Distributionally Robust Optimal Auction Design under Mean Constraints
DDistributionally Robust Optimal Auction Design under MeanConstraints ∗ Ethan W. Che † March 12, 2020
Abstract
We study a seller who sells a single good to multiple bidders with uncertainty over the jointdistribution of bidders’ valuations, as well as bidders’ higher-order beliefs about their opponents.The seller only knows the mean of the marginal distribution of each bidder’s valuation and therange, and an adversarial nature chooses the worst-case distribution within this ambiguity set.We find that a second-price auction with an optimal, random reserve price obtains the optimalrevenue guarantee within a broad class of mechanisms that include all the standard auctionformats. We find that as the number of bidders grows large, the seller’s optimal reserve priceconverges in probability to a non-binding reserve price.
Keywords:
Robust mechanism design, second-price auction, reserve price.
JEL codes:
D82, D44
In the standard auction theory, beginning with Myerson (1981), the seller’s uncertainty over thebidders’ private values is typically modeled as a well-defined prior distribution. While this frameworkis flexible, and allows for a wide range of beliefs, it does not describe how such beliefs are formedin the first place. Indeed, the seller not only faces uncertainty over the realizations of a probabilitydistribution, but also the deeper uncertainty that they may not have the correct model at all.How does a seller, who is concerned about model misspecification, form their beliefs? Perhapsit is unrealistic to assume that a seller begins with a full distribution over types, which is high-dimensional and assigns a specific probability to all possible circumstances. It is more realistic thatthe seller carries low-dimensional “summary statistics,” such as the mean, variance, mode, etc., onwhich he bases his decisions. In such a case, one reasonable criterion, if robustness is desired, forchoosing among the distributions that match these moments and all possible higher order beliefsconsistent with the moments, is choosing the one which makes him worst off. ∗ I am grateful to Ben Brooks, Songzi Du, Vitor Farinha Luz, Jonathan Libgober, Eran Shmaya, Alex Wolitzky,Doron Ravid and Gabriel Carroll for their helpful comments. I am particularly thankful to Chris Ryan and Yeon-KooChe, for their support and generosity with their time. † University of Chicago Booth School of Business. a r X i v : . [ ec on . T H ] M a r e consider a setting in which a seller sells a single good to one of n bidders. The seller lacksknowledge of the joint distribution of valuations, including the correlation structure between bidders’valuations and their exact marginal distributions. Instead, the seller knows the range of the bidders’valuations and the (possibly asymmetric) means of each bidder’s marginal distribution. Additionally,the seller knows that the bidder’s valuations are private and bidders always play undominatedstrategies, but he is uncertain about the information bidders have about their opponents’ values.The objective of this paper is to find the selling mechanism which obtains the optimal revenueguarantee in this setting, the highest revenue the seller is guaranteed to achieve regardless of thejoint distribution and possible information bidders have about their opponents. To solve for therevenue guarantee, we use an equivalent characterization of the revenue guarantee as the equilib-rium revenue of a simultaneous-move, zero-sum game between the seller and an adversarial nature.Nature chooses the joint distribution, consistent with the mean and range constraints, as well asthe information bidders possess on their opponents’ values, in order to minimize the seller’s rev-enue given the mechanism, and the seller chooses the mechanism to maximize his revenue given theworst-case joint distribution.We show that the second-price auction with a random, symmetric reserve price obtains theoptimal revenue guarantee, within a wide class of mechanisms, we call competitive mechanisms.Competitive mechanisms comprise of all Bayesian individually rational and incentive compatiblemechanisms that never assign the good to “noncompetitive” bidder-types – bidder types for whomwith probability one there exists another bidder with higher valuation. Competitive mechanismsinclude all standard auctions with or without reserve prices. It also includes auction mechanismsthat treat different bidders differently, e.g, with binding reserve prices that differ across bidders.The optimality is not trivial as we exhibit the distribution and information about opponents forwhich the first auction generates strictly lower revenue. Furthermore, we find that as the numberof bidders n (with symmetric means) grows large, the optimal reserve price becomes non-binding –effectively zero – with probability that grows to one.This result is appealing in a number of aspects. First, even though competitive mechanismsencompass a broad class of mechanisms, including those that depart from classic auctions, a standardauction format emerges as the optimal mechanism. The robust optimality of standard auction, inparticular, the second price – or its more practical equivalent, English auctions – accords well withits prevalence in practice. Second, the optimal mechanism is symmetric across bidders, even thoughthe seller may know bidders are asymmetric ex-ante (i.e. bidders may have asymmetric meanconstraints) and the class of mechanisms we consider allows for asymmetric treatment. Finally, incontrast with Myerson (1981), the optimal reserve price depends on the number of bidders, andbecomes negligible as the number of bidders increases.This paper joins the growing literature of robust mechanism design, beginning with Scarf (1957).There is much work that addresses robust design in auctions, such as Bergemann et al. (2016), Brooksand Du (2019), and Du (2018). The papers closest to ours are: He and Li (2019), Bergemann et See also Bergemann and Schlag (2011), Wolitzky (2016), Carroll (2015, 2017, 2019), Libgober and Mu (2019), andChen and Li (2018) for work on robust mechanism design in other environments or with alternative specifications. single buyer from an unknowndistribution subject to an arbitrary number of moment constraints, and solve the problem with aknown mean and range as a special case. This paper can be seen as a generalization of this modelto n ≥ potential buyers. While the method based on duality is similar, the application of themethod as a verification tool becomes more demanding. While their method requires verifying thevalue distribution in a single dimensional case, the current setup with multiple bidders requires usto conjecture the full-fledged joint distribution with nontrivial correlation among bidders’ valuation,as will later become clear. Furthermore, our analysis, by varying the number of bidders, allows usto study the impact of competition. More importantly, the choice of a selling mechanism (i.e., theauction format) has no analogue in that paper.Kocyigit et al. (2019) studies Stackelberg version of our robust second price auction problem,in which nature has a second-mover advantage and fully knows the reserve price before choosingthe worst-case distribution subject to the symmetric mean and range constraints. They also findan optimal highest-bidder lottery mechanism, which implements the same outcome as our secondprice auction under nature’s chosen distribution, and they prove the optimality of this mechanismwithin a smaller class of mechanisms – ex-post incentive compatible mechanisms in which only thehighest value bidder is allocated the object.He and Li (2019) study the second-price auction within a setting in which a seller is uncertainover the joint distribution of valuations but knows the exact marginal distributions . Nature in ourmodel can be seen as not only choosing the correlations between bidders’ valuations, but also the marginal distribution for each bidder. This additional choice matters. As will be seen later, theoptimally chosen marginal distribution involves both mass points and a smooth density, which isoutside the set of marginal distributions they consider. Further, the optimal marginal distributionchanges with the number of bidders, which they do not allow for. Most importantly, the auctionrule is fixed in their paper to the second-price auctions, whereas the second-price auction is shownto