \\frac{1}{U^*_i (\\nu)} \\cdot \\frac{1 + \\delta}{(\\theta \/ 2 - \\delta)^2} \\cdot \\lprp{\\frac{\\eta^2 \\delta}{\\alpha} + 2 \\eps_g} < \\frac{1}{4}\n \\end{align*}\n establishing the first claim. Note that the previous argument also\n establishes the second claim as if $x_{T} < 1 - \\theta$, a new\n macro-interval exists. \n\\end{proof}\n\nThe next lemma lower bounds the value of our contractivity proxy, $\\gamma^{(\\tau)}_{\\ell^{(\\tau)}}$, for intervals, $[x_{\\tau - 1}, x_\\tau]$ where $x_\\tau$ is not too large either with respect to $x_{\\tau - 1}$ or the upper threshold $1 - \\theta \/ 2$. We will subsequently establish in the proof of \\cref{lem:T_bnd} that most intervals satisfy this condition and such intervals guarantee substantial growth in the potential functions employed in \\cref{lem:T_bnd}.\n\\begin{lemma}\n \\label{lem:gamma_lt_lb}\n Consider the context of our interval-construction algorithm \n (\\cref{alg:macro_ints}). We have:\n \\begin{equation*}\n \\forall \\tau \\in [T] \\text{ s.t } x_{\\tau - 1, \\ell^{(\\tau)} + 1} \\leq \\min(2 x_{\\tau \n - 1}, 1-\\theta\/2): \\gamma^{(\\tau)}_{\\ell^{(\\tau)}} \\geq \\frac{1}{8}.\n \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n We have:\n \\begin{align*}\n \\gamma_{\\ell^{(\\tau)}}^{(\\tau)} &= \\max_{i \\in [k]} 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\sum_{m = 1}^{\\ell^{(\\tau)}} \\frac{x_{\\tau, m}}{(1 - x_{\\tau, m})^2} \\cdot \\Delta^{(\\tau)}_{i, m} \\\\\n &= \\max_{i \\in [k]} 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\lprp{\\sum_{m = 1}^{\\ell^{(\\tau)} + 1} \\frac{x_{\\tau, m}}{(1 - x_{\\tau, m})^2} \\Delta^{(\\tau)}_{i, m} - \\frac{x_{\\tau, \\ell^{(\\tau)} + 1}}{(1 - x_{\\tau, \\ell^{(\\tau)} + 1})^2} \\Delta^{(\\tau)}_{i, \\ell^{(\\tau)} + 1}} \\\\\n &\\geq \\gamma^{(\\tau)}_{\\ell^{(\\tau)} + 1} - 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\cdot \\frac{1 - \\theta \/ 4}{(\\theta \/ 4)^2} \\cdot \\lprp{\\frac{\\eta^2 \\delta}{\\alpha} + 2 \\eps_g} \\\\\n &\\geq \\gamma^{(\\tau)}_{\\ell^{(\\tau)} + 1} - 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{2}{\\alpha} \\cdot \\frac{1}{U^*_i (x_{\\tau - 1})} \\cdot \\frac{1 - \\theta \/ 4}{(\\theta \/ 4)^2} \\cdot \\lprp{\\frac{\\eta^2 \\delta}{\\alpha} + 2 \\eps_g} \\\\\n &\\geq \\gamma^{(\\tau)}_{\\ell^{(\\tau)} + 1} - 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{2}{\\alpha} \\cdot \\frac{1}{U^*_i (\\nu)} \\cdot \\frac{1 - \\theta \/ 4}{(\\theta \/ 4)^2} \\cdot \\lprp{\\frac{\\eta^2 \\delta}{\\alpha} + 2 \\eps_g} \\geq \\frac{1}{8}\n \\end{align*}\n where the first inequality follows from \\cref{lem:g_bnd} and noting $\\delta < \\theta \/ 4$ and $x_T \\leq 1 - \\theta \/ 2$ and the next two from \\cref{lem:u_appx}, our bounds on $\\eps_g, \\delta, \\theta$ (\\ref{eq:par_set}) and observing $U^*_i (\\nu) \\geq (\\alpha \\nu)^{k - 1}$.\n\\end{proof}\n\nThe final preliminary result guarantees that the conclusion of \\cref{lem:u_appx} holds for all intervals $[x_{\\tau - 1}, x_\\tau]$ constructed by \\cref{alg:macro_ints} and will be used to guarantee large potential growth for intervals satisfying the conclusion of \\cref{lem:gamma_lt_lb}.