Strategic Information Acquisition in Auctions.

Transcription

1 Stategic Infomation Acquisition in Auctions. Kai Hao Yang 12/19/217 Abstact We study a stategic infomation acquisition poblem in auctions in which the buyes have independent pivate valuations and can choose signal stuctues to lean about thei own valuations simultaneously, afte which the selle obseves the chosen signal stuctues and designs optimal selling mechanisms. We chaacteized the unique symmetic equilibium that descibes the buyes infomation acquisition behavios. In this equilibium, each buye andomly selects a signal stuctue unde which the posteio expected value is a tuncated Paeto distibution. Keywods: Mechanism design, optimal auctions, infomation acquisition, evenue maximization, buye suplus, vitual valuation. Jel classification: C72, D44, D82, D83 1 Intoduction Fo a selle who wishes to sell goods to multiple potential buyes, auction is cetainly one of the most common devices. Unde a context that the selle has all the bagaining powe and is able to design any selling mechanism in ode to maximize he evenue, the study Depatment of Economics, Univesity of Chicago, hyang@uchicago.edu. I am indebted to Ben Boos, Doon Ravid and Elliot Lipnowsi fo valuable comments and suggestions fo the stategies of the poofs. I am also gateful fo the discussions and comments fom Roge Myeson, Nancy Stoey, Steven Davis, Emi Kamenica, Alex Fanel, Eic Masin and Stephen Mois. 1

2 2 of auctions and optimal auction designs povides abundant implications and suggestions fo the selle. In canonical studies of auctions and optimal auction designs, it is often assumed á pioi that each buye pefectly nows about thei own valuation and the distibution fom which the valuations ae dawn ae exogenously given. Howeve, unde vaious eal-wold contexts, the buyes themselves may not have complete nowledge about the good they ae bidding fo á pioi. Instead, the buyes have to lean and acquie infomation about thei valuations of the good. Moeove, due to the apid impovement of infomational technology, the buyes ability to acquie infomation has become much moe flexible. Fo instance, in online auctions, buyes typically do not have much infomation about thei own valuation befoe infomation acquisition buyes lean about thei valuations though viewing the advetisements, examining the details of the objects povided on the website o compaing this object with othe countepats that ae sold elsewhee. Similaly, in auctions of atwos o houses, buyes do not now about thei valuations befoe examining the objects o visiting the popeties. On the othe hand, in these scenaios, the selle is often able to obseve the infomation acquisition behavio of the buyes. In the online auction example, the selle can obseve the buyes infomation acquisition behavio by collecting infomation fom the buyes bowsing histoy on the auction webpage. In the atwo o house auctions, the buyes ae often acquiing infomation with the pesence of the selle and theefoe the selle can also monito how the buyes collect infomation. As such, the buyes infomation acquisition decisions not only affect the buye s bidding stategy in auctions but also induce diffeent selling mechanisms as the selle will optimally design a selling mechanism based on the buyes infomation acquisition behavio. Theefoe, buyes infomation acquisition decision will be stategic, taing both the othe buyes infomation acquisition behavio and the selle s best esponse into account. That is, buyes infomation about thei own valuation will be a esultant of ational and stategic inteactions among each othes and against the selle and thus aives endogenously. In this pape, we aim to captue this aspect in auctions by examining the optimal auction design poblem in which the infomation stuctue is also endogenously detemined by ational agents. Specifically, the goal is to undestand and chaacteize the esulting infomation stuctue and the associated optimal selling mechanism if the buyes facing a selle who

3 3 will design optimal selling mechanisms accodingly can ationally engage in infomation acquisition. In a model whee each buye s pivate value is identically and independently dawn fom some pio and the buyes choose signal stuctues simultaneously in ode to lean about thei valuations, anticipating that the selle will set optimal selling mechanism accodingly, we fully chaacteized the unique symmetic equilibium that descibes the infomation acquisition behavio when the pio is binay. We also extend the esult and give a compact chaacteization fo the symmetic equilibia when the pio is egula and has full suppot. The est of the pape poceeds as follows: In section 2, we discuss the elated liteatue and stess the similaities and diffeences of ou model. In section 3 we descibe the model and deive some peliminay esults. Section 4 povides the chaacteization of equilibium when the pio is binay. Section 5 extends to the case when the pio is egula and has full suppot. 2 Related Liteatue This pape is elated to the liteatue between the inteplay of infomation and mechanism design and auctions. In the seminal pape, Myeson 1981) studies the optimal auction unde independent pivate values when the buyes now thei valuations and the distibutions of valuations ae commonly nown. Thee ae seveal stands of liteatue that study the inteplay between infomation stuctue and mechanism designs. In the obust auction design liteatue, Neeman 23) studied the pefomance of the second pice auction with flexible eseve pice unde the wost case infomation stuctue. Chung & Ely 27) studied a max-min poblem in which the selle wishes to set a mechanism against the natue, who would select a wost-case belief stuctue in esponse. Yamashita 217a) studied a max-min poblem in which the natue select the wost infomation stuctue and the selle s feasible mechanism is esticted. Du 217) studied the max-min poblem in which thee is a common pio and the natue selects the wost case infomation stuctue. The max-min solution fo a single-buye poblem is chaacteized and is shown to achieve appoximately full suplus as numbe of buyes tends to infinity. Begemann, Boos & Mois 217) then chaacteizes

4 4 the solution to the max-min poblem with two buyes, common value, and a pio that has binay suppot. Although ou pape is distinct fom this liteatue in the sense that in ou famewo, the infomation stuctue aives endogenously due to buyes ational and stategic concens, the esulting infomation stuctue is simila to some of the solutions in these papes. Specifically, Neeman 23), Du 217) and Begemann, Boos, and Mois 217) all have cetain fom of Paeto distibution as the Natue s wost case solution. In this pape, the equilibium is also a andomized stategy on a class of Paeto distibutions. On the othe hand, the infomation acquisition liteatue and infomation disclosue liteatue also examined the endogenously aived infomation stuctues in auctions. In the liteatue of infomation disclosue in mechanism designs, Esö & Szentes 27) studied an optimal disclosue poblem of a selle who can disclose multiple patially infomed buyes in an independent fashion and showed that full disclosue of the othogonalized shoc is optimal as long as the paticipation constaint is pio to the ealization of the signals. 1 Begemann & Pesendofe 27) also studied an optimal disclosue poblem in which the buye is á pioi uninfomed and the selle can disclose any signal stuctues fo each buye as long as they ae independent and any mechanisms. They showed that the optimal infomation stuctue is a finite patitional signal stuctue. Yamashita 217b) also studied a selle s optimal public disclosue stategy about the infomation of a common value component of the buyes and showed that when such common value component and the pivate values ae independent o affiliated and when budget balance is not equied, full disclosue is optimal. On the othe hand, in the liteatue o infomation acquisition in mechanism designs, Begemann & Välimäi 22) examined an efficiency implementation poblem when the agents can acquie infomation ationally, unde a set of esticted infomation stuctues. Shi 212) also studied the optimal auction poblem in which the buyes and acquie infomation ationally. The diffeence between Shi 212) and this pape is that ou ode is the opposite the buyes hee acquie infomation fist and the selle can obseve the infomation acquisition decision of the buyes, poviding anothe laye of stategic concens to the acquisition poblem. Futhemoe, in the famewo of Shi 212), the signal stuctues ae esticted to a one-dimensional family, wheeas in ou famewo the buyes choice is 1 Li and Shi 217), howeve, showed that full disclosue of all the infomation is not optimal in geneal.

