Bidding in Common‐Value Auctions With an Unknown Number of Competitors

Lauermann, Stephan; Speit, Andre Article — Published Version Bidding in Common‐Value Auctions With an Unknown Number of Competitors Econometrica Provided in Cooperation with: John Wiley & Sons Suggested Citation: Lauermann, Stephan; Speit, Andre (2023) : Bidding in Common‐Value Auctions With an Unknown Number of Competitors, Econometrica, ISSN 1468-0262, Wiley, Hoboken, NJ, Vol. 91, Iss. 2, pp. 493-527, https://doi.org/10.3982/ECTA17793 This Version is available at: https://hdl.handle.net/10419/287831 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ Econometrica, Vol. 91, No. 2 (March, 2023), 493–527 BIDDING IN COMMON-VALUE AUCTIONS WITH AN UNKNOWN NUMBER OF COMPETITORS STEPHAN LAUERMANN Department of Economics, University of Bonn ANDRE SPEIT Private Sector This paper studies a first-price common-value auction in which bidders do not know the number of their competitors. In contrast to the case of common-value auctions with a known number of rival bidders, the inference from winning is not monotone, and a “winner’s blessing” emerges at low bids. As a result, bidding strategies may not be strictly increasing, but instead may contain atoms. Moreover, an equilibrium fails to exist when the expected number of competitors is large and the bid space is continuous. Therefore, we consider auctions on a grid. On a fine grid, high-signal bidders follow an essentially strictly increasing strategy, whereas low-signal bidders pool on two adjacent bids on the grid. The solutions of a “communication extension” based on Jackson, Simon, Swinkels, and Zame (2002) capture the equilibrium bidding behavior in the limit, as the grid becomes arbitrarily fine. KEYWORDS: Common-value auctions, random player games, numbers uncertainty, Poisson games, endogenous tie-breaking, nonexistence. 1. INTRODUCTION IN MOST AUCTIONS, bidders are uncertain about the number of competitors they face: •At auction houses such as Christie’s and Sotheby’s, personal attendance is in decline as bidders prefer to phone in or place their bids online. Therefore, bidders “know even less about who they’re bidding against, which in some cases can leave them wondering how high they should go.”1 •eBay reveals the number of bidders who have placed a bid but does not disclose how many prospective bidders are following the auction. In particular, the platform does not display how many bidders are online to “snipe,” that is, to place their bids in the last seconds of the auction (Roth and Ockenfels (2002)). •In the realm of auction-like trading mechanisms, the continuous order book at the New York Stock Exchange informs market participants about the stream of Stephan Lauermann: [email protected] Andre Speit: [email protected] We thank Alp Atakan, Mehmet Ekmekci, Matthew O. Jackson, Philippe Jehiel, Deniz Kattwinkel, Jan Knöpfle, Nenad Kos, Daniel Krähmer, Wolfram Merzyn, Benny Moldovanu, Stephen Morris, Thomas Tröger, Juuso Välimäki, Gábor Virág, Asher Wolinksy, and Charles Zheng for their helpful comments and suggestions. The paper also benefited from the comments of seminar participants at Bonn, NASMES 2018 (UC Davis), the 29th International Conference on Game Theory (Stony Brook), the EARIE Annual Conference 2018 (Athens), and the CETC 2019 (McGill). Funded by the Deutsche Forschungsgemeinschaft (DFG) through CRC TR 224 Project B04 (Lauermann) and Project B05 (Speit) and under Germany’s Excellence Strategy (EXC 2126/1 - 390838866, Lauermann). Open Access funding enabled and organized by Projekt DEAL. The code for the numerical solutions is in the online Supplemental Material. 1The Wall Street Journal, “Why Auction Rooms Seem Empty These Days” (https://www.wsj.com/articles/ with-absentee-bidding-on-the-rise-auction-rooms-seem-empty-these-days-1402683887), June 15, 2014; cf. Akbarpour and Li (2020). © 2023 The Authors. Econometrica published by John Wiley & Sons Ltd on behalf of The Econometric Society. Stephan Lauermann is the corresponding author on this paper. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. 494 S. LAUERMANN AND A. SPEIT (un-)filled buy and sell orders, but reveals neither the number nor the identity of (potential) buyers and sellers. Although uncertainty about the number of competitors, or “numbers uncertainty,” is ubiquitous, the subject has received little attention in the literature on auction theory. One reason may be its irrelevance in standard auction formats with independent private values and risk neutrality: by a revenue-equivalence argument, equilibrium bids are just a weighted average of the bids that are optimal when the number of rival bidders is known; see Krishna (2010, Chapter 3.2.2) and Harstad, Kagel, and Levin (1990). By contrast, in a common-value setting, numbers uncertainty significantly alters bidding behavior. Recall that when the number of rival bidders is known, classic results going back to Milgrom and Weber (1982) establish that there exists a unique symmetric equilibrium in first-price and second-price auctions, in which bids are strictly increasing in the bidders’ own value estimates. Uniqueness and strict monotonicity facilitate revenue comparison between auction formats, simplify welfare considerations (in general interdependent-value settings), and allow for empirical identification of the bidders’ signals. We show that these classic results no longer hold when the number of competitors is uncertain. Equilibria generally are not strictly increasing but contain atoms. The locations of the atoms are often indeterminate, implying equilibrium multiplicity. Moreover, equilibrium payoffs are discontinuous at the atoms, invalidating standard methods for analyzing bidding behavior in these auctions: with a continuous bid space, equilibrium generally fails to exist. To model an auction with numbers uncertainty, we start with a canonical common-value first-price auction. The value of the good is binary (high or low), and bidders receive conditionally independent and identically distributed signals, with higher signals indicating a higher value. All bidders submit their bids simultaneously; the highest bidder wins and pays her bid. Ties are broken uniformly. We only deviate from the textbook setting in assuming that the number of (rival) bidders is not known but Poisson distributed. Numbers uncertainty affects bidding behavior with common values because it changes the value inference from winning. In a conventional common-value auction with a known number of bidders, the expected value conditional on winning is increasing in the relative position of the bid because a higher bid eases the “winner’s curse.” In fact, there is no winner’s curse at the very top bid. This reduction reinforces price competition and implies the absence of pooling, that is, of atoms in the bid distribution. With numbers uncertainty, winning is also informative about the number of rival bidders. In particular, winning with a low bid is more likely when there are fewer competitors, which eases the winner’s curse. Therefore, winning with a low bid is not necessarily bad news about the value of the good. In our model, the inference is U-shaped: intermediate bids are subject to the strongest winner’s curse, while there is no winner’s curse at the bottom or the top (Lemmas 2and 3). We show that every equilibrium is nondecreasing in the bidder’s signal (Lemma 1), but the non-monotone inference implies that equilibria cannot be strictly increasing unless the expected number of competitors is small (Proposition 