\n\n\\begin{lemma}\n \\label{lem:ust_diff}\n Consider the context of our interval-construction algorithm (\\cref{alg:macro_ints}). We have:\n \\begin{gather*}\n \\forall \\tau \\in [T], i \\in [k]: U^*_i (x_\\tau) - U^*_i (x_{\\tau - 1}) \\geq \\frac{1}{2048} \\cdot \\lprp{\\frac{\\alpha}{\\eta}}^{10} \\cdot \\theta \\cdot \\lprp{\\frac{\\alpha \\nu}{2}}^{k - 1}.\n \\end{gather*}\n\\end{lemma} \n\\begin{proof}\n To start, suppose $\\tau \\in [T]$. We prove the lemma in three cases:\n \n \\paragraph{Case 1:} $x_{\\tau - 1, \\ell^{(\\tau)} + 1} \\leq \\min \\lprp{2 x_{\\tau - 1}, 1 - \\theta \/ 2}$. \\cref{lem:gamma_lt_lb} implies $\\gamma^{(\\tau)}_{\\ell^{(\\tau)}} \\geq 1 \/ 8$. We have for some $i$ from the facts that $x_{\\tau, m} < 1 - \\theta \/ 2$ for all $m \\in [\\ell^{(\\tau)}]$ and $\\hat{U}_{i} (x_{\\tau - 1}) \\geq \\hat{G}_i (\\nu) \\geq (1 - \\eta \\nu) (\\alpha \\nu)^{k - 1} - \\eps_g$ along with our parameter settings \\eqref{eq:par_set}:\n \\vspace{-5pt}\n \\begin{equation*}\n 64 \\lprp{\\frac{\\eta}{\\alpha}}^6 \\frac{1}{(\\alpha \\nu)^{k - 1}} \\frac{1}{\\theta} \\sum_{m = 1}^{\\ell^{(\\tau)}} \\frac{1}{(1 - x_{\\tau, m})} \\Delta^{(\\tau)}_{i, m} \\geq 16 \\lprp{\\frac{\\eta}{\\alpha}}^6 \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\sum_{m = 1}^{\\ell^{(\\tau)}} \\frac{x_{\\tau, m}}{(1 - x_{\\tau, m})^2} \\Delta^{(\\tau)}_{i, m} \\geq \\frac{1}{8}\n \\end{equation*}\n Re-arranging the above and applying \\cref{lem:uh_disc_apx,lem:gamma_lt_lb,lem:ui_uj_comp} yields for all $j \\in [k]$:\n \\begin{align*}\n 2\\lprp{\\frac{\\eta}{\\alpha}}^4 (U^*_j (x_{\\tau}) - U^*_j (x_{\\tau - 1})) &\\geq 2\\eta (U^*_i (x_{\\tau}) - U^*_i (x_{\\tau - 1})) \\\\\n &\\geq \\hat{U}_i (x_{\\tau}) - \\hat{U}_i (x_{\\tau - 1}) \\geq \\frac{1}{1024} \\cdot \\lprp{\\frac{\\alpha}{\\eta}}^6 \\cdot \\theta \\cdot (\\alpha \\nu)^{k - 1}\n \\end{align*}\n proving the claim in this case. \n \n \\paragraph{Case 2:} $x_{\\tau - 1, \\ell^{(\\tau)} + 1} > 2 x_{\\tau - 1}$. As $x_{\\tau - 1} \\geq \\nu$, $x_{\\tau} \\geq \\frac{3}{2} x_{\\tau - 1}$ from our setting of $\\delta$ and the claim follows:\n \\begin{equation*}\n U^*_i (x_{\\tau}) - U^*_i (x_{\\tau - 1}) = \\int_{x_{\\tau - 1}}^{x_{\\tau}} \\sum_{j \\neq i} f_j (z) \\prod_{m \\neq i, j} F_m (z) dz \\geq (k - 1) \\alpha \\int_{0}^{\\frac{\\nu}{2}} (\\alpha z)^{k - 2} dz \\geq \\frac{(\\alpha \\nu)^{k - 1}}{2^{k - 1}}.\n \\end{equation*}\n \n \\paragraph{Case 3:} $x_{\\tau - 1, \\ell^{(\\tau)} + 1} > 1 - \\theta \/ 2$. Note that \\cref{lem:gamma_lt_lb,alg:macro_ints} imply $\\tau = T$, $x_{\\tau - 1} < 1 - \\theta$ and $x_{\\tau} + \\delta \\geq 1 - \\theta \/ 2$. We obtain from our choice of $\\delta$ that $x_\\tau \\geq 1 - (3 \\theta) \/ 4$. Furthermore, since $x_{\\tau - 1} < 1 - \\theta$, we have from \\cref{lem:u_ub_lem} and the constraint on $\\eps$ in \\cref{thm:sec_price_fin_samp}:\n \\begin{align*}\n U^*_i (x_\\tau) - U^*_i (x_{\\tau - 1}) &\\geq U^*_i \\lprp{1 - \\frac{3 \\theta}{4}} - U^*_i (1 - \\theta) = \\int_{1 - \\theta}^{1 - \\frac{3 \\theta}{4}} \\sum_{j \\neq i} f_j (z) \\prod_{m \\neq i, j} F_m (z) dz \\\\\n &\\geq \\alpha \\int_{1 - \\theta}^{1 - \\frac{3 \\theta}{4}} \\sum_{j \\neq i} \\prod_{m \\neq i, j} F_m (z) dz \\geq \\alpha \\int_{1 - \\theta}^{1 - \\frac{3 \\theta}{4}} (k - 1) U^*_i (1 - \\theta) dz \\geq \\frac{\\alpha\\theta}{8}.