5 5 unesticted. Among the liteatue, Rosele & Szentes 217) is the closest to this pape. They studied a single buye s optimal infomation acquisition poblem when facing a monopolist who can chage the buye accodingly. This pape is a multiple-buye vesion of Rosele & Szentes 217), in which the pio of the buyes ae independent and the buyes can acquie infomation simultaneously and stategically. Methodologically, we ely on the contibution of Monteio & Svaite 21) on genealizing the vitual valuation in Myeson 1981) to all the distibutions, as well as some emas on the continuity of optimal auction in Monteio 215). In addition, the mean-peseving spead chaacteization of infomation stuctue has also been used by vaious studies about disclosue and pesuasion e.g. Gentzow & Kamenica 216), Kolotilin et al. 217), Rosele & Szentes 217), Yamashita 217b) and Libgobe & Mu 217).) 3 Model Thee is a single selle, selling an indivisible good to N 2 buyes, each of whom has quasi-linea pefeences u i = v i p i y i, whee p i is the pobability that i wins the object and y i is the amount of tansfe payed by i. Suppose that the valuations {v i } N i=1 ae i.i.d., dawn fom a common pio F, with suppf ) [, 1]. We study the context in which buyes can ationally, and theefoe stategically engage in infomation acquisition. That is, each buye i does not now the ealization of v i á pioi but can choose to acquie infomation about v i. Fomally, each buye can choose a signal stuctue S i, π i ), whee S i is the set of possible signals and π i v i ) S i ) is a conditional distibution on S i given a ealization of valuation v i. 2 We assume that buyes cannot communicate and can only choose thei own signal stuctue. As such, the maginal distibutions of signals of each buye ae also independent. Afte seeing the buyes infomation acquisition behavio, the selle designs a mechanism to maximize expected suplus. 2 Moe fomally, fo each i {1,..., N}, S i is endowed with some σ-algeba S i, S i ) is the set of pobability measues on the measuable space S i, S i ) and π i : [, 1] S i ) is a function that maps each ealization of valuation to distibution of signals such that fo any v i [, 1], π v i ) is a pobability measue on S i and fo any E S i, πe ) is measuable with espect to the Boel algeba on [, 1].

6 6 The timing of the model is as follows: 1. Natue daws valuations {v i } N i=1 identically and independently accoding F. 2. Buyes choose thei signal stuctues {S i, π i )} N i=1 simultaneously, which then becomes publicly obsevable. 3. Selle publicly chooses a mechanism. 4. Natue then independently daws signals {s i } N i=1 accoding to {π i v i )} N i=1 and buye i pivately obseves the ealization of s i. 5. Buyes tae actions in the announced mechanism. Given a pofile of signal stuctues. Fo the selle s optimal mechanism poblem, the selle designs a mechanism and select a Bayes Nash equilibium to maximize expected suplus. Notice that standad evelation pinciple applies in this envionment and thus we may estict attention to diect mechanisms unde which the buyes epot thei signal ealizations. Moeove, due to quasi-lineaity of buyes pefeences, fo each buye i {1,..., N}, the only payoff elevant statistic is the conditional expected valuation given the ealization of the signals, E[v i s i ]. As such, any incentive compatible diect mechanism is payoff equivalent to some mechanism unde which buyes epot only thei inteim expected valuations and tuthfully epoting foms a Bayes Nash equilibium. Theefoe, we may futhe estict attention to the class of mechanisms p, y) that is incentive compatible and individually ational, whee p = p i ) N i=1 : [, 1] N [, 1] N, with N i=1 p i 1 denotes the allocation ule and y = y i ) N i=1 : [, 1] N R N denotes the tansfe ules Consequently, fo any pofile of signal stuctues, each buye s expected suplus depends only on the buyes conditional expected valuation. We can then nomalize the signal stuctues such that the signals ae unbiased estimatos of the valuation. That is, fo each buye i {1,..., N} it is without loss to conside only the signal stuctues such that E[v i s i ] = s i. Equivalently, fo any signal stuctue, F must be a mean peseving spead of the the maginal distibution of signals. As a esult, fo each buye i {1,..., N}, let µ := xf dx) be the pio mean of the valuation, the set of feasible signal stuctues can

7 7 be chaacteized as the set of CDFs: { G F := G : [, 1] [, 1] x x Gt)dt F t)dt, x [, 1] } Gx)dx = 1 µ, G is a CDF x Ξt)dt Ξ = F full infomation) 1 µ Ξ = G intemediate) Ξ = F no infomation) µ x 1 Figue 1: Feasible Signal Stuctues It is convenient to descibe the set G F by Figue 1. In Figue 1, the hoizontal axis is a vaiable x [, 1] and the vetical axis is the integal of a CDF fom to x. Fo any feasible G, since G is a mean peseving spead of F, it coesponds to some convex function the gaph in between) that is majoized by the integal of F and shaes the same end points as the integal of F. Moeove, since the maximum value that G can tae is 1, any feasible G G F must also majoize the convex function associated with F the bottom gaph), whee F is the CDF of a degeneate distibution that puts pobability 1 on µ, which coesponds to a signal stuctue that gives no infomation. On the othe hand, the convex function associated with F the top gaph) epesents a signal stuctue that has full infomation. As such, the set of feasible signal stuctues G F can be chaacteized by a collection of gaphs of inceasing convex functions that ae bounded fom above and below by two gaphs of