1). Hence, the equilibrium bid distribution contains one or more atoms, as bidders with different signals pool on common bids. Numbers uncertainty incentivizes bidders to pool because pooling shields them against the winner’s curse: under a uniform tie-breaking rule, a bid that ties with positive probability is more likely to win when there are fewer competitors, which reduces the winner’s curse. The presence of atoms in the bid distribution substantially alters the analysis of the auction. First, the locations of atoms are often indeterminate, so that there may be multiple equilibria. Second, atoms create discontinuities in the bidders’ payoffs, implying that BIDDING IN COMMON-VALUE AUCTIONS 495 no equilibrium exists when the expected number of bidders is sufficiently large (Proposition 2). To overcome the nonexistence and study the bidding incentives, we consider equilibria on a finite but fine grid. We characterize the equilibrium bidding behavior in Proposition 3. Qualitatively, any equilibrium on a fine grid with increments >0 consists of three regions: bidders with high signals essentially follow a strictly increasing strategy (as the grid permits), while bidders with intermediate signals pool on some bid bp,andbidders with low signals bid one increment below it, bp−. The equilibria are shaped by a severe winner’s curse at bp, and a “winner’s blessing” that arises at bids below bp, so that, at these bids, the expected value conditional on winning is significantly higher than at bp. This induces bidders with low signals to compete for the largest bid strictly below bp.On the grid, this competition leads them to pool on bp−; on the continuous bid space, the nonexistence of a largest bid below bpimplies the nonexistence of an equilibrium. We show that bidding on a fine grid can be approximated by the equilibria of a “communication extension” of the auction, based on Lebrun (1996)andJackson et al. (2002). In the communication extension, bidders submit not only a monetary bid from the continuous bid space but also a message that indicates their “eagerness” to win, which is used to break ties. The communication extension is useful because, in contrast to the case of the standard auction, the limit of any converging sequence of equilibria on ever finer grids corresponds to an equilibrium of the communication extension. Proposition 5shows that all equilibria of the communication extension share the qualitative features of the equilibria on a fine grid. In Section 7, we use the communication extension to discuss the implications of numbers uncertainty for the revenue of the seller and the optimal design of auctions, including reserve prices. Moreover, we show that the seller may benefit from running a generalized clock auction with a non-monotone price path that mirrors the non-monotone expected value conditional on winning. Finally, we discuss the assumptions, especially the Poisson distribution, as well as the related literature, including recent contributions by Murto and Välimäki (2019)andLauermann and Wolinsky (2022). 2. MODEL A single, indivisible good is sold in a first-price, sealed-bid auction. The good’s value is either high, vh,orlow,v,withvh>v ≥0, depending on the unknown state of the world ω∈{h}. The state is ω=hwith probability ρand ω=with probability 1 −ρ,where ρ∈(01). The number of bidders is a Poisson-distributed random variable with mean η, so that there are kbidders in the auction with probability e−ηηk k!. The bidders do not observe the realized number of bidders. Every bidder receives a signal sfrom the compact set [s¯ s]. Conditional on the state, the signals are independent and identically distributed according to the cumulative distribution functions Fhand F, respectively.2Both distributions have continuous densities fω, and the likelihood ratio fh(s) f(s)is strictly increasing in s; that is, the signal distribution satisfies the strict monotone likelihood ratio property (MLRP). Furthermore, 0<fh(s) f(s)<fh(¯ s) f(¯ s)<∞, so that no signal reveals the state. Let ˘ sdenote the unique “neutral” signal, the one for which fh(˘ s) f(˘ s)=1. 2This is the “mineral rights” setup (Krishna (2010)). The signal structure excludes general affiliated signals and signals with shifting support. 496 S. LAUERMANN AND A. SPEIT Having received her signal, each bidder submits a bid b. There is a reserve price at v,3and it is without loss to exclude bids above vh, so that b∈[vvh]. The bidder with the highest bid wins the auction, receives the good, and pays her bid. Ties are broken uniformly. If there is no bidder, the good is not allocated. The Poisson distribution has a number of useful properties; a detailed derivation and discussion of Poisson games can be found in Myerson (1998). In particular, when participating in the auction, a bidder does not change her belief regarding the number of other bidders in the auction: this belief is again a Poisson distribution with mean η.4Moreover, Myerson (1998, p. 377) argued that in a Poisson game, attention can be restricted to symmetric equilibria. Accordingly, we consider symmetric strategies, which are measurable functions β: [s¯ s]→[vvh] mapping the signals into the set of probability distributions over bids. Let πω(b;β) denote the probability of winning the auction with a bid bin state ω, if the rival bidders follow strategy β. Using Bayes’s rule, the interim expected utility for a bidder with signal schoosing bid bis U(b|s;β)=ρfh(s) ρfh(s)+(1 −ρ)f(s)πh(b;β)(vh−b) +(1 −ρ)f(s) ρfh(s)+(1 −ρ)f(s)π(b;β)(v−b) Astrategyβ∗is a best response to a strategy βif, for almost all s,abidb∈suppβ∗(s) implies that b∈argmaxˆ b∈[vvh]U(ˆ b|s;β). 3. MONOTONICITY OF BIDDING BEHAVIOR 3.1. Best Responses Are Weakly Increasing In the Appendix, bidders’ payoffs are shown to satisfy the strict single-crossing condition: for any β,bidsb<b , and signals s<s , U(b|s;β)≤Ub|s;β⇒Ub|s;β<Ub|s;β(1) An immediate consequence of this strict single-crossing condition is that best responses are monotone and pure; see, for example, Athey (2001). LEMMA 1—Best Responses Are Monotone: If βis a strategy and ˆ βa best response to it, then ˆ βis pure and weakly increasing. The proof of the lemma in Appendix A.1 verifies (1). The key observation is that, for any bid b∈(vvh), the bidder’s payoff is positive in state hand negative in . Hence, bidders value a higher winning probability at a higher bid b>bonly if they believe that the state is more likely to be h. 3If v>0 and there is no reserve price, then the single-crossing condition (Lemma 1) may fail and equilibria may not be monotone; see Murto and Välimäki (2019)andLauermann and Wolinsky (2022) for examples. As ηbecomes large, the assumption becomes innocuous because competition drives almost all bids above v. 4This property is analogous to that of a stationary Poisson process, in which an event does not allow for inferences about the number of other events. BIDDING IN COMMON-VALUE AUCTIONS 497 Given Lemma 1, we restrict our attention to pure and nondecreasing strategies, which we also denote by β:[s¯ s]→[vvh]. For any such βand any bid b, the set of signals bidding bis an interval (possibly empty). We denote its boundaries by σ−(b)=infs:β(s)≥bσ +(b)=sups:β(s)≤b where we use the convention that inf{∅}=¯ sand sup{∅}=s.Thus,σ(b)=[σ−(b)σ+(b)] is the generalized inverse of β.5If σ−(b)<σ +(b), then β(s)=bfor all s∈σ(b). In this case, there is an atom in the implied bid distribution at b, and we say that bis a pooling bid and σ(b)isapool. 