\n \\end{align*}\n concluding the proof of the lemma.\n\\end{proof}\n\nA key observation behind our proof is that when $x_{\\tau,0}$ is smaller than an \nappropriately chosen constant $c_{\\eta, \\alpha}$, the rate of growth of \n$U^*_i$ is {\\em doubly exponential}, in that \n$$U^*_i (x_{\\tau+1, 0}) \\geq U^*_i (x_{\\tau, 0})^{1 - \\frac{1}{2 (k - 1)}}.$$\nThis rate of growth allows us to bound $T$\nHence, the rate of growth is \\emph{doubly\nexponential} before $c_{\\eta, \\theta}$ allowing the bound on $T \\approx \\log_2\n(1 \/ \\nu) + k \\log \\log (1 \/ \\nu)$ while the initial error scales is at most\n$\\nu^{k}$. With \\cref{lem:err_prop}, a simple post-processing step proves\n\\cref{thm:sec_price_fin_samp}. Auxiliary results needed in the proof are\nincluded in \\cref{ssec:misc_res_sec_price}. \n\n\\begin{lemma}\n \\label{lem:T_bnd}\n We have:\n \\begin{equation*}\n T \\leq 2(k - 1) \\log \\log (1 \/ (\\alpha \\nu)) + \\log_2 (1 \/ \\nu) + 2^{20} \\lprp{\\frac{\\eta}{\\alpha}}^{14} \\lprp{k^2 \\log (2\\eta \/ \\alpha) + k \\log_2 (2 \/ \\theta)}.\n \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n To bound $T$, we break $[\\nu, 1 - \\theta]$ into three segments and handle each separately:\n\\begin{equation*}\n I_1: \\lsrs{\\nu, \\lprp{\\frac{\\alpha}{2\\eta}}^{32}}, \n I_2: \\lsrs{\\lprp{\\frac{\\alpha}{2\\eta}}^{32}, 1 - \\frac{1}{4\\eta k}}, \n \\text{ and } \n I_3: \\lsrs{1 - \\frac{1}{4\\eta k}, 1 - \\frac{\\theta}{2}}. \n\\end{equation*}\n\nBefore we proceed, we prove a simple claim allowing us to restrict ourselves to intervals, $[x_{\\tau - 1}, x_{\\tau}]$ such that $\\gamma^{\\tau}_{\\ell^{(\\tau)}}$ is large ($> 1 \/ 8$).\n\\begin{claim}\n \\label{clm:few_bad_ints}\n We have:\n \\begin{equation*}\n \\abs{S} \\leq \\log_2 (4 \/ \\nu) \\text{ where } S \\coloneqq \\lbrb{\\tau: \\gamma^\\tau_{\\ell^{(\\tau)}} \\geq \\frac{1}{8}}.\n \\end{equation*}\n\\end{claim}\n\\begin{proof}\n Consider $\\tau \\in S$. Then, we have from \\cref{lem:gamma_lt_lb} that either $x_{\\tau} \\geq 1 - \\theta$ ($\\tau = T$ as \\cref{alg:macro_ints} now terminates) or $x_{\\tau, \\ell^{(\\tau)} + 1} \\geq 2 x_{\\tau - 1}$. In the second case, we get:\n \\begin{equation*}\n x_{\\tau, \\ell^{(\\tau)} + 1} = x_{\\tau} + \\delta \\geq 2 x_{\\tau - 1} \\implies x_{\\tau} \\geq \\lprp{2 - \\frac{\\delta}{x_{\\tau - 1}}} \\cdot x_{\\tau - 1} \\geq x_{\\tau} \\geq \\lprp{2 - \\frac{\\delta}{\\nu}} \\cdot x_{\\tau - 1}.\n \\end{equation*}\n Letting $T_S = \\abs{S \\setminus \\{T\\}}$, we get by iterating the above inequality and noting that $x_\\tau$ are monotonic:\n \\begin{equation*}\n 1 \\geq \\lprp{2 - \\frac{\\delta}{\\nu}}^{T_S} \\cdot \\nu\n \\end{equation*}\n and hence, \\ref{eq:par_set} yields:\n \\begin{equation*}\n T_S \\leq \\frac{1}{\\log_2 (2 - \\delta \/ \\nu)} \\cdot \\log_2 (1 \/ \\nu) \\leq \\frac{1}{1 - \\delta \/ \\nu} \\cdot \\log_2 (1 \/ \\nu) \\leq \\lprp{1 + 2 \\cdot \\frac{\\delta}{\\nu}} \\cdot \\log_2 (1 \/ \\nu) \\leq \\log_2 (2 \/ \\nu).