8 8 inceasing convex functions, espectively, and shae the same end points. 3 Consequently, any pofile of signal stuctues can be descibed by a pofile of CDFs G i ) N i=1 G F ) N. This simplifies the timing of the model to the following two stages. At the infomation acquisition stage, each buye i chooses G i G F simultaneously. Then at the optimal auction stage, the selle obseves the buyes choice of distibutions of signals and each buye obseves thei own signal ealization s i and the othes signal stuctues. The selle then designs an incentive compatible and individually ational diect mechanism to maximize evenue. At the optimal auction stage, since the valuations ae dawn independently and the signals ae dawn independently conditional on ealization of valuations, the maginal distibutions of signals must be independent. Theefoe, given any G i ) N i=1 G F ) N, the selle is essentially facing an optimal auction poblem with independent pivate values. As such, the optimal mechanism fo the selle can be futhe chaacteized, due to Myeson 1981) and a genealization made by Monteio & Svaite 21). Moe specifically, given any CDF, G i G F, thee exists a wealy inceasing function, ψ G i ), that epesents the vitual valuation 4 induced by G i and the selle s optimal mechanism is to allocate the good accoding to the vitual valuations. That is, given any G = G i ) N i=1 and any pofile of epots s = s i ) N i=1 [, 1] N, let W s G) := agmax i {1,...,N} ψs i G i ), the optimal mechanism is given by 5 1 p i s G) :=, if i W s G) and ψs W s G) i G i ). 1), othewise. y i s G) := p i s i G)s i si p i z G)dz, fo each i {1,..., N} and fo any s = s i ) N i=1 [, 1] N, whee p i s i G) := E G i [p i s G)] fo 3 It is notewothy that the set G F is a compact set unde the wea-* topology, by the Lebesgue dominated convegence theoem and compactness of set of pobability measues on [, 1], endowed with the Boel algeba. 4 Fomal definition and some popeties of ψ G i ) can be found in the appendix. Unde this definition, ψ G i ) can be thought of as a genealization of the ioned out vitual valuation as in Myeson 1981), in the sense that it does not put any assumptions on the undelying distibution G. 5 Notice that we ae beaing the ties in favo of the buyes when the selle is indiffeent between selling o not, and unifomly when the selle is indiffeent in whom to sell. We will always focus on this tie beaing ule heeafte.

9 9 each s i [, 1] is the inteim pobability of winning fo buye i when the signal ealization is s i and the infomation stuctue is G = G i ) N i=1. Fom 1) and fom the envelope chaacteization, given any pofile of distibutions G = G i ) N i=1 G F ) N, buye i s expected suplus, anticipating that the selle will set an optimal mechanism accodingly, is given by x ) p i z G)dz G i dx) = p i x G)1 G i x))dx, due to Fubini s theoem, whee p i s i G) := E G i [p i s)] is the inteim pobability of winning of i when the signal ealization is s i unde the opponents chosen distibutions G i = G j ) j i. Fo each i {1,..., N}, fo any G = G i ) N i=1 G F ) N, let U i G) := p i x G)1 G i x))dx. 2) Then at the infomation acquisition stage, the buyes ae essentially playing a mixed extension of a stategic fom game that taes the subsequent plays into account. In this game, the set of playes is {1,..., N}. Each playe has stategy space G F and has a payoff function given by 2). 6 With such chaacteizations, the poblem then educes to solving fo Nash equilibia of the induced mixed extension game. We will estict attention to the symmetic Nash equilibium of this game. Using 2), fo each buye i {1,..., N}, given that all the othe buyes ae using a stategy ν G F ), expected suplus of choosing G i G F ) is p i x G i, G i )1 Gx))dxν N 1 dg i ). 3) G N 1 F Since the optimal mechanism unde any pofile of distibutions is to allocate the good accoding to the induced vitual valuations, 3) can be futhe simplified by intoducing the distibution of the highest vitual valuation among all the othe buyes. That is, fo each i {1,..., N}, fo any pofile of othe buyes distibution G i = G j ) j i G F ) N 1, let 6 Fomally, let Λ be a stategic fom game with playes {1,..., N}, stategy spaces G F and payoff functions {U i } N i=1 and let Λ be the mixed extension of Λ. We endow the set G F with the wea-* topology and let G F ) be the set of pobability distibutions on the measuable space G F, with the σ-algeba being the Boel algeba geneated by the wea-* topology. We will also slightly abuse the notation and let U i denote the VNM on G F ).

10 1 Φ G i denote the distibution of the highest vitual valuations induced by G i, 7 condition on being positive and afte adjustment of tie and let Φ ν q) := G N 1 F all q, 1]. Then 3) can be ewitten as: Φ ν ψx G i ))1 G i x))dx, G i G F. Φ G i q)ν N 1 dg i ) fo As such, a necessay and sufficient condition fo ν,..., ν) G F ) N equilibium is that fo any G suppν), any R G F, to be a symmetic Φ ν ψx G))1 Gx))dx Φ ν ψx R))1 Rx))dx 4) It is notewothy that the assumption about the obsevability of each buye s infomation acquisition behavio is not essential. Indeed, since the buyes signals must be in independent as the valuations ae independent and the buyes can only lean about thei own valuations and since the inteim expected value is one-dimensional, fom the selle s pespective, the mechanism design poblem is always equivalent to an optimal auction poblem with independent pivate values. It is well nown that the maximal evenue that can be achieved by dominant-stategy mechanisms is the same as that can be achieved by Bayesian incentive compatible mechanisms. 8 As a esult, even if we emove the assumption that buyes can see each othes infomation acquisition behavio, all the aguments above still follow except that the selle now uses dominant-stategy mechanisms as the buyes do not now about each othes distibution of signals when taing actions in the mechanism. Nevetheless, this will not change the fomation of the infomation acquisition game since the oiginal optimal allocation ule is itself dominant-stategy implementable and each buye s expected suplus depends only on allocation ules in the optimal mechanism. Specifically, even if the buyes cannot see each othes infomation acquisition behavio, the mechanism given by 1) is still optimal fo the selle at the optimal auction stage and theefoe the payoff function given by 2) is unchanged. As such, the infomation acquisition game is the same. 7 Fomally, given any G G F, define ϕ G q) := {ψx G) q}gdx) fo all q, 1]. Fo any i {1,..., N}, fo any G i = G j ) j i G F ) N 1, let W i := {W {1,..., N} i W } and let Φ G i q) := 1{q } 1 W W i W j W \{i} ϕ G j q) ϕ Gj q )) j / W ϕ G j q ) fo all q, 1]. 8 In auction settings, this follows diectly fom the envelope chaacteization and the monotonicity of the genealized vitual valuation ψ G i ) fo any distibution G i.