be optimal within a large class of mechanisms in the current paper.Finally, Bergemann et al. (2019) perform a similar exercise of comparing standard auction mecha-nisms according to their revenue guarantee. They study interdependent value environments in whichthere is a commonly known value distribution which is symmetric across bidders, but the seller isuncertain about the signals possessed by the bidders. They find a revenue guarantee optimality ofthe first-price auction among standard auctions, and revenue equivalence among standard auctionsunder the restricted domain of common value with symmetric affiliated signals. Our model doesnot assume a common knowledge of value distribution, and we allow for asymmetric means and abroader class of mechanisms. At the same time, we do not allow for interdependent values. In this Strictly speaking, the optimal highest-bidder lottery differs from a second-price auction with a random reserveprice, as it involves a different allocation rule. The difference becomes irrelevant at the optimal distribution chosenby nature, as it puts zero probability to the types where the allocation rules differ. Due to this difference, there is nological relationship between the two results, even in the symmetric means case. They assume that the exogenously given marginal distribution F admits positive density f everywhere. Theyfurther require that xf ( x ) is weakly increasing in x . These conditions fails for our optimally chosen marginal distri-bution. Consider a seller selling a single good to one of n bidders. The valuations of the bidders aredistributed according to a distribution F , which is unknown to the seller. Instead, the seller knowsthat the support of F is contained in V := [0 , n and knows the means for each of bidders’ marginaldistributions { m i } ni =1 , where m i ∈ (0 , for each bidder i . Without loss, we index the bidders andthe means in descending order of the means (unless specified otherwise): m ≥ m ≥ ... ≥ m n . Wedenote the set of probability distributions satisfying these constraints as P ( V, { m i } ni =1 ) .The seller also knows that each bidder i observes his value v i , which makes this a private valueauction. However, he is uncertain about bidders’ information regarding their opponents’ values.We denote I = { I i } ni =1 ∈ I to be the information structure that defines the information possessedby the bidders about their opponents’ values, where I is the set of all private value informationstructures. Formally, each I i is a Blackwell experiment that consists of a set of signals and a systemof conditional distributions of these signals for each realization of v i . Since the analysis does notdepend on the formal representation of the information, we do not specify it for notational ease.We now describe a class of auction rules and the equilibria they induce. An auction induces aBayesian game, and we assume that the seller knows that bidders only play undominated strategies.We represent such an equilibrium as a direct mechanism. A direct mechanism M = ( x , t ) =( { x i } ni =1 , { t i } ni =1 ) specifies an allocation function x i : V → [0 , and a payment rule t i : V → R for each bidder. Each bidder i obtains utility as a function of their reported type: u i ( v | v i ) = v i x i ( v, v − i ) − t i ( v, v − i ) . We interpret ( x , t ) as the outcome that emerge from an auction gamewhen bidders play Bayesian Nash equilibrium strategies which depend on their types as well as theinformation they possess. A direct mechanism is Bayesian individually rational (IR) if for all bidders i , U i ( v i ) ≥ , ∀ i, v i (IR)and Bayesian incentive-compatible (IC) if U i ( v i ) ≥ U i (ˆ v i | v i ) , ∀ i, ˆ v i , v i , (IC)in which U i (ˆ v | v ) := E v − i [ u i (ˆ v, v − i ) | v i = v ] is bidder i ’s interim utility function when their typeis v and their reported type is ˆ v , and U i ( v i ) := U i ( v i | v i ) . Both conditions are necessary for themechanism M to correspond to a Bayesian Nash equilibrium of some auction rule.Let R ( F, I, M ) denote the expected revenue to the seller under value distribution F ∈ P ( V, { m i } ni =1 ) ,information structure I , and mechanism m ∈ ˆ M , which is simply the sum of the expected transfersfrom each bidder: Formally, the object is a projection into the type profile after integrating out the signals they may observe at thetime of play. ( F, I, M ) = n (cid:88) i =1 E ( F,I ) [ t i ( v )] . We now introduce an adversarial nature in order to model the seller’s uncertainty. The seller’sbelief is the worst-case value distribution in P ( V, { m i } ni =1 ) consistent with his knowledge over therange and the means of the marginal distributions, and the worst-case information structure in I .Nature chooses ( F, I ) ∈ P ( V, { m i } ni =1 ) ×I in order to minimize the seller’s expected revenue. Natureand the seller play a zero-sum, simultaneous move game, in which the seller chooses mechanism M in order to maximize his utility given the worst-case distribution ( F, I ) , and nature chooses ( F, I ) to minimize the revenue given M .We define an equilibrium of this game to be a value distribution and information structure ( F ∗ , I ∗ ) and mechanism M ∗ ∈ ˆ M within some admissible class ˆ M , which we shall specify in Section4, such that: R ( F ∗ , I ∗ , M ) ≤ R ( F ∗ , I ∗ , M ∗ ) ≤ R ( F, I, M ∗ ) , (1)for all F ∈ P ( V, { m i } ni =1 ) , information structures I ∈ I , and mechanisms M ∈ ˆ M .Note that the equilibrium condition is equivalent to finding a saddle point of R . A well-knownproperty of saddle points is that if there exists ( F ∗ , I ∗ ) and M ∗ that satisfy (1), then: R ( F ∗ , I ∗ , M ∗ ) = sup M inf F,I R ( F, I, M ) = inf
F,I sup M R ( F, I, M ) . This max-min characterization implies that all equilibria will be payoff equivalent, even though therecould be multiple equilibria that generate this revenue.We proceed our analysis in two steps. In Section 3, we fix the auction rule to be a second-price auction and allow the seller to choose reserve price optimally against worst case distribution F ∗ chosen by nature. We prove the saddle-point inequalities among this class of mechanisms, andillustrate some properties of the equilibrium. In Section 4, we prove that the optimal second-priceauction obtains the optimal revenue guarantee among a wide class of mechanisms we refer to as competitive mechanisms (which we will formally define later). We also provide an explicit counter-example, illustrating that this does not hold generally among all Bayesian mechanisms. In this section, we restrict our attention to second-price auction mechanisms and solve for the optimalreserve price within this class of mechanisms. We also find the worst-case distribution, with which By the max-min inequality, sup M inf F,I R ( F, I, M ) ≤ inf F,I sup M R ( F, I, M ) . From the saddle point condition,we can obtain the reverse inequality: inf F,I sup M R ( F, I, M ) ≤ sup M R ( F ∗ , I ∗ , M ) ≤ R ( F ∗ , I ∗ , M ∗ ) ≤ inf F,I R ( F, I, M ∗ ) ≤ sup M inf F,I R ( F, I, M ) . Combining these two inequalities, we obtain the desired equality: R ( F ∗ , I ∗ , M ∗ ) = sup M inf F R ( F, I, M ) =inf F sup M R ( F, I, M ) . The seller sells a single good in a second price auction to one of n bidders. We denote the second-price auction mechanism as S G , where G is a (possibly random) reserve price that applies to allbidders identically. Formally, G is a probability distribution over [0 , and the mechanism S G is: x i ( v ) = G ( v i ) · { v i = v (1) } t i ( v ) = E r ∼ G (cid:104) max { v (2) , r } · { v (1) >r } (cid:105) · { v i = v (1) } . As mentioned in the general setting, the seller’s belief over valuations and his reserve pricestrategy is determined as the outcome of zero-sum, simultaneous move game between him and anadversarial nature. The seller chooses a (possibly degenerate) reserve price G ∈ P ([0 , given theworst-case distribution F . Nature chooses the distribution of bidders’ valuations F ∈ P ( V, { m i } ni =1 ) given the seller’s reserve price G .Given our restriction in equilibria with undominated strategies, bidders will bid their valuationsin the second-price auction setting. Note also that the information structure I ∈ I chosen by naturedoes not affect the seller’s revenue, since bidders will play their dominant strategies regardless ofthe realized signals. In other words, R ( F, I, S G ) = R ( F, I (cid:48) , S G ) , ∀ F, I, I (cid:48) , G.