\n \\end{equation*}\n Noting that $\\abs{S} \\leq T_S + 1$ concludes the proof of the claim.\n\\end{proof}\n\n\\paragraph{Case 1:} $I_1$. We restrict ourselves to the intervals $[x_{\\tau - 1}, x_\\tau] \\subset I_1$ where $\\gamma^{(\\tau)}_{\\ell^{(\\tau)}} \\geq 1\/8$. From the definition of $\\gamma^{(\\tau)}_{\\ell^{(\\tau)}}$, there exists some $i \\in [k]$ such that: \n\\begin{gather*}\n 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\sum_{m = 1}^{\\ell^{(\\tau)}} \\frac{x_{\\tau, m}}{(1 - x_{\\tau, m})^2} \\cdot \\Delta^{(\\tau)}_{i, m} \\geq \\frac{1}{8} \\\\\n U_i^* (x_{\\tau - 1}) \\geq (\\alpha x_{\\tau - 1})^{k - 1} \\implies x_{\\tau - 1} \\leq \\frac{U_i^*(x_{\\tau - 1})^{1 \/ (k - 1)}}{\\alpha}\n\\end{gather*}\nand as a result, we obtain from \\cref{lem:uh_disc_apx,lem:u_appx,lem:ust_diff}:\n\\begin{align*}\n\\frac{1}{8} &\\leq 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\cdot \\frac{2 x_{\\tau - 1}}{(1 - x_\\tau)} \\sum_{m = 1}^{\\ell^{(\\tau)}} \\frac{1}{(1 - x_{\\tau, m})} \\cdot \\Delta^{(\\tau)}_{i, m} \\\\\n&\\leq 32 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\cdot \\frac{1}{\\alpha} \\cdot \\frac{U^*_i (x_{\\tau - 1})^{1 \/ (k - 1)}}{(1 - x_\\tau)} \\sum_{m = 1}^{\\ell^{(\\tau)}} \\frac{1}{(1 - x_{\\tau, m})} \\cdot \\Delta^{(\\tau)}_{i, m} \\\\\n&\\leq 512 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^8 \\cdot \\frac{1}{U^*_i (x_{\\tau - 1})^{(k - 2) \/ (k - 1)}} \\cdot \\lprp{U^*_i (x_\\tau) - U^*_i (x_{\\tau - 1})}.\n\\end{align*}\n\nRe-arranging the above, two applications of \\cref{lem:ui_uj_comp} with the fact $U^*_j (x) \\leq (\\eta x)^{k - 1}$:\n\\begin{equation*}\n j \\in [k]: U^*_j (x_\\tau) \\geq \\lprp{\\frac{\\alpha}{2\\eta}}^{14} \\cdot U^*_j (x_{\\tau - 1})^{(k - 2) \/ (k - 1)} \\geq U_j^* (x_{\\tau - 1})^{1 - \\frac{1}{2(k - 1)}}.\n\\end{equation*}\nDefining $S_1 \\coloneqq \\{\\tau: [x_{\\tau - 1}, x_\\tau] \\subset I_1 \\text{ and } \\gamma^{(\\tau)}_{\\ell^{(\\tau)}} \\geq 1\/8\\}$, $T_1 \\coloneqq \\abs{S_1}$ and $\\tau_1^* \\coloneqq \\max S_1$, we get by a recursive application of the above inequality for all $j \\in [k]$:\n\\begin{equation*}\ne^{-(k - 1)} \\geq U^*_j (x_{\\tau_1^*}) \\geq (U^*_j (x_0))^{{\\lprp{1 - \\frac{1}{2(k - 1)}}}^{T_1}} \\geq (\\alpha \\nu)^{(k - 1) {\\lprp{1 - \\frac{1}{2(k - 1)}}}^{T_1}}.\n\\end{equation*}\nIteratively taking logs,\n\\begin{equation*}\n \\exp \\lbrb{- \\frac{T_1}{2 (k - 1)}} \\log (\\alpha \\nu) \\geq 1 \\implies T_1 \\leq 2 (k - 1) \\log \\log (1 \/ (\\alpha \\nu)).