3.2. The Inference From Winning Is Non-Monotone Fix some strategy βand some bid bthat is not a pooling bid. The bid bwins when s(1) ≤s=σ+(b), where s(1) =sup{s−i}denotes the highest of the competitors’ signals. If there are no competitors, we set s(1) =−∞. Thus, the cumulative distribution function of s(1) is Fs(1) (s|ω)=e−η(1−Fω(s)).6Since bid bwins if s(1) ≤s, its winning probability in state ω is πω(b;β)=e−η(1−Fω(s)). A characteristic feature of common-value auctions is that winning is informative about the value of the good. In choosing a non-pooling bid b, all that matters for this inference is the relative position of the bid, which is given by s=σ+(b). The position of saffects the likelihood ratio of winning, πh(b;β) π(b;β)=e−η(1−Fh(s)) e−η(1−F(s)) (2) which, in turn, determines the conditional expected value of the good. In particular, E[v|win with b;β]=E[v|s(1) ≤s] is strictly increasing in the ratio (2)andisequalto E[v|s(1) ≤s]=ρe−η(1−Fh(s))vh+(1 −ρ)e−η(1−F(s))v ρe−η(1−Fh(s)) +(1 −ρ)e−η(1−F(s)) (3) LEMMA 2: The conditional expected value E[v|s(1) ≤s]is strictly decreasing in swhen s<˘ s,has its unique global minimum at s=˘ s,and is strictly increasing when s>˘ s. PROOF: Because the expected value is strictly increasing in the likelihood ratio (2), it is sufficient to show that the likelihood ratio is U-shaped around ˘ s. Note that ∂ ∂s e−η(1−Fh(s)) e−η(1−F(s)) =eη(Fh(s)−F(s))ηfh(s)−f(s); therefore, (2) is indeed strictly decreasing below ˘ sand strictly increasing above. Q.E.D. The intuition behind the shape is best explained with the help of Figure 1. 5In the following, we pretend that all pooling intervals are closed to simplify notation. The specification of bids on the boundaries is irrelevant because (a) the set of boundary signals of nontrivial intervals has zero measure, and (b) by the continuity of the likelihood ratio, a pooling bid is optimal for the boundary signals if it is optimal for the interior signals. 6Conditional on state ω, any competitor independently receives a signal larger than swith probability 1−Fω(s). By the decomposition and environmental equivalence properties of the Poisson distribution—see Myerson (1998)—bidders believe that the number of rival bidders with signals larger than sis Poisson distributed with mean η(1 −Fω(s)). 498 S. LAUERMANN AND A. SPEIT FIGURE 1.—The conditional expected value E[v|s(1) ≤s] is U-shaped. First, consider point (i) at the top right, which marks E[v|s(1) ≤¯ s]. By definition, the highest signal, s(1), is always smaller than ¯ s, independent of the state. Hence, the event that s(1) ≤¯ sis uninformative about the state, and so E[v|s(1) ≤¯ s]=E[v]. Therefore, there is no negative inference at the top, just as in an auction with a known number of competitors. Second, consider point (ii) at the top left, denoting E[v|s(1) ≤s]. The highest signal s(1) equals swith zero probability because the signal distribution has no atoms, while there are no competitors and s(1) =−∞with positive probability; consequently, E[v|s(1) ≤s]= E[v|s(1) =−∞]. Since the distribution of bidders is independent of the state,7the event that s(1) ≤sis uninformative about the state, and so E[v|s(1) ≤s]=E[v]. Thus, there is no winner’s curse at the bottom (ii) or at the top (i). In the middle, where s∈(s¯ s), the winner’s curse comes into play. With positive probability, there are competitors, all of whom received signals below s. Hence, for s∈(s¯ s), the conditional expected value is smaller than the unconditional one; that is, E[v|s(1) ≤s]<E[v], with a global minimum at ˘ s,wherefh(˘ s)=f(˘ s).8 The non-monotone inference is not an artifact of the Poisson distribution but a feature of numbers uncertainty in general. Bidders make an inference from winning not only about the others’ signals but also about the number of competitors, which can have opposing effects. In particular, the expected value is again non-monotone if the Poisson distribution is truncated at the bottom, guaranteeing a minimal degree of competition.9 We discuss the distributional assumption in Section 8. 3.3. A Large Auction Has No Strictly Increasing Equilibrium The non-monotone inference from winning can substantially affect the equilibrium behavior of bidders. As a benchmark, consider the common-value auction with a known number of bidders, n≥2. In this setup, the inference is monotone, which implies that there exists a strictly increasing equilibrium and it is unique; see Krishna (2010). With numbers uncertainty, an equilibrium of this form does not exist in general, owing to the non-monotone inference. 7We discuss a state-dependent bidder distribution in Section 8. 8Note that the non-monotonicity occurs with respect to the position of the bid. For a fixed bid, the expected value is monotone in the signal, which is part of the argument for Lemma 1. 9Consider a truncated Poisson distribution with at least n≥2 bidders. At the top, the inference from winning is unaffected by the truncation; at the bottom, the winning bidder updates her belief toward n−1rival bidders, all of whom received signal s. Thus, there is a limited winner’s curse at s, which, however, does not depend on η. In the middle, s∈(s¯ s),thewinner’scursegrowsinη,sothatE[v|s(1) ≤s] is still U-shaped when ηis large. BIDDING IN COMMON-VALUE AUCTIONS 499 PROPOSITION 1: When ηis sufficiently large,no strictly increasing equilibrium exists. The proof in Appendix A.2 relies essentially on two observations. First, for large ηand any strictly increasing β, the expected value conditional on winning at β(s) and on the bidder’s own signal s, given by Ev|win with β(s)s;β=E[v|s(1) ≤ss] inherits the U-shape of E[v|s(1) ≤s]. This means that while the inference from the bidder’s own signal is monotone increasing, the U-shaped inference from winning turns out to be more relevant for the expected value when ηis large. Second, for large η, it must be that β(s)≈E[v|s(1) ≤ss]ifβis an equilibrium; that is, bids must be close to the expected value conditional on winning, because of bidder competition. However, it cannot be that βis simultaneously close to E[v|s(1) ≤ss] and strictly increasing, given that E[v|s(1) ≤ss] is decreasing below ˘ s. The proof formalizes this idea by considering signals s<s <s  <˘ s. We show that when βis strictly increasing and ηis not too small, either a bidder with signal shas an incentive to deviate to β(s), or β(s) is too high to be individually rational for a bidder with signal s. The critical observation of Lemma 6used for the proof is that for any R>1, when ηis sufficiently large, πhβs;β πβs;β>Rπhβs;β πβs;β(4) so that the expected value conditional on winning with β(s) is much higher than with β(s). Hence, if β(s) is low enough to be individually rational, then β(s) is strictly below the expected value conditional on winning at β(s). This gives san incentive to deviate to β(s). 3.4. Strictly Increasing Bidding Strategies: An Example For intuition about what it means for ηto be “sufficiently large” in Proposition 1,we introduce an example that can be solved numerically. EXAMPLE:Letvh=1, v=0, and ρ=1 2. Assume s∈[01] and fh(s)=1, f(s)= 15−s. By standard arguments (Krishna (2010, Chapter 6.4)), a strictly increasing equilibrium exists if and only if the unique solution of the following ordinary differential equation (ODE) is strictly increasing: ∂ ∂sβ(s)=E[v|s(1) =s s]−β(s)fs(1) (s|s) Fs(1) (s|s)with β(s)=v(5) where Fs(1) (s|s) denotes the expected cumulative distribution function of s(1) conditional on observing s.10,11 In Figure 2, we plot the solution to the ODE (5) for the example, with η∈{359}.One can see that a strictly increasing equilibrium still exists for η=5, but this is no longer the case for η=9. 