Because of this, we will ignore the information structure and focus solely on the value distribution.Throughout this section, we will assume that F is equipped with I = { I i, } ni =1 , or the null infor-mation structure, in which all bidders have complete information of their own types, but have noinformation of their opponents’ types.Given this behavior, we denote the expected revenue given F and G , as follows: Ψ( F, G ) := R ( F, I , S G ) = m (cid:88) i =1 E F [ t i ( v )] = (cid:90) V (cid:90) max { v (2) , r } · { v (1) >r } dG ( r ) dF ( v ) , where v ( i ) is the i th highest component of v ∈ V .Since the auction format is fixed and the information structure is irrelevant (and set to I withoutloss), an equilibrium of this restricted game between nature and the seller is defined to be a pair ofstrategies G ∗ ∈ P ([0 , and F ∗ ∈ P ( V, { m i } ni =1 ) such that: Ψ( F ∗ , G ) ≤ Ψ( F ∗ , G ∗ ) ≤ Ψ( F, G ∗ ) , (2)for all F ∈ P ( V, { m i } ni =1 ) and G ∈ P ([0 , . The equilibrium revenue can be interpreted as theoptimal revenue guarantee among reserve price strategies:6 ( F ∗ , G ∗ ) = sup G inf F Ψ( F, G ) = inf F sup G Ψ( F, G ) . Toward finding the revenue guarantee, our strategy will be to present a conjectured strategyprofile and verify that it satisfies the saddle-point inequalities. Before presenting the main result ofthis paper, which provides the solution with n ≥ bidders, it is helpful to first illustrate the solutionin a simpler setting. We illustrate our results with the case of two bidders with symmetric mean m ∈ (0 , . First, observethat as in the single buyer case in Carrasco et al. (2018), there is no pure strategy equilibrium.Assume, for the sake of argument, that the marginal mean is m = 1 / . Suppose, to the contrary,there were a pure strategy equilibrium in which the seller chooses a (deterministic) reserve price p .Clearly p < m , or else nature will put all mass on the value profile ( m, m ) , resulting in no sale.Given p < m , nature’s best response will put positive mass at ( p, p ) , in which case no sale occursand the seller receives zero revenue (assuming ties are broken in favor of nature). However, if theseller lowers his reserve price to some p (cid:48) < p , the reserve price is no longer binding at ( p, p ) and asale occurs with revenue equal to p . The deviation is strictly profitable, so there is no pure strategyequilibrium.Figure 1: φ ( v ; G ∗ ) and the support of F ∗ in the two bidder equilibrium ( m = / ) . Notes:
On the left are the expected revenue function φ ( v ; G ∗ ) (in red) induced by the seller’s equilibrium strategy G ∗ , thesupporting affine function L ( v ) (in blue), and the intersection of the two (in bold red). On the right, the support of F ∗ (in boldred) and the mean point. When m = , α ≈ . . We begin with our conjecture for nature’s strategy. The absence of a pure strategy equilibriumsuggests that nature cannot always prevent a sale from occurring. Hence, a reasonable conjecture isa strategy that minimizes the seller’s revenue when a sale occurs. This means that nature sets thevalue of the second highest bidder to be as low as possible, perhaps to some lower bound α (which This is generally the case for any n ≥ bidders and for any m ∈ (0 , as can be seen in the Stackelbergsecond-price auction equilibrium in Kocyigit et al. (2019).
7s yet to be determined). As for the valuation of the highest bidder, we conjecture that nature willpick this value by randomizing with some probability distribution H ( v ) over ( α, .Of course, this is simply a conjecture at this point, and one will need to verify that it formsan equilibrium. Later, we will prove this is indeed an equilibrium strategy in a general setting.Summing up, this conjectured strategy F ∗ randomly picks one bidder to be the highest bidder,producing value profiles of the form ( v, α ) or ( α, v ) , where α is some fixed value in (0 , and v isdrawn from some H ( v ) over ( α, . Visually, the support of F has an “L”-shape, as can be seen inthe right panel of Figure 1. Formally, supp ( F ∗ ) = S ∪ S , where S := { ( v, α ) : v ∈ ( α, } and S := { ( α, v ) : v ∈ ( α, } . Each segment S i contains therealized value profiles when the i th bidder is chosen to be the highest (and the other the secondhighest).In order to pin down H ( v ) , we consider strategies which make the seller indifferent across all(possible) reserve prices. We define η ( p ; F ) to be the expected revenue facing the seller: η ( p ; F ) = (cid:90) V max { v (2) , p } · { v (1) >p } dF = · p (1 − H ( p )) if p > αα if p ≤ α = p (1 − H ( p )) if p > αα if p ≤ α. For the seller to be indifferent across reserve prices, we require that for all p ∈ ( α, : p (1 − H ( p )) = α, or H ( p ) = p − αp . Since lim p → H ( p ) = 1 − α , there must be a mass of α at . Intuitively, for the seller to earn α even for p arbitrarily close to 1, nature must put a mass of α at . Our conjecture for H is therefore: H ( v ) = if v ≤ α v − αv if v ∈ ( α, if v = 1 . This distribution has density h ( v ) = αv on v ∈ ( α, and a mass point of α at v = 1 .We can pin down α using the mean constraint on F . Each bidder i is chosen to be the highestbidder with probability / , in which case his valuation is determined by H . With the remainingprobability, he is chosen to be the second highest bidder, with valuation α . Hence, the mean of eachbidder’s marginal distribution is: 8 = 12 α + 12 (cid:18) α + (cid:90) α vh ( v ) dv (cid:19) = α + 12 (cid:90) α αv dv = α (cid:18) −
12 ln( α ) (cid:19) , This equation pins down a unique solution for α , as we will later argue in the general n biddersetting. When m = 1 / , α = 0 . , as is depicted in Figure 1.Now we will turn to the problem of finding a G ∗ so that this conjectured F ∗ will be a bestresponse by nature. If we take a step back, we can see that for any given strategy G by the seller,nature faces the following problem: inf F (cid:90) V φ ( v ; G ) dF (3)s.t. (cid:90) V v i ( v ) dF = m (cid:90) V dF = 1 F ≥ , where v i : V → R is the projection map on the i th bidder’s valuation and φ is the expected revenuefrom nature’s point of view, φ ( v ; G ) := (cid:82) max { v (2) , p } · { v (1) >p } dG. It is useful to observe that this is a linear program in F . Given our conjecture for F ∗ , we can usethe dual of this program to verify the optimality of F ∗ with respect to the expected revenue function φ induced by the seller’s strategy G . We use a complementary slackness condition expressed in thefollowing lemma. Note also that this lemma applies generally to the n -bidder case, and in fact, wewill reuse this lemma to prove the optimality of our candidate F ∗ in the general case. Lemma 1.
Given the seller’s strategy G , F ∗ is an optimal solution for nature’s problem if and onlyif F ∗ is feasible and there exists an affine function L : V → R such that L ( v ) ≤ φ ( v ; G ) , ∀ v ∈ V and supp ( F ∗ ) ⊆ { v : L ( v ) = φ ( v ; G ) } . Proof.
We prove the “if” part. The proof of the “only if” part, which requires the dual program, isrelegated to the Appendix.Suppose there exist feasible F ∗ and an affine function L satisfying the requirements. For any F ∈ P ( V, m ) , this implies (cid:90) V φ ( v ; G ) dF ∗ = (cid:90) V L ( v ) dF ∗ = (cid:90) V L ( v ) dF ≤ (cid:90) V φ ( v ; G ) dF. The first equality follows because φ and L coincide on the support of F ∗ . The second equalityfollows because L is affine and F and F ∗ have the same mean. The final inequality follows since L ( v ) ≤ φ ( v ; G ) for all v . Hence F ∗ is an optimal solution for nature.9his lemma states that in order for our conjectured F ∗ to be a best response to the seller’sstrategy G , there must exist an affine function L ( v ) such that expected revenue function φ inducedby G coincides with L on the support of F ∗ , and is everywhere above L . The key implication of thislemma is that if we can find a strategy G ∗ and an affine function L such that the conditions of thelemma are satisfied, then ( F ∗ , G ∗ ) is an equilibrium.This duality lemma borrows from the approach used in Scarf (1957) in the context of inventorymanagement, in which he constructs a supporting quadratic polynomial to solve for the worst-casedemand distribution under a mean and variance constraint. This technique has also been used inthe context of monopoly pricing in Carrasco et al. (2018), in an optimal transport program for anauction setting in He and Li (2019), and in Bayesian persuasion in Dworczak and Martini (2019).One difference relative to Scarf and several other recent work is that the primal objective function φ is itself endogenous. Unlike finding the dual for a fixed φ , we are simultaneously choosing G and L to satisfy complementary slackness.For any given G , it is without loss to first examine φ ( v ; G ) on S , the subset of the “L”-shapesupport in which nature picks the bidder 1 to be the highest bidder. On S , the expected revenuefunction will be: φ ( v ; G ) = (cid:90) max { v (2) , p } · { v (1) >p } dG = (cid:90) v v dG ( p ) + (cid:90) v v pg ( p ) dp, assuming that G admits density g ( p ) , which will be the case as we show below. For the conjectured F ∗ to be optimal, Lemma 1 requires that φ ( v ; G ) must be equal to some affine function L ( v ) = c v + c v + b on this segment. This implies that for all ( v, α ) ∈ S : ∂φ∂v ( v, α ) = vg ( v ) = c , or g ( v ) = c v . This logic applies symmetrically if nature had picked bidder 2 to be the highest bidder instead,so we can consider c = c = c . This pins down the form of the sender’s strategy