\n\\end{equation*}\n\n\\paragraph{Case 2:} $I_2$. As before, we restrict to intervals $[x_{\\tau - 1}, x_\\tau] \\subset I_2$ with $\\gamma^{(\\tau)}_{\\ell^{(\\tau)}} \\geq 1 \/ 8$. Noting that $1 - x_{\\tau} \\geq 1 \/ (4\\eta k)$, we have from \\cref{lem:uh_disc_apx,lem:u_appx,lem:ust_diff} for some $i \\in [k]$:\n\\begin{align*}\n\\frac{1}{8} &\\leq 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\cdot \\frac{2 x_{\\tau - 1}}{(1 - x_\\tau)} \\sum_{m = 1}^{\\ell^{(\\tau)}} \\frac{1}{(1 - x_{\\tau, m})} \\cdot \\Delta^{(\\tau)}_{i, m} \\\\\n&\\leq 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\cdot 8 \\eta k \\cdot \\sum_{m = 1}^{\\ell^{(\\tau)}} \\frac{1}{(1 - x_{\\tau, m})} \\cdot \\Delta^{(\\tau)}_{i, m} \\\\\n&\\leq 1024 k \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^8 \\cdot \\frac{1}{U^*_i (x_{\\tau - 1})} \\cdot \\lprp{U^*_i (x_\\tau) - U^*_i (x_{\\tau - 1})}.\n\\end{align*}\nRe-arranging the above inequality and two applications of \\cref{lem:ui_uj_comp} yield:\n\\begin{equation*}\n\\forall j \\in [k]: U^*_j (x_\\tau) - U^*_j (x_{\\tau - 1}) \\geq \\frac{1}{8192 k} \\cdot \\lprp{\\frac{\\alpha}{\\eta}}^{11} \\cdot U^*_i (x_{\\tau - 1}) \\geq \\frac{1}{8192 k} \\cdot \\lprp{\\frac{\\alpha}{\\eta}}^{14} \\cdot U^*_j (x_{\\tau - 1}).\n\\end{equation*}\nDefine $S_2 \\coloneqq \\{\\tau: [x_{\\tau - 1}, x_\\tau] \\subset I_2 \\text{ and } \\gamma_{\\ell^{(\\tau)}}^{(\\tau)} \\geq 1 \/ 8\\}, T_2 \\coloneqq \\abs{S_2}$ and $\\tau^*_2 \\coloneqq \\max S_2$ as before. As $x_{\\tau - 1} \\geq (\\eta \/ 2\\alpha)^{32}$ for all $\\tau \\in S_2$, recursively applying the above inequality yields:\n\\begin{equation*}\n1 \\geq U^*_j (x_{\\tau^*_2}) \\geq \\lprp{1 + \\frac{1}{8192k} \\cdot \\lprp{\\frac{\\alpha}{\\eta}}^{14}}^{T_2} U^*_j \\lprp{\\lprp{\\frac{\\alpha}{2\\eta}}^{32}}.\n\\end{equation*}\nAgain, noting $U^*_j(x) \\geq (\\alpha x)^{k - 1}$ and taking logs, the current case follows:\n\\begin{equation*}\n1 \\geq \\exp \\lbrb{\\frac{T_2}{16384k} \\cdot \\lprp{\\frac{\\alpha}{\\eta}}^{14}} \\cdot \\lprp{\\frac{\\alpha}{2 \\eta}}^{64 (k - 1)} \\implies T_2 \\leq 2^{20} k^2 \\lprp{\\frac{\\eta}{\\alpha}}^{14} \\log (2\\eta \/ \\alpha).\n\\end{equation*}\n\n\\paragraph{Case 3:} $I_3$. We start by subdividing $I_3$ into $r$ subintervals $\\lbrb{I_{3, p} \\coloneqq \\lsrs{1 - 2^{p} \\theta, 1 - 2^{p - 1} \\theta}}_{p = 0}^r$. Note $r \\leq \\log_2 (2 \/ \\theta) + 1$. We now bound the number of intervals in each of these sub-intervals and restrict ourselves to intervals $[x_{\\tau - 1}, x_{\\tau}] \\subset I_{3, p}$ for some $p$ with $\\tau < T$. Note this excludes at most $2(r + 1)$ intervals. For one such interval $[x_{\\tau - 1}, x_\\tau] \\in I_{3, p}$, note that $\\gamma_{\\ell^{(\\tau)}}^{(\\tau)} \\geq 1 \/ 8$ from \\cref{lem:gamma_lt_lb} and the definition of $I_3$. Similarly, we have for some $i$ using \\cref{lem:ust_diff,lem:u_ub_lem,lem:u_appx,lem:uh_disc_apx}:\n\\begin{align*}\n\\frac{1}{8} &\\leq 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\sum_{m = 1}^{\\ell^{(\\tau)}} \\frac{x_{\\tau, m}}{(1 - x_{\\tau, m})^2} \\cdot \\Delta^{(\\tau)}_{i, m} \\\\\n&\\leq 16 \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^6 \\cdot \\frac{1}{\\hat{U}_i (x_{\\tau - 1})} \\sum_{m = 1}^{\\ell^{(\\tau)}} \\frac{1}{(1 - x_{\\tau, m})(2^{p - 1} \\theta)} \\cdot \\Delta^{(\\tau)}_{i, m} \\\\\n&\\leq \\frac{256}{2^{p - 1} \\theta} \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^7 \\cdot \\lprp{U^*_i (x_\\tau) - U^*_i (x_{\\tau - 1})}\n\\end{align*}\nwhich yields:\n\\begin{equation*}\nU^*_i (x_\\tau) - U^*_i (x_{\\tau - 1}) \\geq \\frac{2^{p - 1} \\theta}{2048} \\cdot \\lprp{\\frac{\\alpha}{\\eta}}^7 \\implies \\forall j \\in [k]: U^*_j (x_\\tau) - U^*_j (x_{\\tau - 1}) \\geq \\frac{2^{p - 1} \\theta}{2048} \\cdot \\lprp{\\frac{\\alpha}{\\eta}}^{10}.