10So, Fs(1) (s|s)=ρfh(s) ρfh(s)+(1−ρ)f(s)e−η(1−Fh(s)) +(1−ρ)f(s) ρfh(s)+(1−ρ)f(s)e−η(1−F(s)). 11For further discussion, see Lauermann and Speit (2019). 500 S. LAUERMANN AND A. SPEIT FIGURE 2.—Equilibrium candidates for an example with different η. 4. POOLING AND EQUILIBRIUM NONEXISTENCE In Section 3, we showed that numbers uncertainty prevents the existence of a strictly increasing equilibrium when ηis large. Hence, if an equilibrium exists for large η,it must contain flat parts. We now examine these flat parts to understand why bidders with different signals may have an incentive to pool on the same bid. 4.1. Pooling Can Ease the Winner’s Curse Fix some nondecreasing strategy β, and suppose that there is a pooling bid bp; that is, β(s)=bpfor all sfrom a pool σ(bp)=[σ−σ+], where we have dropped the argument bpfrom σ+/−for readability. The following lemma compares the inference from winning with the pooling bid bpto the inference from winning with a marginally lower or higher bid. LEMMA 3: Assume that βis a weakly increasing strategy with a pooling bid bp;that is, σ(bp)is a nondegenerate interval.Then the following hold: 1. If σ+(bp)≤˘ s,then E[v|s(1) ≤σ−]>E[v|win with bp;β]>E[v|s(1) ≤σ+]. 2. If σ−(bp)≥˘ s,then E[v|s(1) ≤σ−]<E[v|win with bp;β]<E[v|s(1) ≤σ+]. The proof is in Appendix A.3.2. Combined with Lemma 2, Lemma 3implies that the inference from winning is always U-shaped—even if βcontains atoms. To gain intuition for the inference, note that with positive probability, multiple bidders tie on the pooling bid bp, so that the winner is decided by a uniform tie-break. Consequently, the bid bpis more likely to win when there are fewer competitors who also choose bp—that is, who have signals in [σ−σ+]. If those signals are low, meaning that they are more likely to be realized in the low state, this implies that bpwins less often in the low state than in the high state—a blessing, compared to marginally overbidding bp. Conversely, if the signals are high, so that they are more likely to be realized in the high state, then the bid bpwins more often in the low state—an additional winner’s curse.12 4.2. Atoms Complicate the Equilibrium Analysis In auctions, atoms usually do not occur, because the discretely higher winning probability when overbidding provides a deviation incentive. In common-value auctions with a known number of bidders, this incentive is reinforced by the curse from tying, which fos12Formally, if σ+≤˘ s, the MLRP implies that η[Fh(σ+)−Fh(σ−)] <η[F(σ+)−F(σ−)], so that winning the tie-break is indeed a blessing; if σ−≥˘ s, then all inequalities and the inference from winning the random tie-break are reversed. BIDDING IN COMMON-VALUE AUCTIONS 507 FIGURE 5.—The bounce auction. 7.3. Information Revelation and Contingent Bidding Would it be beneficial for the seller to commit to revealing the number of bidders before the auction? In our running example (η=5) without a reserve price, doing so raises the seller’s expected revenue slightly, from 04260 to 04262. However, in a related setting, Murto and Välimäki (2019) gave an example with the opposite conclusion; thus, it may be interesting to understand what drives the revenue implications more systematically. A related idea in auction design by Harstad, Kagel, and Levin (1990) is to allow bidders to submit bids that are contingent on the actual number of bidders. Fully contingent bidding would replicate revealing the number of bidders. 8. DISCUSSION OF ASSUMPTIONS Auction Format. When ηis large, a second-price auction has no strictly increasing equilibrium either.22 We further conjecture that the nonexistence on the continuous bid space, as well as the equilibrium characterization on the discrete bid space, extend to the second-price auction. State-Independent Competition. One natural modification of our model is statedependent participation, expressed by a state-dependent mean ηω. This combines numbers uncertainty with the deterministic but state-dependent participation in Lauermann and Wolinsky (2017). We analyze this general case in our working paper, Lauermann and Speit (2019). Importantly, we show that our results extend if there is a signal with fh(s) f(s) ηh η=1. Exogenous Participation. Numbers uncertainty arises endogenously with a prior entry stage. For example, suppose uninformed potential bidders decide whether to enter at a cost. With entry cost, the expected number of actual participants needs to be bounded, and when the pool of potential bidders is large, symmetric equilibria need to be in mixed strategies. As the pool of potential bidders grows, the distribution of actual participants will converge to a Poisson distribution. If the potential bidders observe their signal first and condition their entry decision on it, then the distribution of actual participants will 22This is because the expected value conditional on being tied is not monotone when ηis large; see Lauermann and Speit (2019). 508 S. LAUERMANN AND A. SPEIT FIGURE 6.—The expected value conditional on winning with either nor n2bidders. generally be state-dependent, as discussed above.23 By characterizing the equilibrium outcomes for general bidder distributions, our predictions do not depend on the details of the entry stage. Thereby, our analysis remains valid for other settings with numbers uncertainty. Distribution of the Number of Bidders and Large Auctions. The use of the Poisson distribution simplifies the analysis but has several special properties that may raise concerns: it allows for fewer than two bidders, has unbounded support, and is concentrated around its mean for large η(its standard deviation is just √η). As discussed in Footnote 9,truncating the distribution at two bidders does not qualitatively change the main insights; the same is true when the distribution is truncated at the top, for a sufficiently large upper bound. However, the concentration of the Poisson distribution around its mean has implications. As ηbecomes large, the winning bid comes almost surely from a bidder having signal s>˘ s, where the equilibrium strategy is strictly increasing. For distributions that do not become concentrated, the critical non-monotonicity of the expected value will emerge at the top. As a result, the atoms remain part of the winning bid distribution even in large auctions. For instance, suppose the number of bidders is either nor n2with equal probability, and the signal distribution is as in the example from Section 3.4.Fornsufficiently large, the expected value conditional on winning, E[v|s(1) ≤s s], will be non-monotone at the top; see Figure 6.24 In general, one may be interested in the effect of a change in the variance of the number of bidders, holding its mean fixed; this is something the Poisson distribution does not allow for. One may expect that, for a fixed mean, the effects of uncertainty smoothly vanish as the variance decreases, approaching the standard outcome in the limit. Conversely, as the distribution becomes very dispersed, one may suspect that the outcome increasingly diverges from the standard one. Binary State. When there are more than two states, the inference from winning retains its qualitative shape: at the bottom and at the top, there is no winner’s curse, whereas in the middle, winning is bad news about the quality of the good. Thus, no strictly increasing equilibrium can exist when ηis large; instead, bidders with low signals must pool. 23One can show that the implied bidder distribution in a model with signal-dependent entry rates is indeed equivalent to the one from some model with exogenous, state-dependent bidder numbers; see Lauermann and Wolinsky (2022, Section 5.3). 