for p ∈ [ α, . Inorder to satisfy the remaining conditions of the Lemma, we now can find c so that φ ≥ L for allvalue profiles.A necessary condition for φ ≥ L is that on each of the segments S i of supp ( F ∗ ) , the partialderivative of φ with respect to the second highest bidder must be greater than or equal to c . Takinga point ( v, α ) ∈ S we find: ∂φ∂v ( v, α ) = G ( α ) ≥ c. We can set c = G ( α ) , which is the mass the sender assigns for reserve prices below α . Using thefact that G must integrate to , we find: 10 = (cid:90) dG ( p ) = G ( α ) + (cid:90) α G ( α ) p dp = G ( α )(1 − ln( α )) , so we have: c = G ( α ) = 11 − ln( α ) . We prove the optimality of the strategy where the seller places the entire mass G ( α ) at p = 0 ,but there could be other payoff-equivalent equilibria in which the seller may put density on [0 , α ] .Note that unlike the equilibrium for single buyer case in Carrasco et al. (2018), the support of theseller and nature do not fully coincide. In other words, with positive probability, the seller choosesa reserve price that guarantees sale. As we will see later, this probability increases as the numberof bidders increases. The intuition is that as competition increases, the seller can rely less on a“binding” reserve price to guarantee revenue as competition between the bidders limits the abilityof nature to suppress revenue.Finally, using the condition that φ = L at ( α, α ) , we determine the strategy G ∗ and the affinefunction L : G ∗ ( p ) = (cid:82) pα r (1 − ln( α )) dr + − ln( α ) if p > α − ln( α ) if p ≤ αL ( v ) = 11 − ln( α ) ( v + v ) − α − ln( α ) . As it turns out, φ ( v ; G ∗ ) ≥ L ( v ) for all v ∈ V and φ = L on the support of F ∗ , satisfying theconditions of Lemma 1. The left panel of Figure 1 visualizes how φ and L satisfy this requirementfor the case of m = 1 / . Lemma 1 guarantees that F ∗ will be a best response to G ∗ and due tothe construction of F ∗ , which makes the sender indifferent across all reserve prices, G ∗ is a bestresponse to F ∗ . Thus, we have found an equilibrium for the two-bidder game with symmetric meanconstraints.Two differences emerge with the introduction of the second bidder (compared to the monopolypricing equilibrium in Carrasco et al. (2018)). First, as mentioned earlier, the seller puts massstrictly below the support of the marginal distribution of F . Second, the marginal distribution of F has two mass points, one at the upper bound v = 1 (as in the single buyer equilibrium), and one atthe lower bound α (unlike the single buyer equilibrium). As mentioned before, the marginal distribution of the optimal F ∗ is outside the class of marginal distributionsconsidered in He and Li (2019), which does not allow for mass points. The authors also require that xf ( x ) is weaklyincreasing in x , where f is the exogenously given density. Even in the region where the marginal distribution admitsdensity h , the density fails this condition: vh ( v ) = αv v = αv , which is decreasing in v . .3 Optimal reserve price with n bidders In this section, we present a profile of candidate strategies ( F ∗ , G ∗ ) that form a second-price auctionequilibrium in the n -bidder case with (possibly) asymmetric mean constraints. The equilibrium is anatural extension of the two bidder, symmetric mean equilibrium. In Appendix A.3, we will provethat these strategies satisfy the saddle point inequalities. Nature’s strategy.
Nature’s worst case distribution retains the “L”-shape structure, randomlychoosing a bidder to be the highest bidder (with value distributed according to H ( v ) ) while settingthe valuations of the remaining bidders to a lower bound α . However, a few modifications areintroduced when there are bidders with asymmetric means. First, the “L”-shape joint distributionmay not include all n bidders. If there are bidders with mean constraints strictly less than α ,they cannot be included in this distribution without violating these constraints. Instead, naturewill set the marginal distributions of these bidders to a point mass on their means. Second, the“active” bidders who are included in the distribution are selected as the highest bidder with differentprobabilities, reflecting differences in their means.We will proceed to describe the set of active bidders and the joint distribution among the activeand inactive bidders. Selection of Active Bidders.
Given the mean constraints { m i } ni =1 for each of the n bidders,sorted in descending order m ≥ m ≥ ... ≥ m n , we can find the cutoff k for which all bidders withmean m i ≥ m k will be included in the “L”-shape joint distribution, and all remaining bidders willbe excluded. k := min (cid:40) (cid:96) : m j > α (cid:96) ( ¯ m (cid:96) ) , for j ≤ (cid:96)m p ≤ α (cid:96) ( ¯ m (cid:96) ) , for p > (cid:96) (cid:41) , (4)where α (cid:96) ( x ) := (cid:26) α : x = α (cid:18) − (cid:96) ln( α ) (cid:19)(cid:27) and ¯ m (cid:96) := 1 (cid:96) (cid:96) (cid:88) i =1 m i In other words, the cutoff k is determined so that all bidders with means m i ≥ m k will havemeans above the lower bound α = α k ( ¯ m k ) , which is determined by the mean of the mean constraintsof all such bidders. All bidders with m i < m k , will have means strictly below the lower bound andfor the reasons described earlier, will not be included in the “L”-shape distribution. Note that if themean constraints are symmetric for all n bidders, then k = n . Lemma 2.
The cutoff k in (4) always exists.Proof. See Appendix A.2.Lemma 2 states that given any n bidders { m i } ni =1 , we can always separate the bidders into activeand inactive bidders (the proof of Lemma 2 provides a simple algorithm for finding k ).12or example, if there are n = 3 bidders with m = 0 . , m = 0 . , and m = 0 . , the cutoff k will equal since ¯ m = m + m = 0 . , α ( ¯ m ) ≈ . , and m , m > α ( ¯ m ) with m < α ( ¯ m ) .There will be two active bidders, since the mean of the third bidder’s valuation is too low.Figure 2: Support of F ∗ with symmetric and asymmetric mean constraints ( α,α,α ) Bidder 1 Bidder 2 B i dd e r ( α,α,m ) Bidder 1 Bidder 2 B i dd e r Notes:
On the left panel is an example of F ∗ with n = 3 bidders with symmetric mean constraints. On the right panel is anexample of F ∗ with n = 3 bidders but only k = 2 active bidders, in which m < α ( ¯ m ) . Worst-case Distribution.
With the cutoff k from (4) in hand, we can now describe thestructure of Nature’s worst-case distribution: F ∗ ( U ) = k (cid:88) i =1 θ i S i · H ( v i ( U )) , (5)where S i := { v i ( x ) : x ∈ ( α, } , v i ( x ) := (cid:40) v : v i = x, v j = α, j ≤ kv l = m l , l > k (cid:41) α := α k ( ¯ m k ) ,θ i := m i − α − α ln( α ) , and U ⊂ V is any measurable set. v i : V → R is the projection map on the i th bidder’s valuation,and S i is the indicator function of the set S i . The support of F ∗ consists entirely of value profiles v i ( x ) of the form: v i ( x ) = ( α, α, ..., x, ..., α (cid:124) (cid:123)(cid:122) (cid:125) k active bidders , m k +1 , m k +2 , ..., m n (cid:124) (cid:123)(cid:122) (cid:125) n − k inactive bidders ) , where x > α is the value of the highest bidder.Finally, H is the measure defined by the CDF of the highest bidder value distribution (whichremains the same as in the two bidder, symmetric mean case):13 ( v ) = if v ≤ α v − αv if v ∈ ( α, , if v = 1 . Nature selects bidders i = { , ..., k } as active bidders and bidders { j = k + 1 , ..., n } as inactivebidders. Each inactive bidder j is assigned degenerate value of m j . Each active bidder i is chosenwith probability θ i to be the highest bidder, in which case all other active bidders have degeneratevalue α = α ( ¯ m k ) , and bidder i ’s value is distributed according to H . Effectively, we can considerthis to be a second-price auction with only k active bidders.Note that θ i is determined by (5) to satisfy the mean constraint for bidder i . Specifically, themeans of the marginal distributions for each of the k bidders is as follows: (1 − θ i ) α + θ i (cid:18) α + (cid:90) α vh ( v ) dv (cid:19) = α (cid:18) − m i − α − α ln( α ) ln( α ) (cid:19) = m i , Naturally, the higher the mean m i , the higher the selection probability θ i . In other words, θ ≥ θ ≥ ... ≥ θ k . Nature adjusts the selection probabilities to “soak up” all the asymmetries, whileensuring that the value distribution of the highest bidder is the same (given by H ) regardless ofwhich bidder is chosen. In an important sense, Nature eliminates asymmetries from the perspectiveof the seller, which ultimately eliminates the need for the seller to treat bidders differently.The second-highest value, and therefore the auction revenue, will always be equal to α under thisdistribution. The explicit characterization of α can be found by summing up the equations impliedby the mean constraints: k (cid:88) i =1 m i = k (cid:88) i =1 α (1 − p i ln( α ))1 k k (cid:88) i =1 m i = ¯ m k = α (cid:18) − k ln( α ) (cid:19) When α = 0 , the RHS equals and when α = 1 , the RHS is 1. Since it is continuous and strictlyincreasing, there exists a unique α ∈ (0 , such that this equation holds. Furthermore, α < ¯ m k since ln( α ) < . Explicitly, the α that solves this equation is: α = exp (cid:18) k + W − (cid:18) − k ¯ m k e k (cid:19)(cid:19) , where W − is the lower branch of the Lambert W function. Note that if the means of all n biddersare symmetric, then: The mean constraints for the n − k bidders are satisfied automatically, since their valuations are always equal totheir means. = α (cid:18) − n ln( α ) (cid:19) . (6) Seller’s Strategy.