11 11 We end this section by summaizing the single buye poblem as in Rosele & Szentes 217) in ode to illustate pat of the main stategic featues in the multiple buye game. If N = 1, then the optimal auction stage educes to a monopolist s sceening poblem. Due to quasi-lineaity, it is always optimal fo the selle to set a posted pice. Theefoe, given any signal stuctue chosen by the buye, which can be chaacteized by some G G F, the selle solves: max p1 p [,1] Gp )). Let π G denote the value of this poblem. Then we have: fo all p [, 1], p1 Gp)) p1 Gp )) π G, which is equivalent to Gp) 1 π G p. Notice that the ight hand side esembles the CDF of a Paeto distibution. As such, if thee is no feasibility constaint and the buye can choose any valid CDF with suppot [,1], then any distibution is fist-ode stochastic dominated by a Paeto distibution. In fact, Rosele & Szentes 217) shows that thee exists a solution that is a tuncated Paeto distibution:, if x [, π ) Gπ b x) = 1 π, if x x [π, b ), 1, if x [b, 1] fo some π and b. Seveal featues of such tuncated Paeto distibutions ae notewothy. Fist, it can be constucted by the optimality of a posted pice unde some distibution G G F and always taes values below Gx) at any x [, b ). Second, the vitual valuation induced by a Paeto distibution is always zeo on the suppot, except at x = b. Thus, the selle is indiffeent in setting any pices in [π, b ) and buye s expected suplus is maximized when the selle sets the posted pice at π. As a esult, simila to the single buye poblem, even when thee ae multiple buyes, thee ae always benefits to shift thei distibution towad the Paeto shape, as it puts moe weights on highe values and thus inceases expected suplus fom the optimal auction

12 12 conditional on winning. Howeve, with multiple buyes, eveyone choosing the same Paeto distibution cannot be a symmetic equilibium. Indeed, since the vitual valuation is zeo on the suppot except at b, if all the buyes choose the distibution G b π, thee will be a positive pobability of tie in the optimal auction. As such, any buye can petub the distibution G b π locally and bea the tie at an ignoable cost. In fact, fo all the tuncated Paeto distibutions with diffeent location paametes, the vitual valuation equals to the location paamete on the suppot except at the mass points and theefoe the same logic applies. Howeve, since whoeve has the highest location paamete will win the auction fo sue but will have lowe expected valuation conditional on winning, thee is an additional tade-off fo the buyes when choosing among the Paeto distibutions, which is simila to a Betand competition. In the next section, we will chaacteize the symmetic equilibium in the game when the pio has binay suppot. Suggested by the above obsevations, the symmetic equilibium at the infomation acquisition stage is a mixed stategy that andomizes on a family of tuncated Paeto distibutions and the shape of this andomization is detemined by the indiffeence conditions. 4 Equilibium of Infomation Acquisition In this section, we chaacteize the symmetic Nash equilibium of the game at the infomation acquisition stage, which descibes the stategic behavios of the buyes when acquiing infomation. We conside the case when the pio F is discete and has a suppot on {, 1} so that the feasibility constaint educes to only an equality. We will extend to the case when the pio is absolute continuous and egula in the next section. Unde this envionment, we have a complete chaacteization of the unique symmetic equilibium at the infomation acquisition stage. In this equilibium, each buye andomizes on a paticula family of Paeto distibutions that ae detemined by thee paametes and the shape of such andomization is detemined by the indiffeence condition. The chaacteization is obtained in two steps. Fist, we show that fo any symmetic equilibium, the suppot of the equilibium stategy must be on this Paeto family. This implies that any

13 13 symmetic equilibium must also be an equilibium in a esticted game whee the buyes ae only allowed to choose fom the Paeto family. As it is finite-dimensional, we can then completely chaacteize the unique symmetic equilibium in the esticted game. Finally, we veify that this is indeed an equilibium in the oiginal game Suppose that evey buye s valuation can eithe be high v i = 1) o low v i = ). The pio is then of fom: 1 µ, if x [, 1) F x) = 1, if x = 1 fo some µ [, 1]. To ule out tivial cases, we assume that µ, 1). Unde such pio, the set G F can be simplified to { G F = G : [, 1] [, 1] } Gx)dx = 1 µ, G is a CDF. As suggested by the obsevations about Paeto distibutions in the pevious section, a family of tuncated Paeto distibutions plays a cucial ole in ou analysis. Fomally, conside the following family of distibutions: Fo any [, 1], π [, 1 ], η [, 1 µ], let with a convention that / :=. Let G F [ ) π η, if x, + 1 η G η π, x) = [ ) 1 π, if x π +, 1, 5) x 1 η 1, if x = 1 be the set of distibutions of fom 5) that ae also feasible. Fom the pevious obsevation about feasibility unde a binay pio, fo any π,, η [, 1], G η π, G F if and only if: ) 1 )1 η) π log + π + 1 η) = µ, η [, 1 µ], π [, 1 ]. 6) π As such, the set G F is essentially descibed by thee paametes π,, η that satisfy 6). Figue 2 illustates a typical element of the set GF. Since F has suppot on {, 1}, the uppe bound of the convex functions as in Figue 1 is linea and has slope 1 µ. Any feasible G G F coesponds to a unique convex function that has end points and 1 µ and has slope no geate than 1. An element in the family G F taes the following fom: It has a mass point with pobability η at ; on [π/1 η)+, 1) the distibution has a Paeto shape with π

14 14 x Ξt)dt 1 µ Ξ = F Ξ = G η π, slope=1 π x slope=η π + 1 η 1 x Figue 2: An Element of G F and being the paametes. Finally, at 1, it has a mass point of which the size is detemined by π and, which in tuns depend on η though 6). By constuction, the vitual valuation of any G η π, G F is given by: ) ψ x ψ η Gπ,), if x π [, π, 1 η [ + ) η = π, if x +, 1 1 η 1, if x = 1 fo some ψ η π, <. In paticula, the vitual valuation induced by Gη π,, is always on its suppot except at the atoms. Theefoe, if thee is only a single buye, given any G η π, G F, the selle will optimally set a posted pice so that the vitual valuation is positive wheneve the good is sold and thus the expected suplus fo the single buye will be 1 { ψ x ) } G η π, 1 G η π, x)) dx = µ π + 1 η)). 7) As in the single-buye poblem, with multiple buyes, each buye still has incentives to shift the distibution towad Paeto shapes locally. Indeed, fo any CDF R G F, suppose