24When the number of bidders is either 10 or 100, a bidder having signal 096 (the local minimizer) wins with probability 044, and when the number of bidders is either 15 or 225, the signal s=098 wins with probability 047. BIDDING IN COMMON-VALUE AUCTIONS 509 However, when there are more than two states, we cannot exclude decreasing strategies (cf. proof of Lemma 1). Without monotonicity, however, the analysis becomes technically much more challenging. 9. RELATED LITERATURE There is a small literature on numbers uncertainty with independent types—notably Matthews (1987), McAfee and McMillan (1987), and Harstad, Kagel, and Levin (1990)— studying, for example, the interaction of numbers uncertainty and risk aversion. Moreover, there is a recent strand of literature on numbers uncertainty in commonvalue auctions with correlated types. Murto and Välimäki (2019) considered a commonvalue auction with costly entry.25 After observing a binary signal, potential bidders decide whether to pay a fee to bid in the auction. When the pool of potential bidders is large, the number of participating bidders is approximately Poisson distributed with a statedependent mean. Their interest was in the information revelation incentives of the seller; see Section 7.3. They concentrated on parameters for which the entry pattern implies atomless bidding strategies, excluding the effects we are interested in here.26 In Lauermann and Wolinsky (2017,2022) the participation is deterministic but statedependent due to a solicitation decision by an informed auctioneer. In the paper, atoms are the result of a “participation curse” that arises when there are far fewer bidders in the high than in the low state. In a double-auction setting with many goods, Harstad, Pekec, and Tsetlin (2008)and Atakan and Ekmekci (2021) considered the effect of numbers uncertainty on the information aggregation properties of a kth-price auction (Pesendorfer and Swinkels (1997)). In Harstad, Pekec, and Tsetlin (2008), the distribution of bidders is exogenously given. Harstad, Pekec, and Tsetlin (2008) found that even if the equilibrium strategy is strictly increasing (which aids aggregation), information aggregation fails unless the numbers uncertainty is negligible. They also provided an example in which equilibrium is not strictly increasing, but they did not study this question further. In Atakan and Ekmekci (2021), bidders have a stateand type-dependent outside option so that numbers uncertainty arises endogenously via entry, and participation is correlated with the state. They studied how the winning bid in the auction is affected by the opportunity cost of forgoing the outside option. 10. CONCLUSION We have studied a canonical common-value auction in which the bidders are uncertain about the number of their competitors. Such “noise” in participation is ubiquitous in auctions, and in price competition more generally, so that the forces studied are present across a wide range of settings. We find that the numbers uncertainty invalidates classic findings for common-value auctions (Milgrom and Weber (1982)). In particular, it breaks the affiliation between the first-order statistic of the signals and the value of the good. As a consequence, bidding strategies generally are not strictly increasing but contain atoms. The locations of the atoms are indeterminate, implying equilibrium multiplicity. Moreover, no equilibrium exists in the standard auction on the continuous bid space when the expected number of bidders is sufficiently large. 25For auctions with endogenous entry, see also Levin and Smith (1994)andHarstad (1990). 26Basically, in terms of our model, only bidders with the highest signals enter and bid above v. 510 S. LAUERMANN AND A. SPEIT Many of the known failures of equilibrium existence in auctions require careful crafting of the setup, and rely on a discrete type space to generate atoms in the bid distribution (Jackson (2009)). By contrast, we have identified a failure of equilibrium existence in an otherwise standard auction setting in which the type space is continuous, and atoms in the bid distribution arise endogenously. The pooling and the equilibrium multiplicity that arise from numbers uncertainty have interesting implications. For example, even though the model is purely competitive, bidders with low signals behave “cooperatively” to reduce the winner’s curse; unlike in the case of a common-value auction with affiliation, they have an incentive to coordinate on certain bids. Consequently, equilibria resemble collusive behavior, even though they are the outcome of independent, utility-maximizing behavior on the bidders’ part. Moreover, the presence of atoms in the bid distribution invalidates empirical identification strategies that rely on the bidder’s first-order condition (cf. Athey and Haile (2007)) and, hence, on a strictly increasing strategy. Further analysis may examine more systematically the consequences of pooling and equilibrium multiplicity for classic auction design questions that we touched on in Section 7. In Section 7.2, we discussed how a clock auction with a non-monotone price path can be used to increase revenue given the non-monotone value conditional on winning. The underlying idea is that, when the good has not been sold even after a long delay, a bidder believes that she is the sole participant, rather than that the other bidders are all pessimistic about the value of the good. These considerations for dynamic trade with adverse selection may be worth further study. Some generalizations may also be worthwhile. We noted that the Poisson distribution becomes highly concentrated on its mean for large numbers. It may be interesting to study the bidding behavior in large auctions for other distributions that are less concentrated. Similarly, one could consider general interdependent values for which the random allocation within a pool has efficiency implications. Last, an unknown ratio of goods to buyers might also stem from a random supply of goods (cf. Harstad, Pekec, and Tsetlin (2008)). Future research could dig deeper into the implications of such random market tightness more generally. APPENDIX A: CONTINUOUS AUCTION A.1. Proof of Lemma 1 We prove the strict single-crossing condition, (1). Since b>b≥v, it follows that (v−b)<(v−b)≤0. Because the winning probability πωis weakly increasing and never zero (the bidder is alone with positive probability), πω(b;β)≥πω(b;β)≥ πω(v;β)>0. Together, these observations yield π(b;β)(v−b)<π (b;β)(v−b)≤0. Hence, U(b|s;β)≥U(b|s;β) requires that πh(b;β)(vh−b)>π h(b;β)(vh−b). Rearranging U(b|s;β)≥U(b|s;β) gives ρ (1 −ρ) fh(s) f(s)πhb;βvh−b−πh(b;β)(vh−b)≥π(b;β)(v−b)−πb;βv−b From s>s,wehavefh(s) f(s)>fh(s) f(s). Since the left-hand side is strictly positive, it follows that the left-hand side is strictly larger for sthan for s, which is equivalent to the claimed inequality, U(b|sβ)>U(b|sβ). BIDDING IN COMMON-VALUE AUCTIONS 511 A.2. Proof of Proposition 1 Consider two signals s,s with s<s <s  <˘ s. The proof shows that if ηis large enough, and β(s) is low enough to be individually rational for s, a bidder with signal shas an incentive to deviate to β(s). Step 1: Individual rationality. Suppose β(s) is optimal for some signal s.Thenitmustbe individually rational: β(s)≤Ev|win with β(s)s;β(6) Otherwise, swould be strictly better off bidding v, which ensures nonnegative