The seller’s strategy is to randomize his reserve price according to the CDF: G ∗ ( p ) = (cid:82) pα r ( k − − ln( α )) dr + k − k − − ln( α ) if p > α k − k − − ln( α ) if p ≤ α. (7)In other words, the seller chooses p ∈ [ α, with density g ( p ) = p ( k − − ln( α )) and places a point massof k − k − − ln( α ) on the reserve price p = 0 .Note that from the seller’s perspective, the particular asymmetries between bidders do not mat-ter, only insofar as the number of active bidders k and the revenue α are affected. This is because allthe payoff-relevant features remain the same regardless of which bidder is picked as the winner: thehighest bidder’s value is always drawn from H ( v ) and the second-highest value is always α . Hence,the seller’s optimal reserve price is symmetric across bidders.The pair of strategies ( F ∗ , G ∗ ) form an equilibrium of this game. We will later verify that thesestrategies satisfy the saddle point condition. Theorem 1.
The strategy profile ( F ∗ , G ∗ ) satisfies the saddle point condition in (2) and thereforeare an equilibrium of the zero sum game. The equilibrium revenue is Ψ( F ∗ , G ∗ ) = α .Proof. See Appendix A.3.We explore several implications of this theorem. First, suppose there are n bidders with sym-metric means { m i } ni =1 = m . Let G ∗ n and F ∗ n be the equilibrium strategy in this auction. Corollary 1.
Given a sequence of auctions with symmetric mean bidders { m i } ni =1 = m , thereexists a sequence of equilibria ( G ∗ n , F ∗ n ) such that the random reserve price G ∗ n converges to zero inprobability as n → ∞ . Furthermore, the equilibrium revenue Ψ( F ∗ n , G ∗ n ) increases in n and convergesto m .Proof. We define α ( n ) to be the α which satisfies equation (6) when there are n bidders and sym-metric means equal to m . First, we observe that α ( n ) is increasing in n . This is because the RHSof (6) is decreasing in n , for any α ∈ (0 , . Hence, as n increases, the RHS shift down. Since RHSis a increasing function of α , the α satisfying (6) must increase as n increases. Furthermore, as n goes to infinity the RHS converges to α , which implies that α converges to m .The second part of the statement results from observing that the reserve price associated with G ∗ defined in (7) converges in probability to zero as n → ∞ , as the probability assigned to reserveprices above zero vanishes in the limit: G ∗ ((0 , G ∗ ([ α ( n ) , − ln( α ( n )) n − − ln( α ( n )) → . We consider the symmetric mean case to analyze asymptotics, since this allows us to take the number of bidders n → ∞ , without worrying about the number of active bidders, which may remain finite as n → ∞ . orollary 2. Given an auction with n (cid:48) bidders and another with n bidders with n (cid:48) > n , whereall bidders have the same mean m , the equilibrium distribution of the highest bidder’s value H n (cid:48) first-order stochastically dominates H n .Proof. The difference H n ( v ) − H n (cid:48) ( v ) for all v ∈ [0 , is as follows: H n ( v ) − H n (cid:48) ( v ) = if v ≤ α ( n ) v − α ( n ) v if v ∈ ( α ( n ) , α ( n (cid:48) )] α ( n (cid:48) ) − α ( n ) v if v ∈ ( α ( n (cid:48) ) , if v = 1 . As shown in the proof of Corollary 1, α ( n ) is increasing in n . Hence, for all v ∈ [0 , , H n ( v ) ≥ H n (cid:48) ( v ) .Corollary 1 says that as the number of bidders grows large, the equilibrium revenue convergesto the mean of a single bidder’s valuation, which implies that the seller extracts a single bidder’ssurplus in the limit. Additionally, it states that there exists a sequence of equilibria such that theseller’s reserve price converges to zero in probability, as the number of bidders goes to infinity (Figure3 shows the effect of increased competition from n = 2 to n = 10 ). The asymptotic behavior ofthe equilibrium reserve price strategy is consistent with the widespread observation that reserveprices tend to be lower than is prescribed by the standard (Bayesian) theory under the estimateddistribution of value profiles, as noted by Haile and Tamer (2003) in timber auctions, Bajari andHortacsu (2003) in eBay auctions, and McAfee et al. (2002) in real estate auctions.A key feature of this equilibrium is that the seller randomizes in reserve price. Since naturecan only pick one worst-case distribution, randomization allows the seller to hedge against nature.Randomization protects the seller against states of the world that are adverse for a particular reserveprice, and it forces nature to minimize revenue across multiple realizations.As the number of bidders increases however, the role of reserve price disappears, as the probabilityof binding reserve prices vanishes. Competition between bidders appears to “substitute” for a reserveprice. The intuition of this result is that competition constrains the seller’s worst case belief: asthe number of bidders increases, nature is must increase the value of the second-highest bidder α tomaintain the mean constraint. Furthermore, the marginal distribution of each bidder converges toa point mass on α (as each bidder is the second-highest bidder with probability n − n ). In the limit,the worst-case joint distribution becomes the point mass distribution on the value profile consistingof α for all bidders, which itself converges to m . The optimal strategy for the seller facing a pointmass on m = ( m, ..., m ) is to charge a reserve price of zero. By contrast, the optimal reserve price in the Stackelberg case converges deterministically to as n → ∞ , as seenin Kocyigit et al. (2019) and in the Supplemental Appendix. n = 2 and n = 10 ( m = / ). α m n − n Bidder i ’s valuation αm n − n Bidder i ’s valuation α m G ( α )1 Reserve Price αm G ( α )1 Reserve Price
Notes:
These are the equilibrium strategies for nature and the seller when n = 2 and n = 10 and all bidders have symmetricmeans m = 1 / . In red is the CDF of the marginal distribution of nature’s equilibrium strategy F ∗ . In blue is G ∗ , or the CDFof the seller’s strategy over reserve price. When n = 2 , α ≈ . and when n = 10 , α ≈ . . Observe that as the number ofbidders increases, the seller’s mass on p = 0 increases towards 1, and α increases towards the mean m . In this section, we explore the performance of the optimal second-price auction within a broad classof mechanisms and show that the second-price auction with an optimal reserve price is revenue-guarantee optimal within this class.We begin with some definitions. For each bidder, we say a type v i of bidder i is non-competitive if v i is the minimum of all of bidder i ’s possible valuations, and with probability one there is anotherbidder j with a higher valuation. A direct mechanism is a competitive mechanism if it is BayesianIR and IC and if it never assigns the good to a non-competitive bidder type. Formally, the set ofcompetitive mechanism is defined as ˆ M := { ( x, t ) : ( IR ) , ( IC ) , and x i ( v i , v − i ) = 0 if v i ∈ γ i ( F ) , ∀ F } , where γ i ( F ) := { v : v ∈ inf supp ( F i ) and ∃ j, F j ( v j > v | v i = v ) = 1 } , where inf supp ( F i ) refers to the minimum of the support of bidder i ’s valuation and F is the distri-bution of bidders’ valuations.The class of competitive mechanisms encompasses all standard auction mechanisms without17eserve prices or with a common reserve price. This includes all efficient mechanisms (which allocatethe good to the highest value bidder) but also includes some inefficient mechanisms, such as afirst-price auction without a reserve price. Example 1.