\n\\end{equation*}\nAgain, defining $S_{3, p} = \\{\\tau < T: [x_{\\tau - 1}, x_\\tau] \\subset I_{3, p} \\}, T_{3, p} \\coloneqq \\abs{S_{3, p}}$, we have from the above:\n\\begin{align*}\nT_{3, p} &\\leq \\frac{2048}{2^{p - 1} \\theta} \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^{10} \\lprp{U^*_i \\lprp{1 - 2^{p - 1} \\theta} - U^*_i \\lprp{1 - 2^{p} \\theta}} \\\\\n&= \\frac{2048}{2^{p - 1} \\theta} \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^{10} \\cdot \\int_{1 - 2^p \\theta}^{1 - 2^{p - 1} \\theta} \\sum_{j \\neq i} f_j (x) \\prod_{q \\neq i,j} F_q(x) dx \\\\\n&\\leq \\frac{2048}{2^{p - 1} \\theta} \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^{10} \\cdot (\\eta k \\cdot 2^{p - 1} \\theta) \\leq 2048 k \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^{11}.\n\\end{align*}\nSumming up over $p$, we get that:\n\\begin{equation*}\n T_3 \\coloneqq \\sum_{p = 0}^r T_{r, p} \\leq 4096 k \\cdot \\lprp{\\frac{\\eta}{\\alpha}}^{11} \\log_2 (2 \/ \\theta).\n\\end{equation*}\nFinally, \\cref{clm:few_bad_ints}, accounting for intervals $[x_{\\tau - 1}, x_\\tau] \\subsetneq I_j$ for all $j \\in [3]$ and the previous three cases establish the lemma as:\n\\begin{equation*}\n T \\leq T_1 + T_2 + T_{3} + 2(r + 1) + \\log_2 (4 \/ \\nu) + 2.\n\\end{equation*}\n\\end{proof}\n\n\\subsubsection{Completing the Proof of \\cref{thm:sec_price_fin_samp}}\nTo complete the proof of \\cref{thm:sec_price_fin_samp}, we have from\n\\cref{lem:err_prop,lem:T_bnd} and our setting of the parameter, $\\nu$, that\n\\begin{equation*}\n \\norm{\\bar{U} - \\wt{U}}_\\infty \\leq 2^T (2 \\eta \\nu)^k \\leq (2 \\eta \\nu)^{k \/ 8} \\leq \\lprp{{\\alpha \\cdot \\eps\/32 \\eta}}^{2k}.\n\\end{equation*}\n\nWe now recover estimates of $F_i$ from estimates of $U^*_i$. Note, when $x \\leq \\theta$, $F_i (x) \\leq \\eps \/ 16$ and hence, $0$ is suitable in this range. Likewise, when $x \\geq 1 - \\theta$, $F_i (x) \\geq 1 - \\eps \/ 16$ and $1$ is correspondingly accurate. For the final case, assume $\\theta \\leq x \\leq 1 - \\theta$. We will first estimate $F_i$ on the grid points, $x_{\\tau, l}$. Suppose now that $x = x_{\\tau, l}$ for some $\\tau, l$. We have:\n\\begin{equation*}\n U^*_i (x) \\geq \\int_{0}^x \\sum_{j \\neq i} f_j (z) \\prod_{m \\neq i, j} F_m (z) dz \\geq (k - 1) \\alpha^{k - 1} \\int_0^x z^{k - 2} dz \\geq \\lprp{\\frac{\\alpha \\eps}{16}}^{k - 1}.\n\\end{equation*}\nAnd, as a consequence, we get:\n\\(\n \\lprp{1 - \\frac{\\eps}{16}} \\leq \\frac{U^*_i(x)}{\\wt{U}^{(\\tau)}_{i, l}} \\leq \\lprp{1 + \\frac{\\eps}{16}}.