payoffs.27 The inequality (6) can be written in a convenient “ratio form” as β(s)−v vh−β(s)≤ρ 1−ρ fh(s) f(s) πhβ(s);β πβ(s);β(7) Step 2: Competitive bidding. The following lemma shows that as competition becomes fierce, bids must be close to the expected value conditional on winning. LEMMA 5—Competitive Bidding: Take any two signals s1and s2,with s1<s 2.For every η,there exists some C(η)such that,if βis strictly increasing and s1prefers β(s1)to β(s2), that is,if Uβ(s1)|s1;β≥Uβ(s2)|s1;β then β(s2)−v vh−β(s2)≥ρ 1−ρ fh(s1) f(s1) πhβ(s2);β πβ(s2);βC(η)(8) and limη→∞ C(η)=1. Because the winning probability is much lower at β(s1) than at β(s2), a bidder with signal s1will deviate to β(s2) if the payoff conditional on winning there is strictly positive. This requires the bid to be very close to or higher than the expected value conditional on winning. The lemma states this requirement in ratio form, analogously to (7): when C(η)=1, the inequality (8) is equivalent to β(s2)≥E[v|s(1) ≤s2s1]. PROOF OF LEMMA 5: Expanding U(β(s1)|s1;β)≥U(β(s2)|s1;β)gives ρfh(s1)πhβ(s1);βvh−β(s1)+(1 −ρ)f(s1)πβ(s1);βv−β(s1) ρfh(s1)+(1 −ρ)f(s1) ≥ρfh(s1)πhβ(s2);βvh−β(s2)+(1 −ρ)f(s1)πβ(s2);βv−β(s2) ρfh(s1)+(1 −ρ)f(s1) Since β(s1)≥v, a necessary condition for the inequality is that ρfh(s1)πβ(s2);β(vh−v) ≥ρfh(s1)πhβ(s2);βvh−β(s2)+(1 −ρ)f(s1)πβ(s2);βv−β(s2) 27In fact, payoffs are strictly positive at vsince there is a chance of being the only bidder. 512 S. LAUERMANN AND A. SPEIT Rearranging the inequality gives a lower bound on β(s2): β(s2)−v vh−β(s2)≥ρ 1−ρ fh(s1) f(s1) πhβ(s2);β πβ(s2);β1−πhβ(s1);β πhβ(s2);β vh−v vh−β(s2)(9) Let C(η)=1−πhβ(s1);β πhβ(s2);β vh−v vh−E[v|s1] Note that C(η) is independent of β,sinceπh(β(s);β) depends only on sfor strictly increasing β.Moreover,C(η)<1forallηbecause 0 <πh(β(s1);β) πh(β(s2);β)≤1and vh−v vh−E[v|s1]>0. Now, the inequality in the lemma holds: First, if β(s2)<E[v|s1], then the bracketed term in (9) is larger than C(η). Second, if β(s2)≥E[v|s1], then this is equivalent to β(s2)−v vh−β(s2)≥ρ 1−ρ fh(s1) f(s1) and so, in particular, β(s2)−v vh−β(s2)≥ρ 1−ρ fh(s1) f(s1) πh(β(s2);β) π(β(s2);β),sinceπh(β(s2);β) π(β(s2);β)≤1. The claim follows because C(η)≤1. Finally, limη→∞ C(η)=1sinceπh(β(s1);β) πh(β(s2);β)=e−η(Fh(s2)−Fh(s1)) →0. Q.E.D. Step 3: Non-monotone expected values. The next lemma shows that when ηbecomes large, the inference from winning grows arbitrarily strong. LEMMA 6—U-Shaped Values: For any two signals s1,s2with s1<s 2<˘ sand any R>1, there is some ηlarge enough so that,if βis strictly increasing, πhβ(s1);β πβ(s1);β>Rπhβ(s2);β πβ(s2);β(10) PROOF: Using (2), the ratio of the outer terms of (10)is πhβ(s1);β πβ(s1);βπhβ(s2);β πβ(s2);β=eη[(F(s2)−Fh(s2))+(F(s1)−Fh(s1))](11) From s1<s 2<˘ s,wehavef(s)>f h(s)foralls∈(s1s2); therefore, F(s2)−Fh(s2)+F(s1)−Fh(s1)>0 Hence, the result follows from (11)andη→∞.Q.E.D. Combined, the three steps imply the proposition. PROOF OF PROPOSITION 1: Pick some strictly increasing bidding strategy βfor every η, and a pair of signals sand s with s<s <s  <˘ s. From (7), individual rationality at s implies an upper bound for β(s): βs−v vh−βs≤ρ 1−ρ fhs fs πhβs;β πβs;β(12) BIDDING IN COMMON-VALUE AUCTIONS 513 Conversely, (8) from Lemma 5implies a lower bound on β(s) to disincentivize deviations of sfrom β(s)toβ(s): βs−v vh−βs≥ρ 1−ρ fh(s) f(s) πhβs;β πβs;βC(η)(13) We now show that these bounds cannot hold simultaneously when ηis large. For both bounds to hold, given β(s)<β(s), we must have ρ 1−ρ fh(s) f(s) πhβs;β πβs;βC(η)≤ρ 1−ρ fhs fs πhβs;β πβs;β(14) Since C(η)→1forη→∞from Lemma 5, this requires that for any R>fh(s) f(s)/fh(s) f(s), πhβs;β πβs;β≤Rπhβs;β πβs;β for all ηlarge enough. However, the inequality (10) from Lemma 6implies that this inequality fails for ηlarge enough. Thus, we have reached a contradiction. So, if β(s) is individually rational for s, then (8) from Lemma 5fails for ηlarge enough;thatis,abidderwithsignalsstrictly prefers to bid β(s). It follows that, when η is large, no strictly increasing bidding strategy is an equilibrium. Q.E.D. A.3. Auxiliary Results for Section 4 A.3.1. Characterization of the Winning Probability We show that the winning probability with a pooling bid bpis πω(bp;β)=P[s(1) ∈σ|ω] E[#s∈σ|ω]=e−η(1−Fω(σ+)) −e−η(1−Fω(σ−)) ηFω(σ+)−Fω(σ−)(15) Let s+=σ+(bp)ands−=σ−(bp). Then πω(bp;β)=P(no bid >b p|ω)∞  n=0 1 n+1P(ncompetitors bid bp|ω) =e−η(1−Fω(s+))∞  n=0 1 n+1e−η(Fω(s+)−Fω(s−)) ηFω(s+)−Fω(s−)n n! =e−η(1−Fω(s+))∞  n=0 e−η(Fω(s+)−Fω(s−)) ηFω(s+)−Fω(s−)n (n+1)! =e−η(1−Fω(s+)) ηFω(s+)−Fω(s−)∞  n=1 e−η(Fω(s+)−Fω(s−)) ηFω(s+)−Fω(s−)n n! =e−η(1−Fω(s+)) ηFω(s+)−Fω(s−)1−e−η(Fω(s+)−Fω(s−)))=e−η(1−Fω(s+)) −e−η(1−Fω(s−)) ηFω(s+)−Fω(s−) 514 S. LAUERMANN AND A. SPEIT The numerator is P[s(1) ∈[s−s+]|ω], and the denominator is the expected number of signals from [s−s+] in state ω, that is, E[#s∈[s−s+]|ω]. A.3.2. Proof of Lemma 3 We prove Lemma 3for the case of σ+(bp)=s+≤˘ s; the case of σ−(bp)=s−≥˘ sis symmetric and is omitted. In particular, we show that e−η(1−Fh(s−)) e−η(1−F(s−)) >πh(bp;β) πl(bp;β)>e−η(1−Fh(s+)) e−η(1−F(s+)) (16) Let xω=E[#s∈[s−s+]|ω]; that is, xh=ηFh(s+)−Fh(s−)and x=ηF(s+)−F(s−)(17) and note that s+≤˘ simplies xh<x . Further, πω(bp;β) e−η(1−Fω(s−)) =1 e−η(1−Fω(s−)) e−η(1−Fω(s+)) −e−η(1−Fω(s−)) ηFω(s+)−Fω(s−)=exω−1 xω  and so dividing (16)byπh(bp;β) πl(bp;β)gives ex−1 x exh−1 xh = e−η(1−Fh(s−)) e−η(1−F(s−)) πh(bp;β) πl(bp;β) >1> e−η(1−Fh(s+)) e−η(1−F(s+)) πh(bp;β) πl(bp;β) = 1−e−x x 1−e−xh xh  This holds because ez−1 zis strictly increasing in z,1−e−z zis strictly decreasing in z,and xh<x . Thus, (16) holds for the case of s+≤˘ s, as claimed. A.3.3. Zero-Profit Condition and U-Shaped Limit Values We first generalize Lemma 5to Lemma 7, to allow for weakly increasing βkand bids that are not in the image of βk. We then similarly generalize Lemma 6to Lemma 8. LEMMA 7—Competitive Bidding: Let (βk)be a sequence of bidding strategies and ηk→ ∞.Fix some s1and some sequence (bk), with bk>β k(s1)for all k.If lim k→∞ πhβk(s1);βk πhbk;βk=0 and s1prefers βk(s1)to bk,that is, Uβk(s1)|s1;βk≥Ubk|s1;βkfor all k (18) then there is some sequence (Ck)such that Ck→1and bk−v vh−bk≥ρ 1−ρ fh(s1) f(s1) πhbk;βk πbk;βkCkfor all k (19) BIDDING IN COMMON-VALUE AUCTIONS 515 PROOF: By the same argument as in the proof of Lemma 5,(18) implies that bk−v vh−bk≥ρ 1−ρ fh(s1) f(s1) πhbk;βk πbk;βkCk for Ck=1−πhβk(s1);βk πhbk;βk vh−v vh−E[v|s1] Finally, Ck→1 follows from the hypothesis that πh(βk(s1);βk) πh(bk;βk)→0. Q.E.D. LEMMA 8—U-Shaped Limit Values: Let (βk)be a sequence of bidding strategies,ηk→ ∞,and (bk 1bk 2)a pair of bids with limσk +(bk 1)<limσk +(bk 2)≤˘ s.Then,for every R>1, for all klarge enough, πhbk 1;βk πbk 1;βk>Rπhbk 2;βk πbk 2;βk(20) The condition limσk +(bk 1)<limσk +(bk 2) ensures that the winning probability is significantly higher at bk 2than at bk 1. The condition limσk +(bk 2)≤˘ sensures that we are on the decreasing branch of the expected value conditional on winning. PROOF OF LEMMA 8: If there is an atom at some bk, then πhbk;β πbk;β= e−η(1−Fh(s+)) −e−η(1−Fh(s−)) ηFh(s+)−Fh(s−) e−η(1−F(s+)) −e−η(1−F(s−)) ηF(s+)−F(s−) =ηF(s+)−F(s−) ηFh(s+)−Fh(s−) 1−e−η(Fh(s+)−Fh(s−)) 1−e−η(F(s+)−F(s−)) e−η(1−Fh(s+)) e−η(1−F(s+))  and it follows from 1−e−x xbeing decreasing in xthat min1F(s+)−F(s−) Fh(s+)−Fh(s−)≤ηF(s+)−F(s−) ηFh(s+)−Fh(s−) 1−e−η(Fh(s+)−Fh(s−)) 1−e−η(F(s+)−F(s−)) ≤max1F(s+)−F(s−) Fh(s+)−Fh(s−), which is uniformly bounded (we used a similar argument in the proof of Lemma 3). Finally, from limσk +(bk 1)<limσk +(bk 2)≤˘ s,wehave lim e−η(1−Fh(σk +(bk 2))) e−η(1−F(σk +(bk 2))) e−η(1−Fh(σk +(bk 