Suppose there are two bidders with valuations v ∼ U nif orm [1 , and v ∼ U nif orm [2 , iid . The only non-competitive bidder type is the first bidder with v = 1 , as is the minimum ofbidder ’s support and bidder will always have a higher valuation. In a first price auction withouta reserve price, bidder with v = 1 will never bid above , as that would be a dominated strategy.If bidder-type v = 1 wins with positive probability, then so too will all bidder types above (andwill enjoy strictly positive payoffs). This implies that a positive measure of bidder types will bidbelow , and not win the good. However, this is a contradiction as bidder always has v > , andtherefore must enjoy a strictly positive payoff. Therefore, the auction is a competitive mechanismin this setting.It also includes auction mechanisms in which the seller applies an asymmetric reserve price foreach bidder, as long as the reserve prices are binding. Note that we are not assuming a symmetricequilibrium across bidders. The mechanism may discriminate between bidders, as long as thiscompetitiveness constraint is satisfied.At the same time, this competitiveness constraint is not completely innocuous. As seen inExample 2 below, a first price auction with asymmetric reserve prices may not be competitive. Example 2.
Consider the two bidder setting with a first price auction, with valuations drawnindependently from
U nif orm [1 , and U nif orm [2 , as before. Now suppose the the seller imposesa reserve price of r = 2 . on the second bidder, but no reserve price for the first. Bidder withvalue v = 1 is a non-competitive bidder. However, the non-competitive bidder type can guaranteea positive payoff by bidding strictly below , as the binding reserve price for bidder ensures thatbidder will win with probability at least / , irrespective of his bid. Hence, this mechanism is notcompetitive.We find that the optimal second-price auction found in Section 3 obtains the optimal revenueguarantee within the class of competitive mechanisms. In other words, the equilibrium betweennature and the seller found in the second-price auction setting still holds even if we expand the setof possible selling mechanisms beyond second-price auctions. Theorem 2.
The second-price auction mechanism S G ∗ with the optimal reserve price G ∗ in (7)obtains the optimal revenue guarantee out of all competitive mechanisms. In other words, for any M ∈ ˆ M and ( F, I ) ∈ P ( V, { m i } ni =1 ) × I . R ( F ∗ , I , M ) ≤ R ( F ∗ , I , S G ∗ ) ≤ R ( F, I, S G ∗ ) , where F ∗ is the worst-case distribution in (5) and I is the null information structure. Theseinequalities imply: S G ∗ ∈ arg max m ∈ ˆ M inf F,I R ( F, I, M ) . roof. We have already proven that the inequality R ( F ∗ , I , S G ∗ ) ≤ R ( F, I , S G ∗ ) = R ( F, I, S G ∗ ) for all F ∈ P ( V, { m i } ni =1 ) and I ∈ I in the proof of Theorem 1. It remains to show that R ( F ∗ , I , M ) ≤ R ( F ∗ , I , S G ∗ ) for all M ∈ ˆ M .Take any mechanism M ∈ ˆ M . First, all of the n − k inactive bidders in F ∗ are non-competitivebidders, since they all have valuations below α and have a point mass support under F ∗ . They arenever allocated the good under any competitive mechanism M . Hence, we can focus solely on therevenue from the k active bidders.The expected revenue from any active bidder i is as follows: E ( F ∗ ,I ) [ t i ( v )] = E v i [ T i ( v )] = (cid:90) α vX i ( v ) − U i ( v ) dF ∗ i ( v ) = (cid:90) α vX i ( v ) − (cid:18)(cid:90) vα X i ( s ) ds (cid:19) dF ∗ i ( v ) , where F ∗ i is the marginal distribution for bidder i under F ∗ , T i ( v ) is the interim transfer to bidder i , X i ( v ) is the interim allocation, and U i ( v ) is the interim utility of bidder i , all when bidder i ’svaluation is v . Note that under the null information structure, bidders do not gain any additionalinformation of their opponents’ values from their signals. Hence, the interim values are solelydetermined by averaging over the opponents’ valuations according to F ∗ . Of course, since F ∗ doesnot exhibit independent marginal distributions for each bidder’s valuation, the interim values willdepend on bidder i ’s realized type.Recall that each active bidder in F ∗ is chosen as the highest bidder with probability θ i , withvaluation distributed according to H , and is a second-highest bidder with probability − θ i , with avaluation set to α . To simplify the expected revenue expression, we denote Z i ( v ) := (cid:90) vα X i ( s ) ds, and write (cid:90) α vX i ( v ) − Z i ( v ) dF ∗ i ( v ) = θ i · (cid:18)(cid:90) α vX i ( v ) − Z i ( v ) dH (cid:19) + (1 − θ i ) · ( αX i ( α ) − Z i ( α ))= θ i · (cid:18) α ( X i (1) − Z i (1)) + (cid:90) α [ vX i ( v ) − Z i ( v )] h ( v ) dv (cid:19) + (1 − θ i ) · ( αX i ( α )) We can simplify the expression 19 α [ vX i ( v ) − Z i ( v ) ds ] h ( v ) dv = (cid:90) α vX i ( v ) − (cid:90) α (cid:90) v h ( s ) ds · X i ( v ) dv = (cid:90) α (cid:18) v − H (1 − ) − H ( v ) h ( v ) (cid:19) X i ( v ) h ( v ) dv i = (cid:90) α [ J ( v ) X i ( v )] h ( v ) dv where H (1 − ) is the left limit of H at and J ( v ) is the virtual value. We can calculate the virtualvalue using the definition of H : J ( v ) = v − H (1 − ) − H ( v ) h ( v ) = v − (1 − α ) − v − αvαv = v + v − v = v . Plugging this expression back into the expected revenue, we obtain: E F ∗ [ t i ( v )] = θ i · (cid:18) α (cid:18) X i (1) − (cid:90) α X i ( s ) ds (cid:19) + (cid:90) α v X i ( v ) h ( v ) dv (cid:19) + (1 − θ i ) · ( αX i ( α ))= θ i · (cid:18) αX i (1) − α (cid:90) α X i ( s ) ds + α (cid:90) α X i ( v ) dv (cid:19) + (1 − θ i ) · ( αX i ( α ))= θ i · ( αX i (1)) + (1 − θ i ) · ( αX i ( α )) . Any active bidder with valuation equal to α is a non-competitive bidder. Conditional on bidder i having a valuation of α , some bidder j (cid:54) = i must have been selected as the highest bidder, sowith probability another bidder has a higher valuation. Furthermore α is the minimum possiblevaluation for any active bidder i . Hence, for any competitive mechanism M ∈ ˆ M , X i ( α ) = 0 . Wecan now sum up the revenue from all active bidders: R ( F ∗ , I , M ) = k (cid:88) i =1 E ( F ∗ ,I ) [ t i ( v )] = k (cid:88) i =1 θ i · ( αX i (1)) ≤ α = R ( F ∗ , I , S G ∗ ) . Hence, the optimal second price auction obtains the optimal revenue guarantee, out of mecha-nisms in ˆ M .What about general Bayesian mechanisms? One can find that facing ( F ∗ , I ) , the seller can dostrictly better with a mechanism that is not competitive. Consider a setting with two bidders withidentical means equal to m ∈ (0 , , with valuations distributed by F ∗ . Both bidders will be activebidders, and each will be selected as the highest bidder with probability / . Consider a mechanismwith the following allocation rule for the first bidder: x ( v ) = if v = ( α, α ) or ( q, q ) or ( y, α ) , q > α, y ∈ ( α, if v = ( α, y ) and y ∈ ( α, if v = (1 , α ) , α ) isfavored in the allocation unless the high value bidder has v = 1 , in which case the latter bidder getsthe good. This handicapping of the high value bidder eliminates the rents accruing to the high valuebidder, and generates higher revenue.In this mechanism, the interim allocation for bidders with v = 1 and v = α will be X i (1) = 1 and X i ( α ) = 1 − α respectively. Using the revenue expression above, the seller will receive revenue: (cid:88) i =1 E F ∗ [ t i ( v )] = (cid:88) i =1 (cid:20) X i (1) + 12 X i ( α ) (cid:21) = (cid:88) i =1 (cid:20) α + 12 α (1 − α ) (cid:21) = α + α (1 − α ) > α. The seller performs strictly better than in the optimal second price auction.