\n\\)\nDefining our estimate:\n\n\\begin{center}\n\\(\n \\hat{F}_i (x) \\coloneqq \\prod_{j \\neq i} (\\wt{U}^{(\\tau)}_{j, l})^{1 \/ (k - 1)}\n \/\n \\lr{(U^{(\\tau)}_{i, l})^{(k - 2) \/ (k - 1)}}\n\\)\n\\end{center}\n\n\\noindent we get by noting that $F_i (x) = \\frac{\\prod_{j \\neq i} (U^*_j (x))^{1 \/ (k - 1)}}{(U^*_i (x))^{(k - 2) \/ (k - 1)}}$:\n\\begin{align*}\n \\hat{F}_i(x) \\leq F_i (x) \\cdot \\lprp{1 + \\frac{\\eps}{16}} \\cdot \\lprp{1 - \\frac{\\eps}{16}}^{-1} \\leq \\lprp{1 + \\frac{\\eps}{4}} \\cdot F_i (x) \\\\\n \\hat{F}_i(x) \\geq F_i (x) \\cdot \\lprp{1 - \\frac{\\eps}{16}} \\cdot \\lprp{1 + \\frac{\\eps}{16}}^{-1} \\geq \\lprp{1 - \\frac{\\eps}{4}} \\cdot F_i (x).\n\\end{align*}\nFinally, for any $\\theta \\leq x \\leq 1 - \\theta$, there exists $x_{\\tau, l}$ such that $\\abs{x - x_{\\tau, l}} \\leq \\delta$. And we have:\n\\begin{equation*}\n \\frac{\\abs{F_i (x) - \\hat{F}_i (x_{\\tau, l})}}{F_i (x)} \\leq \\frac{\\abs{F_i (x) - F_i (x_{\\tau, l})} + \\abs{F_i (x_{\\tau, l}) - \\hat{F}_i (x_{\\tau, l})}}{F_i(x_{\\tau, l}) - \\abs{F_i(x_{\\tau, l}) - F_i (x)}} \\leq \\frac{\\delta \\eta + (\\eps \/ 4) F_i (x_{\\tau, l})}{F_i (x_{\\tau, l}) - \\delta \\eta} \\leq \\frac{\\eps}{2}\n\\end{equation*}\nfrom our setting of $\\delta$ and $\\theta$. This concludes the proof of the theorem.\n\\qed\n\n\\subsection{Estimation from Partial Observations}\n\\label{sec:sp_bid_insert}\nFinally, we explore a second-price analogue of the ``partial observability'' \nsetting (introduced by \\citet{blum2015learning}) that we studied in the context \nof first-price auctions in Section \\ref{sec:1stPrice:additionalBidder}. \nIn particular, in this setting we observe {\\em the winner} of each auction\nand a binary indicator of whether the reserve price was triggered, but\nnot the price that the winner pays for the auctioned good.\nOn the other hand, as in \\citep{blum2015learning}, \nin this setting the econometrician is given the ability to \nset the reserve price (or equivalently, insert bids into the auction). \nWe formally define the \nsetting below:\n\\begin{definition}[Partial Observation Data -- Second-price]\n \\label{def:sp_partial_obs}\n Let $\\{F_i\\}_{i = 1}^k$ be $k$ cumulative distribution functions with support \n $[0, 1]$, i.e. $F_i(x) = 0\\ \\forall x < 0$ and $F_i(1) = 1$. A sample $(r, Y, Z)$ from a\n first-price auction with bid distributions $\\{F_i\\}_{i = 1}^k$ is generated as\n follows:\n \\begin{enumerate}\n \\item we, the observer, pick a price $r \\in [0, 1]$, and let $X_{k + 1} = r$\n \\item generate $X_i \\ts F_i$ independently for all $i \\in [k]$,\n \\item observe a winner $Z = \\argmax_{i \\in [k+1]} X_i$ and an indicator $Q$\n indicating whether the reserve price $r$ was triggered.\n \\end{enumerate}\n\\end{definition}\n\nWe again operate in the {\\em effective-support setting} (cf. \n\\cref{thm:1stPrice:likely}), and---that is, we\na tuple $p, \\gamma \\in [0, 1]$ such that for all $j \\in [k]$, \n$\\prod_{l \\neq j} F_l(p) \\geq \\gamma$. \nIn other words, the transaction price of\nthe auction will be less than $p$ with probability at least $\\gamma$.\n\nIt turns out that in this seemingly limited observation model, a very\nsimple algorithm suffices for recovering agents' value distributions.