1))) e−η(1−F(σk +(bk 1))) =0 which was observed in Lemma 6. The claim follows. Q.E.D. 516 S. LAUERMANN AND A. SPEIT A.4. Proof of Proposition 2 We consider two cases: first, that for every pair of signals s1,s2with s<s 1<s 2<˘ s,and ηlarge enough, β(s1)=β(s2) (Case 1); second, that there exists some pair s1,s2for which β(s1)<β(s2)forallη(Case 2). The following two subsections show that a sequence of bidding strategies satisfying the assumptions of either case cannot be an equilibrium for large η. A.4.1. Case 1: Pooling of All Signals Below ˘ s The following lemma shows that there can be no equilibrium in which there is a pooling bid bpfor which the pool σ(bp)=[σ−σ+]startsatsomeσ−<˘ sand extends to some σ+ close to or beyond ˘ swhen ηis large: either bpis too high to be individually rational for σ−,orσ+will have a strict incentive to marginally overbid. LEMMA 9: Take any s1<˘ s.Then there exist s2∈(s1˘ s)and η∗such that,for all η≥η∗and for all βwith β(s1)=β(s2)=bp,the following cannot hold simultaneously: U(bp|σ−(bp);β)≥0and Ubp|σ+(bp);β≥lim ε→0+Ubp+ε|σ+(bp);β(21) The main observation of the proof is that, for large enough η, E[v|win with bpσ−;β]<E[v|s(1) ≤σ+σ+;β](22) with σ+/−=σ+/−(bp). For bpto be individually rational for σ−, it must be smaller than the left-hand side of (22). However, when σ+marginally overbids bp, the expected value conditional on winning is equal to the right-hand side of (22). Thus, if bpis low enough to be individually rational for σ−, then σ+obtains strictly positive profits conditional on marginally overbidding bp. Since the winning probability at bp+εis significantly larger than at bp, this will imply that (21) fails, that is, that σ+strictly prefers the deviation. To gain intuition for (22), recall from Lemma 3that, for σ+≤˘ s, it holds that E[v|win with bp;β]>E[v|s(1) ≤σ+;β]; that is, there is a winner’s blessing at bp. Hence, (22) shows that this winner’s blessing is weaker than the change in the signal inference going from σ−to σ+. Since it is a central piece of the argument, we now derive (22). First,28 πh(bp) π(bp)≈F(σ+)−F(σ−) Fh(σ+)−Fh(σ−)lim ε→0 πh(bp+ε) π(bp+ε)(24) Second, from the MLRP, fh(σ−) f(σ−)<Fh(σ+)−Fh(σ−) F(σ+)−F(σ−)(25) 28 We say that f(η)≈g(η)iflim η→∞ f(η) g(η)=1. The claim follows from πω(bp) lim ε→0πω(bp+ε)ηFω(σ+)−Fω(σ−)=e−η(1−Fω(σ+)) −e−η(1−Fω(σ−)) e−η(1−Fω(σ+)) (23) which converges to 1 as η→∞. BIDDING IN COMMON-VALUE AUCTIONS 523 To prove (41), pick any kand any sk>¯σk +(s).Then ¯σk −(sk)≥¯σk +(s)>˘ s, and Lemma 11 implies that ¯σk +(sk)=¯σk −(sk)=sk. Since this is also true for the signal sk+¯σk +(s) 2, it follows that βki(sk)>β ki(s)+ki for ilarge. Hence, e−ηk(1−Fh(¯σk +(s))) e−ηk(1−F(¯σk +(s))) =lim i→∞ e−ηk(1−Fh(σki +(s))) e−ηk(1−F(σki +(s))) ≤lim i→∞ πki hβkis+ki πki βkis+ki ≤lim i→∞ πki hβkisk πki βkisk=e−ηk(1−Fh(sk)) e−ηk(1−F(sk))  where the first equality is from σki +(s)→¯σk +(s), the two inequalities are from Lemma 3 and βki(sk)>β ki(s)+ki, and the final equality is from ¯σk +(sk)=¯σk −(sk)=sk.Since skcan be chosen arbitrarily close to ¯σk +(s) and the two outer expressions are continuous, (41) holds, as desired. Q.E.D. PROOF OF PART 2OF LEMMA 12:Iflim k→∞ ¯σk +(s)=˘ s, then for ski +/−=σki +/−(βki(s)+ ki), lim k→∞ lim i→∞xki ω=lim k→∞ lim i→∞ηkFωski +−Fωski −=∞(42) As in Lemma 3, πki hβki(s)+ki πki βki(s)+kie−ηk(1−Fh(σki +)) e−ηk(1−F(σki +)) =xki  xki h exki h−1 exki −1 with xki ω=ηk(Fω(ski +)−Fω(ski −)), where we note that σki +=ski −. If limk→∞ limi→∞ xki h=¯ xh<∞, then limk→∞ limi→∞ ski +=limk→∞ limi→∞ ski −=˘ s.Since fh(˘ s) f(˘ s)=1 and the likelihood ratio is continuous, this implies limk→∞ limi→∞ xki =¯ x=¯ xh. Together, lim k→∞ lim i→∞ xki  xki h exki h−1 exki −1=1 Thus, the necessary condition (36) from Claim 4fails if ¯ xh<∞. Hence ¯ xh=∞and so ¯ x=∞; that is, (42) holds. This completes the proof of the lemma. Q.E.D. LEMMA 13: Let (βki)be a convergent equilibrium sequence.If s<s 1<s 2<˘ sand βki(s1)<β ki(s2)for all k,i,then,for all large kand ilarge enough given k, βki(s2)=βki(s1)+ki(43) PROOF OF LEMMA 13:Takeanys2<ˆ s2<˘ s.Since(βki) is a sequence of equilibria, Ukiβki(ˆ s2)ˆ s2;βki≥0and lim k→∞ ¯σk +(ˆ s2)≤˘ s where the first inequality states individual rationality and the second follows from Lemma 12.Now,suppose(43) does not hold. Then, choosing a further subsequence if 524 S. LAUERMANN AND A. SPEIT necessary, there is some sequence of bids (ˆ bki)withˆ bki ∈Dki such that, for all k,i, βki(s1)<ˆ bki <β ki(s2) However, for any fixed i,forklarge enough, swould deviate to ˆ bki, that is, Uki(ˆ bkis;βki)>U ki(βki(s1)s;βki). This follows from the same argument as in Case 2 in the proof of Proposition 2. This is because the proof does not utilize the continuum of feasible bids but only compares the payoffs at the bids in question. Q.E.D. B.1.3. Proof of Proposition 3 First, we show the characterization holds for convergent sequences of equilibria (βki). Assertion (iv): Take any s>˘ s. By Lemma 12,lim k→∞ ¯σk −(s)>˘ s, and by Lemma 11, this implies that, for all k,lim i→∞ σki −(s)=limi→∞ σki +(s)=s. Hence, for any kand ilarge enough, the expected number of bids that tie with svanishes. For assertions (i)–(iii), pick some s∈(s˘ s). Case 1:¯σ−(s)=sand ¯σ+(s)=˘ s.The proposition holds with A=σki(s)andB= σki(βki(s)+ki). By hypothesis, assertions (i) and (ii) hold. Assertion (iii) (many bidders are pooled) holds for Aby σki(s)→[s ˘ s], and for Bby Part 2 of Lemma 12. Case 2:There are s<s 1<s 2<˘ ssuch that ¯σ+(s1)≤¯σ−(s2).By Lemma 13,fork,ilarge enough, βki(s2)=βki(s1)+ki. Moreover, for all s 1∈(ss1), s 2∈(s2˘ s), and high k,i, βki(s1)=βki (s 1)andβki(s2)=βki(s 2). Hence, assertions (i)–(iii) hold for A=σki (s1) and B=σki(s2). Since Case 1 and Case 2 are exhaustive, the characterization indeed holds for all convergent sequences of equilibria. Now, take some arbitrary sequence of equilibria (βki). We prove the characterization by contradiction: If the characterization is not true, then, for every K, there is some k≥Ksuch that, for every I, there is some i≥Isuch that at least one of the four assertions, (i)–(iv), fails for βki. Thus, we can pick a new sequence (βki) of equilibria such that some assertion fails for all k,i. Of course, this sequence has a convergent subsequence, and, as just shown, all of the assertions hold for k,ilarge enough—contradicting the starting hypothesis. Q.E.D. APPENDIX C: COMMUNICATION EXTENSION,SECTION 6 C.1. Proof of Proposition 4 Given the profile (M∗β∗μ∗) from the proposition and an extended bid (br), let π∗ ω(br) denote the winning probability in state ω. Below, we show that for all s, π∗ ωβ∗(s)s=lim k→∞πωβk(s);βk(44) Moreover, for every extended bid (br), there exists a sequence bk∈Dkwith bk→bsuch that π∗ ω(br)=lim k→∞πωbk;βk(45) The displayed equations imply the proposition as follows. Take any sand any (b r)with asequencebk→bthat satisfies (45). Since βkis an equilibrium, Uk(βk(s)|s;βk)≥ Uk(bk|s;βk)forallbk.Moreover,(44)and(45) imply that U∗(β∗(s)r∗(s)|s;β∗μ∗)= BIDDING IN COMMON-VALUE AUCTIONS 525 limUk(βk(s)|s;βk)andU∗(br|s;β∗μ∗)=limUk(bk|s;β∗). Taken together, these yield U∗(β∗(s)μ∗(s)|s;β∗μ∗)≥U∗(b r|s;β∗μ∗); thus, (β∗μ∗) is a best response. Proof of (44): First, consider some sfor which β∗(s) is not a pooling bid. Then β∗(s−ε)<β ∗(s)<β ∗(s+ε)forallε>0, and so the winning probability π∗ ω(β∗(s)s) is sandwiched by P(s(1) ≤s−ε|ω)andP(s(1) ≤s+ε|ω). Since βk(s−ε)→β∗(s−ε) and βk(s+ε)→β∗(s+ε), this is also true for πk ω(βk(s);βk). Choosing εsmall implies the claim. Second, consider some sfor which β∗(s) is a pooling bid. Let ˆσ∗ +(s)= inf{s|ˆσ∗ −(s)>ˆσ∗ −(s)}, so