We have studied the seller’s optimal selling mechanism to multiple potential buyers, subject to mo-ment conditions of the distribution of bidders’ values. We have identified the worst-case distributionchosen by nature and the optimal reserve price strategy of the seller as the equilibrium of a zero-sum,simultaneous move game. We have found that the optimal second-price auction obtains the optimalrevenue guarantee out of a wide class of mechanisms, which includes the standard auction mecha-nisms. We have shown that in the optimal second-price auction, the seller sets a binding reserveprice with vanishing probability as the number of bidder increases.
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A.1 Proof of “Only if ” of Lemma 1
Proof.
To prove the converse of Lemma 1, we construct the dual of nature’s minimization problemin (3): max γ i ,η n (cid:88) i =1 γ i · m + η s.t. n (cid:88) i =1 γ i v i ( v ) + η ≤ φ ( v ; G ) , ∀ v ∈ V. The dual variables γ i and η together belong in R n +1 . This representation follows the dual of thesemi-infinite linear program in Anderson and Nash (1987).The primal is bounded above by , and the measure F = δ m , which assigns all mass to the meanpoint m = ( m, m, ..., m ) , is in the interior of the primal cone. By Theorem 3.12 in Anderson andNash (1987), strong duality holds.The dual solution ( γ ∗ , η ∗ ) defines an affine function L ( v ) = (cid:104) γ ∗ , v (cid:105) + η ∗ . Given the primalsolution, we can rewrite the dual in terms of this affine function. max L (cid:90) V L ( v ) dF s.t. L ( v ) ≤ φ ( v ; G ) , ∀ v ∈ V. The constraint of the dual guarantees that the affine function is below the expected revenuefunction. There is no duality gap, which means: (cid:90) V L ( v ) dF ∗ = (cid:90) V φ ( v ; G ) dF ∗ . Therefore, points at which L ( v ) < φ ( v ; G ) must only occur on a measure zero set with respectto F ∗ . By the definition of the support, we obtain: ∀ v ∈ supp ( F ∗ ) , L ( v ) = φ ( v ; G ) . A.2 Appendix: Proof of Lemma 2
Proof.
We can use a straightforward algorithm to find k .First, we begin with i := 2 , the bidder with the second-highest mean constraint. We calculate ¯ m i − and α i − ( ¯ m i − ) and check whether m i ≤ α i − ( ¯ m i − ) . If so, then we set k = i − . If not, thenwe increment the index i := i + 1 and repeat the same procedure. If i reaches n + 1 , then k = n .23here are finitely many bidders so this algorithm will terminate. Since k is returned by thealgorithm only if m k +1 ≤ α k ( ¯ m k ) (or if k is the last bidder), for all bidders with index p greaterthan k , m p ≤ m k +1 ≤ α k ( ¯ m k ) .It remains to show that m j > α k ( ¯ m k ) for all j ≤ k . It must be true that m k > α k − ( ¯ m k − ) ,otherwise k − would have been the outputted cutoff. It suffices to show that m k > α k ( ¯ m k ) .Suppose to the contrary that: α k − ( ¯ m k − ) < m k ≤ α k ( ¯ m k ) . The RHS of the equation x = α (cid:0) − (cid:96) ln( α ) (cid:1) is increasing in α , which means that for theequation satisfied by α k − ( ¯ m k − ) (shortening notation to α k − ): ¯ m k − = α k − (cid:18) − k − α k − ) (cid:19) < m k (cid:18) − k − m k ) (cid:19) , (8)and likewise ¯ m k = α k (cid:18) − k ln( α k ) (cid:19) ≥ m k (cid:18) − k ln( m k ) (cid:19) . (9)We can multiply (8) by k − and (9) by k and subtract the former from the latter to obtain: k ¯ m k − ( k −
1) ¯ m k − > m k m k > m k , which is a contradiction. Hence for all j ≤ k , m j ≥ m k ≥ α k ( ¯ m k ) .The k returned by the algorithm satisfies both conditions required for the cutoff and furthermore,it will be the minimum index that does so since we iterate from the smallest index . A.3 Appendix: Proof of Theorem 1
To prove Theorem 1, we will first show in Section A.3.1 that the seller’s strategy G ∗ is optimal given F ∗ . In Section A.3.2, we will prove that nature’s strategy F ∗ is optimal given G ∗ . A.3.1 Seller’s problem
First, we show that for the F ∗ described above, G ∗ is a best response by the seller. In other words,we will show the following inequality holds for all G ∈ P ([0 , : Ψ( F ∗ , G ) ≤ Ψ( F ∗ , G ∗ ) . Proof.
Recall that F ∗ has the following form: F ∗ ( U ) = k (cid:88) i =1 θ i S i · H ( P i ( U )) , k is the cutoff index from Lemma 2, which determines the number of active bidders. Thedistribution of the highest bidder’s value H ( v ) is: H ( v ) = if v ≤ α v − αv if v ∈ ( α, if v = 1 . Using the same notation as in the two bidder case, we let η ( p ; F ) denote the seller’s expectedrevenue under reserve price p given nature’s strategy F.η ( p ; F ) := (cid:90) V max { v (2) , p } · { v (1) >p } dF. Similar to the two bidder example, we will proceed to show that our candidate F ∗ makes theseller indifferent across all reserve prices (at least in [0 , ). Under F ∗ , if the seller sets the reserveprice below α , a sale will always occur and the seller will receive the second highest value, α . If theseller sets the reserve price above α , then he will make a sale with probability H ( p ) and receive thereserve price p as revenue. Hence, the seller’s payoff under F ∗ from a given reserve price is: η ( p ; F ∗ ) = (cid:80) ki =1 θ i · p (1 − H ( p )) if p > αα if p ≤ α = p (1 − H ( p )) if p > αα if p ≤ α. Note that the n − k inactive bidders do not affect the seller’s expected revenue, as the second-highest value α is always greater than their valuations. Note also that the number of bidders andtheir selection probabilities θ i only affect the seller through their impact on α . With the exceptionof α , the expected revenue to the seller η for any reserve price p is exactly the same as in thetwo-bidder, symmetric mean example.For any reserve price p ∈ [0 , , the revenue to the seller will be: p (1 − H ( p )) = p (cid:18) − p − αp (cid:19) = α. When p = 1 , the revenue may be smaller, depending on the tie breaking rule. Observe howeverthat G ∗ does not put any mass at 1, so Ψ( F ∗ , G ∗ ) = α ≥ Ψ( F ∗ , G ) , for any G ∈ P ([0 , . Hence, G ∗ in Theorem 1 is a best response.25 .3.2 Nature’s problem Now we will prove that for the G ∗ in Theorem 1, F ∗ is a best response by nature. We will showthat for any F ∈ P ( V, m ) , Ψ( F ∗ , G ∗ ) ≤ Ψ( F, G ∗ ) . Equivalence with a k -bidder auction. Since the n − k inactive bidders do not affect the revenue,we can prove the saddle point inequality using the F ∗ distribution containing only the k activebidders, which simplifies the problem. Consider a new auction ( V k , { m i } ki =1 ) , in which we includeonly the k active bidders in the “L”-shape joint distribution of F ∗ .For any distribution of value profiles in the original n -bidder auction F ∈ P ( V, { m i } ni =1 ) , we canproject this distribution on the k active bidders and obtain F k ∈ P ( V k , { m i } ki =1 ) . This distributionis exactly identical to F for the k active bidders, but no longer includes the n − k inactive bidders.We will now show the following relations: Ψ( F ∗ , G ∗ ) = Ψ( F ∗ k , G ∗ ) and Ψ( F k , G ∗ ) ≤ Ψ( F, G ∗ ) . With these in hand, it suffices to show that Ψ( F ∗ k , G ∗ ) ≤ Ψ( F k , G ∗ ) in order to prove that Ψ( F ∗ , G ∗ ) ≤ Ψ( F, G ∗ ) . Proof.