15 15 that the vitual valuation ψ R) is continuous and stictly inceasing. 9 Then fo any x [, 1] such that ψ := ψx R), if we conside a suogate poblem in which the selle has a cost of poduction ψ and is facing only one buye whose signal follows R, then the selle will optimally set the posted pice at x as this is the cut-off pice at which the vitual valuation net of cost will be all positive conditional on selling. Thus, x ψx R))1 Rx)) x ψx R))1 Rx )) x ψx R))1 Rx )), fo all x [, 1]. Let ζ := x ψx R))1 Rx )), and eaange, we have { } ζ Rx) max 1 x ψx R),, 8) fo all x [, 1]. Notice the ight hand side is exactly a CDF of a Paeto distibution with paametes ζ and ψ. On the othe hand, ecall that given all the othe buyes stategy ν G F ), buye i s expected suplus of choosing signal stuctue R G F is Φ ν ψx R))1 Rx))dx. Theefoe, if R is not of Paeto shape and if buye shifts R to the Paeto-shaped distibution 1 ζ /x ψ ) aound x, i s suplus inceases since 1 Rx)) inceases aound x. Contay to the single-buye poblem, such behavio is not costless, as shifting the distibution to Paeto-shaped will yield a vitual valuation that is almost eveywhee) constant wheneve it is positive, which means that fo some ealization of signals, vitual valuation will be educed and hence pobability of winning will be educed. Howeve, since vitual valuation is inceasing, combining the two obsevations above, at x, deviating to a Paetoshaped distibution locally on some inteval [ x, x ] can still be pofitable. As it inceases both the pobability of winning and the expected valuation conditional on winning. The following poposition fomalizes the above intuition and gives a necessay condition fo a stategy ν to constitute a symmetic equilibium. Poposition 1. Suppose that ν,..., ν) is a symmetic Nash equilibium. Then suppν) G F. 9The logic fo any geneal CDF is simila, the fomal poof can be found in Lemma 6 in the appendix.

16 16 Fomal poof of Poposition 1 can be found in the appendix, we povide a gaphical illustation below. The main idea of the poof is exactly the local deviation logic descibed above. Suppose that the suppot of a stategy ν G F ) is not a subset of the Paeto family GF. Then thee must be some distibution R suppν) that does not always have a Paeto shape. Fo the ease of exposition we suppose that ψ R) is continuous and stictly inceasing, although simila aguments hold fo any distibution R, as shown in the poof. Fo any such distibution R, we can constuct a local petubation of R, denoted by R, such that deviating fom R to R gives a buye stictly highe expected suplus given that opponents ae choosing ν. Recall that we may epesent the feasible signal stuctues by the gaph of the integal of CDF as in Figue 1 so that R will be a convex function bounded by the gaphs that epesent full infomation and no infomation. Conside fist the case in which R is associated with a convex function as the lowe cuve in Figue 3. Specifically, let := ψ 1 R) be the eseve pice induced by R. Suppose that R ) > and that the line segment tangent to the cuve at intesects with the hoizontal axis at some positive numbe. By 8), with x =, if we shift the CDF R to the Paeto distibution 1 ζ /x, it will be smalle than R eveywhee. As such, if R is shifted to 1 ζ /x locally on some inteval [x, ], the slope of the associated convex function deceases locally, as illustated by the cuve in between. Since the tangent line at intesects with the hoizontal axis at some positive numbe, the intemediate value theoem then ensues that thee exists some x such that the line segment that tangent to the petubed cuve intesects with the hoizontal axis at zeo, as illustated by the dotted line. We may then constuct a new signal stuctue by connecting the dotted line to the point whee it is tangent to the petubed cuve, x, and eep the oiginal distibution wheneve x. The deivative of this convex function is then a petubed signal stuctue R G F. To see that R stictly impoves a buye s suplus when compaing to R when the opponents ae choosing ν. Notice that on the inteval [x, ], the vitual valuation of R is negative and thus this buye could not win the good when the ealization of signal lies in [x, ]. On the othe hand, the vitual valuation of R is on [x, ] and thus thee is a positive pobability of winning even when the ealization of signal is in [x, ], povided that Φ ν ) >. Moeove, R and R coincide on [, 1] and thei vitual valuations also coincide on [, 1]. Togethe, expected

17 17 suplus must stictly incease povided that Φ ν ) >. The agument is then completed by obseving that since R suppν), and R ) > and the stategy pofile is symmetic, it must be that Φ ν ) >. x Ξt)dt 1 µ Ξ = F slope=1 ζ x a x x 1 Figue 3: Local Deviation 1 If, on the othe hand, the line segment that tangents to the lowe cuve at intesects with the hoizontal axis at, o R ) =, Figue 4 illustates a pofitable local deviation in this case, whee we conside the gaphs of the feasible CDF diectly. Since R is not of Paeto shape and thee is positive pobability that the ealized distibution fom ν is not of Paeto shape eithe, it must be that thee exists x, 1) such that the vitual valuation induced by R at x is > and that the pobability that the opponent s highest vitual valuation is less than is stictly geate than being negative i.e. Φ ν ) > Φ ν )). 1 Then again fom 8), we may locally petub the oiginal CDF R to be Paeto-shaped, both at x, and at any point x, x ), as illustated by the two lowe cuves in Figue 4. Fo any x selected, thee will be a unique x such that if we emove all the pobability weights of ealizations below x to, the integal of the petubed CDF will be exactly 1 µ so that 1 Since if not, if ψ R) ψ on, 1), thee exists some G η π, ψ G F that gives a stictly highe expected suplus given ν being unchanged, see the appendix fo fomal aguments.