that sis in the element m∗(s)=[ˆσ∗ −(s)ˆσ∗ +(s)] of the partition M∗(or its equivalent). Given any ε>0 small enough, pointwise convergence of σk −and σk +implies that, for all klarge enough, βkˆσ∗ −(s)−ε<β kˆσ∗ −(s)+ε=βk(s)=βkˆσ∗ +(s)−ε<β kˆσ∗ +(s)+ε if sis in the interior of m∗(s) (we may ignore the boundary types for this proof). As →0, the winning probability at (β∗(s)s) is sandwiched and the result follows. Proof of (45): Suppose bis not a pooling bid according to β∗, so that ris irrelevant. The cases b≤β∗(s)andb≥β∗(¯ s) are immediate. If β∗(s)<b<β ∗(¯ s), we have β∗(s−ε)< b<β ∗(s+ε)forsomesand all εsmall enough. Hence, the winning probability at bis sandwiched between P(s(1) ≤s−ε|ω)andP(s(1) ≤s+ε|ω), and the claim follows as before. Second, suppose bis a pooling bid according to β∗.Withσ∗ −and σ∗ +the (generalized) inverse of β∗,wearedoneifσ∗ −(b)≤r≤σ∗ +(b), since then (b r)=(β∗(s)s)forsomes and (44) applies. So, suppose σ∗ +(b)<r.Then(br) wins if and only if s(1) ≤σ∗ +(b).Moreover, for εsmall enough, β∗(σ∗ +(b)+ε)>b=β∗(σ∗ +(b)−ε). It follows that there is a sequence bk→bfor which βk(σ∗ +(b)+ε)>b k>β k(σ∗ +(b)−ε)forallklarge enough, implying the claim by sandwiching the winning probabilities as before. An analogous argument applies if r<σ ∗ −(b). Q.E.D. C.2. Proof of Proposition 5 Given some ε>0andηk→∞,let(Mkβkμk) be a sequence of truthful solutions. We argue that, for klarge enough, (Mkβkμk) satisfies the properties of Proposition 5.To do so, we retrace the proof of Proposition 3.Letmk(s)∈Mkdenote the element of the partition containing s,with ˆσk −(s)=inf{s|mk(s)=mk(s)}and ˆσk +(s)=sup{s|mk(s)= mk(s)}. We already argued the analogue of Lemma 13 in the main text: for any s<s 1<s 2<˘ s and any klarge enough, if ris in between the signals, that is, if s1<r<s 2, then either r∈mk(s1)orr∈mk(s2). Thus, either mk(s1)=mk(s2) or the two elements mk(s1)and mk(s2) are adjacent. The analogue of Lemma 11 (no pooling of signals above ˘ s) is that (i) for all k,if ˆσk −(s)≥ ˘ s, then mk(s)=s, and (ii) limk→∞ ˆσk −(s)≥˘ simplies limk→∞ ˆσk +(s)=˘ s. Like the original lemma, this absence of pooling above ˘ sfollows from Lemma 3. The analogue of Lemma 12 is that, whenever limk→∞ ˆσk −(s)<limk→∞ ˆσk +(s), we have the following: (i) limk→∞ ˆσk +(s)≤˘ s, and (ii) if limk→∞ ˆσk +(s)=˘ s, then for all k large enough, there is some Bk∈Mkfor which Bkis adjacent to Ak=mk(s)and limk→∞ Bkηkfω(z)dz =∞. For this, we argue the following analogue of Claim 4.Let limδ→0+ πk h(βk(s)ˆσk ++δ) πk (βk(s)ˆσk ++δ)=Lkbe the likelihood ratio conditional on overbidding the extended bid (βk(s)s) by bidding (βk(s)ˆσk ++δ). Then, by exactly the same arguments as in the 526 S. LAUERMANN AND A. SPEIT original proof, individual rationality of (βk(ˆσk −)ˆσk −) and optimality of (βk(ˆσk +)ˆσk +)require that lim k→∞ Lk e−ηk(1−Fh(ˆσk +)) e−ηk(1−F(ˆσk +)) ≤fh(ˆσ−) f(ˆσ−) F(ˆσ+)−F(ˆσ−) Fh(ˆσ+)−Fh(ˆσ−)<1(46) with ˆσ+/−=limk→∞ ˆσk +/−(s). From this, it follows that there must be some significant pool Bkthat is directly adjacent to mk(s). By the analogue of Lemma 11, it cannot be that Bk starts above ˘ s; hence (i) holds: limk→∞ ˆσk +(s)≤˘ s. Moreover, if limk→∞ ˆσk +(s)=˘ s, then (46) requires that limk→∞ Bkηkfω(z)dz =∞for Bk=mk(ˆσk ++δ), for all δsmall enough. Finally, the analogues of Lemmas 11–13 imply Proposition 5.29 Thus, as claimed, Proposition 5indeed follows from the same arguments as Proposition 3.Q.E.D. REFERENCES AKBARPOUR,MOHAMMAD,AND SHENGWU LI(2020): “Credible Auctions: A Trilemma,” Econometrica, 88, 425–467. [493] ATAKAN,ALP E., AND MEHMET EKMEKCI (2021): “Market Selection and the Information Content of Prices,” Econometrica, 89, 2049–2079. [509] ATHEY,SUSAN (2001): “Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information,” Econometrica, 69, 861–889. [496,501,502] ATHEY,SUSAN,AND PHILIP HAILE (2007): “Nonparametric Approaches to Auctions,” in Handbook of Econometrics, ed. by J. J. Heckman and E. E. Leamer. Elsevier. [510] HARSTAD,RONALD M. (1990): “Alternative Common-Value Auction Procedures: Revenue Comparisons With Free Entry,” Journal of Political Economy, 98, 421–429. [509] HARSTAD,RONALD M., JOHN H. KAGEL,AND DAN LEVIN (1990): “Equilibrium Bid Functions for Auctions With an Uncertain Number of Bidders,” Economics Letters, 33, 35–40. [494,507,509] HARSTAD,RONALD M., ALEKSANDAR SASA PEKEC,AND ILIA TSETLIN (2008): “Information Aggregation in Auctions With Unknown Number of Bidders,” Games and Economic Behavior, 62, 476–508. [509,510] JACKSON,MATTHEW O. (2009): “Non-Existence of Equilibrium in Vickrey, Second-Price, and English Auctions,” Review of Economic Design, 13, 137–145. [510] JACKSON,MATTHEW O., LEO K. SIMON,JEROEN M. SWINKELS,AND WILLIAM R. ZAME (2002): “Communication and Equilibrium in Discontinuous Games of Incomplete Information,” Econometrica, 70, 1711–1740. [493,495,503-505] KRISHNA,VIJAY (2010): Auction Theory (Second Ed.). Elsevier. [494,495,498,499] LAUERMANN,STEPHAN,AND ANDRE SPEIT (2019): “Bidding in Common-Value Auctions With an Uncertain Number of Competitors,” CRC TR 224 Discussion Paper No. 136. [499,507] (2023): “Supplement to ‘Bidding in Common-Value Auctions With an Unknown Number of Competitors’,” Econometrica Supplemental Material, 91, https://doi.org/10.3982/ECTA17793.[504] LAUERMANN,STEPHAN,AND ASHER WOLINSKY (2017): “Bidder Solicitation, Adverse Selection, and the Failure of Competition,” American Economic Review, 107, 1399–1429. [507,509] (2022): “A Common-Value Auction With State-Dependent Participation,” Theoretical Economics, 17, 841–881. [495,496,508,509] LEBRUN,BERNARD (1996): “Existence of an Equilibrium in First Price Auctions,” Economic Theory, 7, 421– 443. [495,503,504] LEVIN,DAN,AND JAMES L. SMITH (1994): “Equilibrium in Auctions With Entry,” American Economic Review, 84, 585–599. [509] 29Assertion (iv): Take any ˘ s+ε. By Lemma 12,forklarge enough, ˆσk −(˘ s+ε)>˘ s,andso,byLemma11, mk(s)=sfor all s≥˘ s+ε,implyingthatβkis strictly increasing above ˘ s+ε. Assertions (i)–(iii): Pick some s∈(s˘ s). Case 1: limk→∞ ˆσk −(s)=sand limk→∞ ˆσk +(s)=˘ s.LetAk=mk(s). By Lemma 12, there is an adjacent element Bkthat becomes large, as required. Case 2: There are s<s 1<s 2<˘ ssuch that ˆσk +(s1)≤ˆσk −(s2) for all klarge enough. The desired properties then follow from Lemma 13. Since all sequences of solutions can be partitioned into subsequences that satisfy either Case 1 or Case 2, these cases are exhaustive. BIDDING IN COMMON-VALUE AUCTIONS 527 MATTHEWS,STEVEN (1987): “Comparing Auctions for Risk Averse Buyers: A Buyer’s Point of View,” Econometrica, 55, 633–646. [509] MCAFEE,R.PRESTON,AND JOHN MCMILLAN (1987): “Auctions With a Stochastic Number of Bidders,” Journal of Economic Theory, 43, 1–19. [509] MILGROM,PAUL R., AND ROBERT J. WEBER (1982): “A Theory of Auctions and Competitive Bidding,” Econometrica, 50, 1089–1122. [494,509] MURTO,PAULI,AND JUUSO VÄLIMÄKI (2019): “Common Value Auctions With Costly Entry,” Mimeo. [495, 496,507,509] MYERSON,ROGER B. (1998): “Population Uncertainty and Poisson Games,” International Journal of Game Theory, 27, 375–392. [496,497] PESENDORFER,WOLFGANG,AND JEROEN M. SWINKELS (1997): “The Loser’s Curse and Information Aggregation in Common Value Auctions,” Econometrica, 65, 1247–1281. [509] ROTH,ALVIN E., AND AXEL OCKENFELS (2002): “Last-Minute Bidding and the Rules for Ending Second-Price Auctions,” American Economic Review, 92, 1093–1103. [493] Co-editor Barton L. Lipman handled this manuscript. Manuscript received 28 October, 2019; final version accepted 31 August, 2022; available online 15 September, 2022.