Characterization of Pure-Strategy Equilibria in Bayesian Games

Transcription

1 Characterzaton of Pure-Strategy Equbra n Bayesan Games We He Yeneng Sun Ths verson: December 16, 2017 Abstract A genera condton caed coarser nter-payer nformaton s ntroduced and shown to be necessary and suffcent for the vadty of severa fundamenta propertes on purestrategy equbra n Bayesan games, such as exstence, purfcaton from behavora strateges, and convergence for a sequence of games. Our suffcency resuts cover varous earer resuts on pure-strategy equbra n Bayesan games as speca cases. New appcatons are presented as ustratve exampes, ncudng auctons wth externates and rsk-neutra bdders, Bertrand prcng games wth asymmetrc nformaton, and apay auctons wth rsk-averse bdders and nterdependent vaues. Keywords: Bayesan game, characterzaton, coarser nter-payer nformaton, purestrategy equbrum, equbrum exstence, purfcaton, cosed graph property, auctons, Bertrand competton The authors are gratefu to Y-Chun Chen, Jngfeng Lu, Wojcech Oszewsk and Bn Wu for hepfu conversatons. Earer versons of ths paper were presented n 6th Workshop on Stochastc Methods n Game Theory, Erce, May 2017; Chna Meetng of the Econometrc Socety, Wuhan, June 2017; 17th SAET Conference, Faro, June Ths paper was prevousy crcuated under the tte Bayesan games wth coarser nter-payer nformaton. Department of Economcs, The Chnese Unversty of Hong Kong, Shatn N.T., Hong Kong. E-ma: hewe@cuhk.edu.com. Department of Economcs, Natona Unversty of Sngapore, 1 Arts Lnk, Sngapore Ema: ynsun@nus.edu.sg 1

2 Contents 1 Introducton 3 2 Bayesan Games wth Coarser Inter-payer Informaton Mode Coarser nter-payer nformaton Man Resuts Exstence of pure-strategy equbra Condtona purfcaton Cosed graph property Appcatons 13 5 Dscussons 15 References 19 A Proofs of Man Resuts 21 A.1 Condtona dstrbuton of correspondences va vector measures A.2 Proofs of Theorem 1 and Proposton A.3 Proofs of Theorem 2 and Proposton A.4 Proof of Theorem B Proofs of Cams B.1 Proof of Cam B.2 Proofs of Cams B.3 Proofs n Cams

3 1 Introducton Bayesan games n the statc settng mode the nteracton of mutpe payers who make decsons smutaneousy wth ony parta nformaton about the payoffs of the other payers. Harsany formuated a genera Bayesan game by ntroducng a type mode, where a payer knows her own type but not others types. The payoff of a payer depends on the type and acton profes. A payer chooses an acton among mutpe choces, based on her observed type. Thus, a strategy of a payer s a compete pan of actons to be taken contngent on her types. Nature randomy chooses a type for each payer. A payer then evauates her expected payoff based on her beef about other payers types. Such a mode for ncompete nformaton has become a basc component of game theory and a standard too wth wdespread appcatons n many areas. Harsany s fundamenta work and many foow-up contrbutons on the genera theory of Bayesan games have been based on the concept of behavora strategy. Besdes Nature s randomzaton for the choce of types, a behavora strategy of a payer nvoves one more eve of randomzaton. After knowng her own type, the payer needs to choose some randomzaton devce to seect an acton. As noted n Radner and Rosentha 1982 and Mgrom and Weber 1985, ths knd of further randomzaton has been crtczed for ts mted appea n many practca stuatons. 1 Indeed, varous economc appcatons of Bayesan games have focused on pure strateges wthout the addtona randomzaton. Gven the centra mportance of Bayesan games and the assocated noton of purestrategy equbrum, a basc theoretca queston arses: can one fnd a genera condton to guarantee severa fundamenta propertes of pure-strategy equbrum n Bayesan games? 2 The frst property s about the exstence of pure-strategy equbra. The second s on the purfcaton from behavora strateges, whch s a usefu method to reate behavora strateges to pure strateges preservng the equbrum property. The thrd concerns the senstvty of equbrum outcomes to modeng assumptons; namey, whether the equbrum property remans under sma perturbatons on the game structure.e., the cosed graph property of equbrum. In ths paper, we ntroduce the condton of coarser nter-payer nformaton, and show ts suffcency for a the three propertes to hod. What s surprsng s that ths condton s aso necessary for the vadty of any of the propertes. As a resut, we have provded a defntve answer to the queston. In the speca case of a Bayesan game wth nterdependent payoffs and ndependent types, the ntuton behnd the condton of coarser nter-payer nformaton s that each 1 There are aso extensve dscussons on how to nterpret an equbrum wth randomzed strateges; see, for exampe, Osborne and Rubnsten 1994, p Fudenberg and Troe 1991, p. 236 ndcated n Remark 1 the need to fnd reguarty condtons for workng wth pure-strategy equbrum n genera Bayesan games beyond the speca case of condtonay ndependent types wth prvate vaues as n Mgrom and Weber Wthout sutabe condtons on Bayesan games, there may not be any pure-strategy equbrum. Radner and Rosentha 1982 presented a smpe 2-payer Bayesan game of matchng pennes n whch the ndvdua type space s the unt nterva, each payer s payoff does not depend on the types, and there s no pure-strategy equbrum. 3