18 18 Ξx) Ξ = R 1 ζ x 1 ζ x ψ x R) x x ˆx x x 1 Figue 4: Local Deviation 2 it will be feasible, as illustated by the flat line in Figue 4, whee x is chosen so that the integal of the oiginal CDF and the petubed CDF is exactly the same. Togethe, we then have a feasible local petubation, R x, as a function of x. Clealy the deviation gain fom R to R x, is zeo when x =. To see that R x stictly impoves a buye s suplus fo some x > when all the opponents ae choosing ν, eaanging the deviation gain yields: Φ ν )1 R)) x1 Rx))) + Ψx), fo some almost-eveywhee) diffeentiable function Ψ with Ψ) =. Since is the eseve pice induced by R, 1 R)) x1 Rx)) fo any x [, 1]. Moeove, it can be veified that the deivative of Ψ at x conveges to Φ ν ) Φ ν ))1 R)) >, as x appoaches to and thus Ψx) > fo x > close enough to. Togethe, Rx is a pofitable deviation when x > is sufficiently close to. Combining the above two cases, thee is always a pofitable deviation fom R, but R suppν), which means that ν,..., ν) cannot be a Nash equilibium.

19 19 Poposition 1 shows that a necessay condition fo an infomation acquisition stategy ν G F ) to constitute a symmetic equilibium is that evey buye must be andomizing on the family of Paeto distibutions GF. As such, any equilibium stategy must has suppot on a family of distibutions that is pinned down by thee paametes π,, η satisfying 6). In fact, in what follows, we will show that any symmetic equilibium stategy can be descibed by a distibution on the vitual valuation,. To see this, obseve that given any infomation acquisition stategy ν G F ), and any G η π, choosing G η π, G F, expected suplus of a buye fom while othe buyes andomize accoding to ν is given by: )) Φ ν ψ x G η π, 1 G η π, x)) dx = Φ ν ) µ π + 1 η))), 9) due to 7) and the popety that ψx G η π, ) = fo all x [π/1 η) +, 1). The ight hand side of 9) suggests that when only consideing the Paeto family G F, the only stategic elevant component of one s choice is the pobability of winning Φ ν ), which depends only on one s own vitual valuation and the distibution of the opponent s highest vitual valuations. Futhemoe, it affects one s expected suplus in a multiplicatively sepaable way. Theefoe, egadless of the opponent s stategy, fix any vitual valuation, a buye s optimal choice of the othe two paametes within the Paeto family, π and η, must be the same. Moe pecisely, fo any [, 1], we may fist solve the following constaint minimization poblem that does not depend on the opponent s stategy: min [π + 1 η)] 1) π [,1],η [,1 µ] ) 1 )1 η) s.t. π log + π+1 η) = µ, η [, 1 µ], π [, 1 ]. π and let π), η) denote the solutions and ω) denote the value. This poblem can be solved by standad fist-ode appoach and solutions have the following popeties: Fo each [, 1], the solution is unique and theefoe π ), η ) ae functions; the value ω is stictly inceasing and is diffeentiable eveywhee except at at most one point and lim 1 ω) = µ We note that G η) π), is exactly the optimal signal stuctue of the single-buye poblem in Rosele & Szentes 217).

20 2 By uniqueness of the solution in 1), fo any [, 1], if π π) o η η), then )) Φ ν ψ x G η π, 1 G η π, x)) dx =Φ ν )µ π + 1 η))) <Φ ν )µ ω)) = Φ ν ψ x )) ) η) G π), 1 G η) π), x) dx. Thus, if ν G F ) constitutes a symmetic equilibium, accoding to 4) and Poposition 1, it must be that: suppν) { G η) π), } [, 1] GF. As a esult, any symmetic equilibium is then effectively a distibution of the induced vitual valuation. Futhemoe, fo any vitual valuation, given that the opponent is choosing ν, expected payoff is Φ ν )µ ω)). 11) Notice that 11) esembles the payoff function of a fim in the Betand competition model when the opponent is using andomized stategy. Indeed, the ole of vitual valuation is simila to the ole of pice in a Betand competition model: Fo highe vitual valuation, pobability of winning, Φ ν ), is highe but the expected valuation conditional on winning, µ ω)) would be lowe, just as lowe pice inceases the pobability that a fim wins the whole maet but the pofit conditional on winning will be lowe. In what follows, we will use aguments that ae simila to the deivation of a mixed stategy equilibium in the Betand competition model to fully descibe the necessay condition fo ν to be an equilibium. Specifically, we will fist use the indiffeence condition to detemine the shape of Φ ν so that Φ ν will be pinned down up to a scala. It is not had to see that the suppot of any equilibium must be an inteval and its lowe bound must be. Finally, we will use the consistency condition, which equies that the equilibium stategy must exactly eplicate Φ ν when we use it to deive the highest vitual valuation among N 1 buyes. Such consistency condition, togethe with the fact that the equilibium is a pobability measue, completely chaacteize the uppe bound of the suppot and the undetemined scala.

21 21 As noted above, the multiplicative sepaability of the expected suplus and the constaint minimization poblem 1) imply that any equilibium stategy is essentially a distibution of the vitual valuation induced by Paeto distibutions. Thus, any such stategy can be epesented by a distibution on [, 1]. Moe pecisely, fo any andomized stategy ν G F ) such that suppν) {G η) π), [, 1]} we can define the push-fowad measue { }) Γ) := ν G ηz) πz),z G F z, [, 1], and let := inf{z [, 1] Γz) > }, := sup{z [, 1] Γz) < 1} be the lowe bound and the uppe bound of the suppot, espectively. Such Γ [, 1]) then epesents the stategy ν. Moeove, if ν,..., ν) is a Nash equilibium, it must be that Φ ν )µ ω)) = Φ ν )µ ω )) 12) fo all, suppγ). Notice fist that since buyes can always guaantee a positive expected suplus, Φ ν ) >. 12 inceasing and Φ ν is constant on [, ], Futhemoe, it must be that =, since othewise, as ω is stictly Φ ν )µ ω)) > Φ ν )µ ω)) fo all [, ), which is a contadiction since G η) π), / suppν) if <. In fact, the same agument implies that suppγ) = [, ]. Togethe, let ν := Φ ν ), 12) can be stated as: fo all [, ]. Φ ν ) = νµ ω )) µ ω), 13) On the othe hand, since Φ ν is the distibution of the highest vitual valuation among N 1 buyes, togethe with 13), it must be that νµ ω )) = Φ ν ) = 1 πz) ) N 1 Γdz) + ηz)γdz)) 14) µ ω) 1 z 12 Othewise, thee exists { n }, 1], { n } such that the limit of buye s expected suplus when choosing along the sequence G n π n),η n) is lim n Φ ν n )µ ω n )) = and thus by 12), Φ n )µ ω n )) =, fo all n N, which implies that the equilibium suplus is zeo. This is a contadiction since a buye can always guaantee a positive expected suplus by choosing, say a distibution that gives pobability µ/1 ε) on 1 ε and pobability 1 µ/1 ε) on fo some ε > sufficiently small.