\nWe \nbegin with the Lemma demonstrating pointwise recovery of the bid\ndistributions for any $x \\in [p, 1]$:\n\\begin{lemma}\n\\label{lem:sp_reserve_conc}\nFix any $x \\in [p, 1]$ and any $\\epsilon > 0$. Using $n$ samples from the \nwe can obtain as estimate $\\widehat{F}_{j \\in [k]}(x)$ satisfying,\nfor all $j \\in [k]$,\n\\[\n \\abs*{\\widehat{F}_j(x) - F_j(x)} \\leq \\epsilon\n \\quad \n \\text{ with probability at least } 1 - \\delta, \n\\]\nas long as $n \\geq \\frac{48}{\\gamma \\epsilon^2}\\log(2k\/\\delta)$.\n\\end{lemma}\n\\begin{proof}\nFirst, suppose we set the reserve price of the auction to $x$, and define the\nrandom variable $Z_{j}$ as the indicator of whether either (a) \nagent $j$ won the auction {\\em and} the reserve price was triggered; or \n(b) no one won the auction and the reserve price was triggered.\nBy construction (and since (a) and (b) are disjoint),\n\\begin{align*}\n\\mathbb{P}(Z_j = 1) &= \\mathbb{P}(X_j \\geq x, X_{-j} \\leq x)\n + \\mathbb{P}(X_{[k]} \\leq x) \n = (1 - F_j(x)) \\prod_{l \\neq j} F_l(x)\n + \\prod_{l \\in [k]} F_l(x)\n = \\prod_{l \\neq j} F_l(x).\n\\end{align*}\nThus, applying a (multiplicative) Chernoff bound combined with the lower bound \n$\\prod_{l \\neq j} F_l(x) \\geq \\gamma$ given by our effective support assumption,\n\\[\n \\mathbb{P}\n \\lr{\n \\abs*{\\sum_{i=1}^n Z_j^{(i)} - \\prod_{l \\neq j} F_l(x)} \n \\geq \\epsilon \\cdot \\prod_{l \\neq j} F_l(x)\n } \\leq 2\\exp\\lbrb{-\\epsilon^2 \\gamma n\/3}\n\\]\nNow, by our effective support assumption, as long as $k \\geq 2$, \\\\\n\\[\n \\widehat{F}_j(x) \\coloneqq \n \\frac{\\prod_{l \\in [k]}\\lr{\\sum_{i=1}^n Z_j^{(i)}}^{\\frac{1}{k-1}}}\n { \\lr{\\sum_{i=1}^n Z_j^{(i)}} }\n \\leq \\frac{(1 + \\epsilon)^{k\/(k-1)} \\prod_{l \\in [k]} F_l(x)}\n {(1 - \\epsilon) \\prod_{l \\neq j} F_l(x)} \n \\leq (1 + 4\\epsilon) F_j(x) \n\\]\nand an identical argument for the lower bound shows that \n$\\abs{\\widehat{F}_j(x) - F_j(x)} \\leq 4\\epsilon$. Applying a union bound over\nall agents completes the proof.\n\\end{proof}\n\nWe can use this result to construct piecewise-constant approximations of\n$F_j(x)$ that is $\\epsilon$-close to the true bid distributions:\n\\begin{theorem}\n \\label{thm:sp_bid_insert}\n Assume the partially observed second-price setting, and suppose the cumulative\n density functions $F_{[k]}$ are all Lipschitz-continuous with Lipschitz\n constant $L$. For any pair $p, \\gamma \\in [0, 1]$ that define an effective\n support,\n we can find piecewise-constant functions $\\widehat{F}_j(\\cdot)$ satisfying\n \\[\n \\sup_{x \\in [p, 1]} \\abs*{\\widehat{F}_j(x) - F_j(x)} \\leq \\epsilon\n \\quad \n \\text{ with probability at least $1 - \\delta$,}\n \\] \n using \\(\n n = \\Theta\\lr{\\frac{k \\log(k\/\\epsilon) \\log(L\/\\epsilon)^2}{\\epsilon^3\\gamma}}\n \\)\n samples from the partially observed second-price model.\n\\end{theorem}\nGiven \\cref{lem:sp_reserve_conc}, we can use the exact binary search and\nestimation procedure from \\cref{sec:1stPrice:additionalBidder} to\nprove \\cref{thm:sp_bid_insert}---we give the full proof in\n\\cref{app:sp_partial_info_proof}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}