4 payer s type can nfuence her own payoff fuy but other payers payoffs partay; more precsey, t means that the nter-payer nformaton of each payer s aways ess nformatve than her tota prvate nformaton, gven any nontrva event n her prvate nformaton. 3 When one aows the types to be ndependent condtoned on some common states, the condton says that the same knd of fu/parta nfuence s satsfed for each common state. The condton can aso be apped to the genera nformaton structure, where payers types can be correated and the common pror on the jont type space has a densty functon wth respect to the product dstrbuton of payers ndvdua type dstrbutons. 4 Such an nformaton structure s normay adopted n appcatons of Bayesan games. We frst consder the exstence of pure-strategy equbra. Theorem 1 characterzes ths exstence property by the condton of coarser nter-payer nformaton for each payer. Next, we ntroduce a new prncpe of condtona purfcaton Lemma 2 whch says that any randomzed decson rue can be purfed to yed the same expected payoffs and dstrbutons condtoned on some gven nformaton. In the settng of a Bayesan game, when a payer has ess nformaton nfuencng the other payers payoffs than her own, ths prncpe aows us to obtan n Theorem 2 a purfcaton from a behavora-strategy profe wth the same expected payoffs and dstrbutons condtoned on her nter-payer nformaton. 5 In addton, we show that the condton of coarser nter-payer nformaton for each payer s aso necessary for the purfcaton property n Bayesan games. The cosed graph property of equbrum means that any sequence of pure-strategy equbra, for a correspondng sequence of Bayesan games convergng to a mt game, has a subsequence convergng to a pure-strategy equbrum of the mt game. Thus, f the prmtves that determne a Bayesan game vary contnuousy, then the set of pure-strategy equbra shoud aso vary upper hemcontnuousy. The condton of coarser nter-payer nformaton s used agan n Theorem 3 to characterze ths cosed graph property. The proofs of the suffcency resuts n Theorems 1 3 are provded va estabshng a new connecton between Bayesan games and condtona dstrbuton of correspondences. For the necessty resuts n Theorems 1 and 2, a sequence of Bayesan games s carefuy constructed so that the equbrum exstence or purfcaton for these games mpes, for each payer, the exstence of many ndependent events n her prvate nformaton beyond her nter-payer nformaton, whch eads to the condton of coarser nter-payer nformaton. Furthermore, the proof of the necessty resut n Theorem 3 nvoves a nove 3 The forma defnton of ths condton s gven n Defnton 1 beow. 4 Ths assumpton, whch s standard n the terature, s to guarantee the contnuty of payers expected payoffs and the exstence of behavora strategy equbra; see, for exampe, Mgrom and Weber Wthout such an assumpton n the case of nfnte types, 1 the contnuty property may fa see Stnchcombe 2011a,b for detaed dscussons; 2 a Bayesan game may not have an equbrum even n behavora strateges see Smon 2003, Heman 2014 and Fredenberg and Meer Carbone-Ncoau and McLean 2017a,b have studed Bayesan-Nash equbra n behavora strateges when the payoffs are dscontnuous n actons as n Reny Such dscontnuty n actons s dfferent from the faure of contnuty n payers expected payoffs as n 1. 5 It means that a behavora strategy may not provde any addtona mert from the game-theoretc pont of vew snce t can aways be repaced by ts purfcaton wth the same propertes. 4