22 22 fo all [, ]. Indeed, since the tem in the paenthesis on the ight hand side is the pobability of one buye s vitual valuation being less than o equal to when andomizing accoding to Γ, the ight hand side is the pobability that the highest vitual valuation among N 1 buyes is less than o equal to when all of them ae andomizing accoding to Γ. As Γ epesents ν, the equality then follows fom the definition of Φ ν To sum up, we have shown that fo any ν G F ), ν,..., ν) is a symmetic equilibium only if the associated Γ solves the functional equation 14). By 14) and by diffeentiability of ω, Γ must be absolutely continuous with a density γ. Hence, diffeentiating both sides of 14) and eaanging yields: γ) := 1 ) N 2 µ ω) N 1 N 1 νµ ω )) νµ ω ))ω ) ), µ ω)) 2 1 π) η) 1 fo almost) all [, ]. As a esult, γ is now detemined up to a scala ν and the uppe bound of it s suppot,. To completely pin down the equilibium stategy Γ, fist notice that when evaluating 14) at =, we have ν 1 N 1 = 1 πz) ) Γdz). 1 z Togethe with the fact that Γ is a CDF and thus γ must integate to 1, we have two equations so that ν and can be completely detemined. Moe pecisely, let ξ) := 1 ) N 2 µ ω) N 1 N 1 νµ ω )) with, ν, 1) being the unique solution of the system 13 1 πz) ) ξz)dz = ν 1 N 1 1 z Let ξz)dz = 1. ξ), if [, ] γ ) :=, othewise νµ ω ))ω ) ), µ ω)) 2 1 π) η) 1 15), 16) 13 Existence and uniqueness of solutions is ensued by the popety that lim 1 ω) = µ, that ω is stictly inceasing and the intemediate value theoem.

23 23 and Γ ) := Then Γ is the unique solution fo 14). stategies, fo ν G F ) defined as { ν G ηz) πz),z G F γ z)dz. 17) Theefoe, by the epesentation of equilibium }) z := Γ ), [, 1] 18) ν,..., ν ) is then the only possible candidate fo a symmetic equilibium. To veify that ν,..., ν ) is indeed an equilibium, we will fist veify that fo each buye, any distibution in the suppot of ν is a best esponse among the Paeto family G F when the opponents andomize accoding to ν. Second, we will ague that given the opponents stategy ν, a best esponse among all feasible distibutions must be in the Paeto family. Specifically, fist ecall that ω ) is the value function of the constaint minimization poblem 1) and is stictly inceasing. Thus, fo any G η π, G F : Φ ν )) ψ x G η π, 1 G η π, x)) dx = Φ ν )µ π + 1 η)) Φ ν )µ ω)) Φ ν )µ ω )), whee the fist inequality follows fom optimality of π) and η) and the second inequality follows fom the facts that Φ ν ) = Φ ν ) fo any [, 1), that ω is stictly inceasing and that Φ ν )µ ω)) = Φ ν )µ ω )) fo all [, ] by the indiffeence condition. Theefoe, fo each buye, fo any G η) π), suppν ) thee is no pofitable deviation to any G η π, G F. The next Poposition then ensues that thee is no pofitable deviation to any othe distibutions G G F eithe. Poposition 2. Fo each buye, the set of best esponses when the opponents andomize accoding to ν is nonempty and is a subset of GF. That is, agmax R G F Φ ν ψx R))1 Rx))dx G F.

24 24 The poof of Poposition 2 contains two steps, which ae included in the appendix as Lemma 7 and Lemma 8. Specifically, Lemma 7 fist shows that fo any distibution R / G F, thee exists anothe distibution R G F that gives a stictly highe expected suplus when the opponents ae andomizing accoding to ν. The poof of this Lemma is simila to the local petubation agument in the poof of Poposition 1. This establishes the statement that the best esponse against ν must be a subset of GF. Lemma 8 then shows that the expected suplus, as a function of a buye s chosen distibution, is uppe-semicontinuous when the opponents andomize accoding to ν and theefoe the best esponse against ν exists. As a esult, ν,..., ν ) is indeed the unique symmetic equilibium at the infomation acquisition stage. We summaize the esult in this section by the following theoem. Theoem 1. Thee exists a unique symmetic Nash equilibium ν,..., ν ) at the infomation acquisition stage, whee ν G F ) with suppν ) G F 18) is defined by 15), 16), 17), With the chaacteization of the symmetic equilibium, we can examine the limiting behavios of the outcomes as numbe of buyes tends to infinity. Specifically, fix µ, 1), fo any N N, N 2, let ν N be the symmetic equilibium with N buyes given by Theoem 1 and Γ N [, 1]) be the associated epesentation of the equilibium stategy. Notice that since µ is fixed, G F is unchanged when N inceases. Let β N, σ N be the buyes total expected suplus and the selle s expected evenue and let τ N := β N + σ N be the total expected suplus, unde the equilibium ν N, espectively. The following Poposition summaizes the limiting behavio of equilibium outcomes. Poposition 3. As N, 1. {Γ N } conveges to the degeneate distibution δ {1} unde the wea-* topology. 2. lim N βn =. 3. lim N σ N = lim N τ N = 1. Pio to the chaacteization of the unique symmetic equilibium given by Theoem 1, it is not obvious what expected suplus will be and how it will be divided between the selle

25 25 and the buyes as numbe a buyes tend to infinity. Fo instance, given that all the buyes ae eceiving a non-infomative signals that ae independently dawn fom the maginal distibution F G F, the selle s evenue is bounded by the highest expected value dawn fom this distibution, namely µ < 1. As such, achieving efficient suplus and full suplus extaction does not necessaily follow when the numbe of buyes tend to infinity. Howeve, thans to the chaacteization in Theoem 1, Poposition 3 ensues that as numbe of buyes tend to infinity, due to competition among buyes at the infomation acquisition stage, all the buyes ae diven to acquie full infomation and buyes total suplus ae diven to zeo. Also, the selle does attain full suplus extaction. The poof of Poposition 3 can be found in the appendix. We conclude this section by consideing a numeical example. Conside the case when µ = 1/2 and N = 2. Using 15),.513 and ν.83. Figue 5 and Figue 6 depicts the equilibium density γ and CDF Γ. In equilibium, buyes total suplus is: 2 νµ ω )).18 wheeas the selle s evenue is [ 2 Φ ν )ψ x ) ) G η) π) [ 1 η),1) π), G η) π), dx) Γ d) and hence total suplus is.7. ] π) Φ ν ) 1 η) Γ d).52. Compaing to Rosele & Szentes 217), with N = 1 and µ = 3/4, 14 buye s suplus is appoximately.38 and selle s evenue is appoximately.37. This suggests that competition at the infomation acquisition stage among buyes educes buyes suplus and enhances selle s evenue. Howeve, when thee is moe than one buye, the esulting mechanism is no longe efficient. Futhemoe, with positive pobability, the signal stuctue that buyes choose is moe infomative then the optimal signal stuctue when N = 1, which suggests that competition among buyes also uges the buyes to acquie moe infomation. 14 Since when N = 2, thee ae two independent daws of valuation and thus the efficient suplus mechanically inceases compaing to the case with N = 1, we nomalize the pio mean so that the efficient suplus ae the same.