For any general F and F k , Ψ( F, G ) ≥ Ψ( F k , G ) for any reserve price G . All else being equal,including more bidders can only increase the revenue for the seller.First, using the same notation as in the two bidder case, we let φ ( v ; G ) denote nature’s expectedvalue for any realization of value profiles v , given the seller’s strategy G : φ ( v ; G ) := (cid:90) max { v (2) , p } · { v (1) >p } dG. Furthermore, let P k : V → V k be the projection of a value profile on to the valuations of the k active bidders in F ∗ . For any realization v ∈ supp ( F ) and its projection v k := P k ( v ) ∈ supp ( F k ) ,the expected revenue φ ( v ; G ) is greater than φ ( v k ; G ) , since v (1) ≥ v (1) k and v (2) ≥ v (2) k due to thefact that v k includes only the valuations from a subset of bidders in v . It follows then that: Ψ( F, G ) = (cid:90) V φ ( v ; G ) dF ≥ (cid:90) V φ (P k ( v ); G ) dF = (cid:90) V k φ ( v k ; G ) dF k = Ψ( F k , G ) . Additionally, Nature’s optimal strategy F ∗ and the projected distrbution F ∗ k are payoff-equivalent.For any v ∗ ∈ supp ( F ∗ ) and its projection v ∗ k := P k ( v ∗ ) ∈ supp ( F ∗ k ) , φ ( v ∗ ; G ) = φ ( v ∗ k ; G ) as the n − k inactive bidders in v ∗ all have valuations m j ≤ α . Therefore it follows that: Ψ( F ∗ k , G ) = (cid:90) V φ ( v ∗ ; G ) dF = (cid:90) V φ (P k ( v ∗ ); G ) dF = (cid:90) V k φ ( v k ; G ) dF ∗ k = Ψ( F ∗ k , G ) . addle point inequality in the k -bidder auction. Now we will show that F ∗ k is a best responseamong distributions F k ∈ P ( V k , { m i } ki =1 ) : Ψ( F ∗ k , G ∗ ) ≤ Ψ( F k , G ∗ ) Proof.
Recall that F ∗ k follows the “L”-shape distribution for the k bidders. In other words, supp ( F ∗ k ) consists of value profiles v ki ( x ) , in which i th bidder’s value is some x ∈ [ α, , and the values of theremaining k − bidders are set to α .We use Lemma 1 to verify the optimality of F ∗ k . Recall that Lemma 1 states that if there existsan affine function L ( v ) such that L ( v ) ≤ φ ( v ; G ∗ ) for all v ∈ V , and in addition, L ( v ) = φ ( v ; G ∗ ) on the support of F ∗ k , then F ∗ k is an optimal strategy for nature. For our purpose, consider thefollowing affine function: L ( v ) = k (cid:88) i =1 k − − ln( α ) v i − αk − − ln( α ) . We will first show that φ ( v ; G ∗ ) = L ( v ) for all v ∈ supp ( F ∗ k ) . First, for any realization v , it iswithout loss to reorder the bidders’ valuations in v from highest to lowest v ≥ v ≥ ... ≥ v n . Wecan write: φ ( v ; G ∗ ) = (cid:90) max { v , p } · { v >p } dG ∗ = v G ∗ ( v ) + (cid:90) v v rg ∗ ( r ) dr = v G ∗ ( v ) + v − v k − − ln( α ) . Recall that the candidate strategy for the seller G ∗ is: G ∗ ( p ) = (cid:82) pα g ∗ ( r ) dr + k − k − − ln( α ) if p > α k − k − − ln( α ) if p ≤ α, where g ∗ ( r ) = r ( k − − ln( α )) if r > α if r ≤ α. Take any point v ki ( x ) in the “L-shape” support. It is without loss to consider v k ( x ) = ( x, α, α, ..., α ) ,as the following holds symmetrically in all v ki ( x ) : While the current equilibrium puts mass of k − k − − ln( α ) on p = 0 , there could be other equilibria in which the sellermay put density on [0 , α ] . All of these equilibria are payoff equivalent but may not yield the same behavior. ( x, α, ..., α ; G ∗ ) = αG ∗ ( α ) + (cid:90) xα rg ∗ ( r ) dr = ( k − αk − − ln( α ) + xk − − ln( α ) − αk − − ln( α ) L ( x, α, ..., α ) = ( k − αk − − ln( α ) + xk − − ln( α )) − αk − − ln( α ) . Hence, L and φ are identical (or intersect) on the support of F ∗ .Now we will show that for all v ∈ V , φ ( v ; G ∗ ) ≥ L ( v ) . We define the following functions inorder to simplify the analysis: ϕ ( u, q ) := φ ( u, q, q, ..., q ; G ∗ ) ,(cid:96) ( u, q ) := L ( u, q, q, ..., q ) .ϕ ( u, q ) and (cid:96) ( u, q ) are respectively φ and L when the highest bidder’s value is u and the valuesof the other bidders are equal to q ≤ u .Since φ only depends on the highest and second highest bidder, the realization of φ will not bealtered if the rest of the valuations v , ..., v n were set to the second highest bidder’s value v . Hence, φ ( v ) = ϕ ( v , v ) . Additionally, since L is increasing in each coordinate, (cid:96) ( v , v ) ≥ L ( v ) . These functions allow us to compare φ and L at various value profiles without having to specify v , ..., v k .There are three separate cases to consider in order to show φ ≥ L : (1) the second highest bidder’svalue is above the lower bound α , (2) the highest bidder’s value is above α but the second highest’sis below α , and (3) all values are below α . Case (1): v > α .Observe first that for any u > q , ∂ϕ∂r ( u, q ) = G ∗ ( q ) . For any v such that v > α , 28 ( v ) ≤ (cid:96) ( v , v ) = (cid:90) v ∂(cid:96)∂v dv = (cid:90) v k − k − − ln( α ) dv = (cid:90) v G ∗ (0) dv φ ( v ; G ∗ ) = ϕ ( v , v ) = (cid:90) v ∂ϕ∂v dv = (cid:90) v G ∗ ( v ) dv , where G ∗ (0) = k − k − − ln( α ) refers to the mass of the atom at p = 0 . Since v > α , G ∗ ( v ) ≥ G ∗ ( α ) = G ∗ (0) and therefore φ ( v ; G ∗ ) = ϕ ( v , v ) ≥ (cid:96) ( v , v ) ≥ L ( v ) . Case (2): v ≥ α ≥ v .For v less than α , G ∗ ( v ) = G ∗ (0) = k − k − − ln( α ) , since G ∗ does not have any mass in (0 , α ) .Hence, L ( v ) ≤ (cid:96) ( v , v ) = v k − − ln( α ) + ( k − v k − − ln( α ) − αk − − ln( α ) φ ( v ; G ∗ ) = ϕ ( v , v ) = v G ∗ (0) + ( v − α ) k − − ln( α )= v k − − ln( α ) + ( k − v k − − ln( α ) − αk − − ln( α ) . Therefore, φ ( v ; G ∗ ) ≥ L ( v ) . Case (3): α > v .Since as mentioned above, G ∗ only has mass at p = 0 within the interval [0 , α ] , the expectedrevenue is φ ( v ; G ∗ ) = v G ∗ (0) . In other words, the expected revenue is simply the second highestbidder’s value times the probability that a sale occurs, which is the probability that p < α , since v < α . Note L ( v ) ≤ (cid:96) ( v , v ) = v k − − ln( α ) + ( k − v k − − ln( α ) − αk − − ln( α ) ≤ ( k − v k − − ln( α ) , and φ ( v ; G ∗ ) = ϕ ( v , v ) = v G ∗ (0) = ( k − v k − − ln( α ) . Therefore, φ ( v ; G ∗ ) ≥ L ( v ) for all v ∈ V .By Lemma 1, we have proven that F ∗ k is an best response for nature out of distributions F k ∈P ( V k , { m i } ki =1 ) . Hence, we have proved the optimality of F ∗ : Ψ( F ∗ , G ∗ ) ≤ Ψ( F ∗ k , G ∗ ) ≤ Ψ( F k , G ∗ ) ≤ Ψ( F, G ∗ ) ..