26 26 Gamma Figue 5: Cumulative Distibution Γ ) gamma Figue 6: Density γ ) 5 Chaacteizing the Equilibia with Regula Pio In this section, we will conside the case when the pio is egula and has full suppot on [, 1] and chaacteize the symmetic equilibia. Suppose that the pio F is absolutely continuous and admits a density f, has full suppot on [, 1] and that the function x x 1 F x) fx)

27 27 is stictly inceasing and continuous. As such, the vitual valuation induced by F is simply ψx F ) = x 1 F x), x [, 1] fx) and is stictly inceasing and continuous. With a pio that has full suppot on [, 1], the feasibility constaint is no longe an equality but a continuum of inequalities. Specifically, the second-ode stochastic dominance constaints must also be satisfied pointwisely. Recall that the set of feasible signal stuctues can be chaacteized by { x x G F := G : [, 1] [, 1] Gt)dt F t)dt, x [, 1] } Gx)dx = 1 µ, G is a CDF. As in the case when F has binay suppot, we fist obseve that thee is a paticula subset G F such that fo any symmetic equilibium ν,..., ν), the suppot of ν must be a subset of G F by using the local petubation aguments. Moe specifically, define G F collection of distibutions in G F that tae fom of: G x,x x) = 1 F x), if x [x, ˆx] x )1 F x)) x, if x [ˆx, x] with ˆx := x )1 F x))/1 F x)) +, x and x such that and which implies: x x G x,x G x,x x x)dx = F x)dx, x x)dx = F x)dx, x )1 F x)) x )1 F x))+ x )1 F x)) log ) 1 F x) = 1 F x) as the, 19) x x 1 F x))dx. Gaphically, as illustated in Figue 7, the set GF coesponds to the collection of distibutions whose integal is always below the gaph of the convex function x F z)dz and tangents to x F z)dz at x, x, being an affine function on [x, ˆx] and has Paeto-shape slope, with paametes x )1 F x)) and on [ˆx, x]. The paametes x and x detemines the ange

28 28 x Ξt)dt 1 µ slope = 1 x )1 F x)) x Ξ = G x,x Ξ = F slope = F x) x x )1 F x)) 1 F x) + x x 1 Figue 7: Typical Element in G F on which the distibution has Paeto shape and is again its constant vitual valuation on the inteval [ˆx, x]. By using simila local petubation aguments as in the poof of Poposition 1, we may conclude that fo any ν G F ), ν,..., ν) is a symmetic equilibium only if suppν) GF. Howeve, unlie the case when the pio is binay, G F is not pinned down by the paametes x, x and, as the condition given by 19) only applies fo the ealizations on the inteval [x, x]. Fo x [, 1]\[x, x], thee ae no futhe estictions fo shape of an element in G F othe than the second-ode stochastic dominance constaint. Futhemoe, as shown in the appendix, fo any symmetic equilibium ν,..., ν) and fo any G x,x suppν), given that the opponents ae andomizing accoding to ν, the expected suplus taes fom of: )) Φ ν ψ x G x,x 1 G x,x x)) dx = Lx, x, Φ ν ) + x Φ ν ψ x G x,x )) 1 G x,x x)) dx, fo some function L of the paametes x, x, ) that depends on the opponents stategies only though Φ ν. In paticula, although 2) does imply that the only stategic elevant object is the function Φ ν, it does not give multiplicative sepaability between the paametes x, x and 2)

29 29. As such, even though the local petubation agument gives a necessay condition that the suppot of any symmetic equilibium must be on GF, the equilibium stategy cannot be pinned down only by the indiffeence between choosing diffeent vitual valuations on the Paeto-shaped egion. Instead, the chaacteization of equilibium becomes a solution of a fixed point poblem. Fomal constuction of such fixed point poblem that chaacteizes the symmetic equilibia unde a full suppot and egula pio can be found in the appendix. Specifically, Theoem 2 shows that any symmetic equilibium in the infomation acquisition stage is essentially a fixed point of a functional. Convesely, fo any solution of such fixed point poblem, one can constuct a distibution ν G F ) such that ν,..., ν) is a symmetic equilibium. In the emainde of this section, we biefly descibe the constuction of the fixed point poblem and explain the equivalence in a less-fomal fashion. Fom the above obsevations, the main difficulty fo chaacteizing the equilibia unde a egula pio is twofold: Fist, the local petubation agument is not enough to pin down the equilibium stategies by finitely many paametes. Second, the paametes affects each buye s expected suplus in a non-multiplicative sepaable way. Fo the fist issue, notice that a distibution G x, x G F is not pinned down by the paametes only on the disjoint intevals [, x) and x, 1]. Fo the inteval [, x), it can be shown that maing the distibution identical to the pio will not educe expected suplus fo any stategy the opponents ae using. On the othe hand, on the inteval x, 1], given any induced distibution of the opponents highest vitual valuation Φ ν, a necessay condition fo G x, x is that it must be the best among all the feasible distibutions in G F to be in the suppot of the equilibium conditioning on the ealization being in the inteval x, 1], which is a well-defined maximization poblem and the solution of this poblem can then be chaacteized by the paametes x and. These then enable us to identify the equilibium stategies by using finitely many paametes. Howeve, due to the second issue, thee ae no futhe ways to sepaate the paametes and to futhe educe dimensionality. Nevetheless, any symmetic equilibium must still be a solution of a functional equation that consists of thee conditions: 1) The indiffeence condition. That is, the expected suplus must be the same fo any paametes x, x, such that G x, x is in the suppot of ν. 2) The consistency condition. That is, the highest vitual