5 constructon of a doube sequence of Bayesan games. To ustrate how the condton of coarser nter-payer nformaton coud be used n specfc economc envronments, we present three exampes n Secton 4, whch cannot be covered by the prevous terature. We frst consder a cass of genera aucton games wth externates and rsk-neutra bdders, where the bdders have nterdependent vaues, and ndependent types condtoned on some common sgnas. 6 The second exampe s a Bertrand prcng game n whch frms wth prvate costs face unobservabe demand shocks and have nterdependent payoffs, where the frms receve prvate sgnas ndependenty, condtoned on the reazed shock. 7 It s easy to show that the condton of coarser nter-payer nformaton s satsfed by the Bayesan games n both exampes. Thus, the exstence of pure-strategy equbra n these two exampes foows from our genera resut n Theorem 1 for Bayesan games wth mutpe actons. A natura queston s whether Theorem 1 can be extended beyond the settng of mutpe actons. In genera, ths s not possbe even for the case of Bayesan games wth ndependent types and prvate vaues. 8 However, t s a standard procedure for obtanng pure-strategy equbra n Bayesan games wth contnuous choces and speca structures, based on mt arguments for pure-strategy equbra n a correspondng sequence of Bayesan games wth dscretzed mutpe actons. 9 We ustrate ths pont by another exampe of an a-pay aucton wth nterdependent vaues, mutdmensona types and CARA preferences. In ths exampe, a bdder chooses her bd from an nterva and pays what she bds. Our Theorem 1 ndcates the exstence of pure-strategy equbra n a correspondng sequence of a-pay auctons wth dscretzed bddng sets. A mt argument s then used to show that the sequence of pure-strategy equbra converges to a monotone pure-strategy equbrum for the orgna a-pay aucton. There has been an actve terature for fndng condtons to guarantee the exstence of pure-strategy equbra n Bayesan games. Radner and Rosentha 1982 worked wth the condtons of ndependent atomess types 10 and prvate vaues. Mgrom and 6 There s a substanta terature on auctons wth nterdependent vaues but ndependent types; see, for exampe, Jehe and Modovanu 2001 and the references theren. Auctons wth condtonay ndependent types and prvate vaues have been used n both the emprca and theoretca teratures; see, for exampe, L, Perrgne and Vuong 2000 and Athey Our departure from those papers s to aow both nterdependent vaues and condtonay ndependent types, whch goes beyond the case of condtonay ndependent types wth prvate vaues as n Mgrom and Weber To show the exstence of pure-strategy equbra n ogopoy prcng games wth asymmetrc nformaton, one often works n the settng of prvate vaue wth ndependent types, or mposes rch structures such as compementarty and supermoduarty on the games; see Vves 2001, Secton 8 for more dscussons. 8 Khan, Rath and Sun 1999 presented a two-payer Bayesan game wth unformy dstrbuted jont types on unt square and the nterva [ 1, 1] as the acton space for both payers, where the payoff of each payer ony depends on her own type and the types are drawn ndependenty. The condton of smpe coarser nter-payer nformaton s trvay satsfed n ths case. However, t was shown n Khan, Rath and Sun 1999 that such a game does not possess any pure-strategy equbrum. Therefore, the resuts n our Theorems 1-3 cannot be drecty apped to genera Bayesan games wth nfnte acton spaces. 9 See, for exampe, Reny 1999, 2011, Athey 2001 and Gentry, L and Lu The atomess types of a payer generay mean that every partcuar type has zero probabty; see Footnote 5

6 Weber 1985 and Fu et a aowed for payoffs wth prvate vaues and correatons among the payers by workng wth condtonay ndependent types. 11 Bare and Duggan 2015 consdered Bayesan games wth product structures. Our suffcency resuts n Theorems 1 and 2 cover the exstence and purfcaton resuts on pure-strategy equbra n those papers; 12 see Secton 5 for the detaed dscussons. As mentoned above, ths paper aso shows the condton of coarser nter-payer nformaton to be necessary for the reevant propertes of pure-strategy equbra to hod. There s another stream of terature on Bayesan games wth addtona order structures on the payoffs and acton sets. Vves 1990 showed the exstence of a argest and a smaest pure-strategy equbra. 13 Athey 2001 frst estabshed suffcent condtons for the exstence of a monotone pure-strategy equbrum, whch was generazed by McAdams 2003 to settngs wth mutdmensona actons and mutdmensona types. Reny 2011 dscovered contractbty to be automatcay satsfed gven any nonempty monotone best responses. A more powerfu fxed-pont theorem was adopted n Reny 2011 than those empoyed n Athey 2001 and McAdams 2003 based on contractbty rather than convexty to estabsh the genera exstence of a monotone pure-strategy equbrum. 14 The paper s organzed as foows. Secton 2 ntroduces the mode of Bayesan games wth coarser nter-payer nformaton. The man resuts are presented n Secton 3. Two appcatons to auctons and ogopoy prcng games are provded n Secton 4. Secton 5 dscusses how our suffcency resuts cover varous earer resuts as speca cases. The proofs of Theorems 1, 2 and 3 are gven n Appendx A. The proofs of a the cams n Sectons 2, 4 and 5 are eft n Appendx B. 2 Bayesan Games wth Coarser Inter-payer Informaton In ths secton, we frst present a forma mode of Bayesan games, and then ntroduce the key noton of coarser nter-payer nformaton to capture the nfuence of a payer s prvate nformaton on other payers densty weghted payoffs. 16 for more detas. In genera, t s not expected to have a pure-strategy equbrum n a Bayesan game wth fnte or atomc type spaces. Appcatons of Bayesan games often use ntervas or rectanges as the type spaces wth densty functons. Infnte types arse naturay even n a Bayesan game wth fntey many choces of payoff uncertanty, snce the set of payers beef types may st have the cardnaty of contnuum n such games; see, for exampe, the dscussons n Brandenburger and Deke 1993 and Hammond The reevant resuts on exstence and purfcaton foow from a genera purfcaton prncpe as dscussed n Dvoretsky, Wad and Wofowtz 1951 and Khan, Rath and Sun We may pont out that those papers dd not consder the cosed graph property for pure-strategy equbra of Bayesan games. 13 See aso Mgrom and Roberts 1990 and Mgrom and Shannon 1994 for further anayss of such supermoduar games. 14 For further appcatons and deveopments of monotone equbra, see, for exampe, Quah and Struovc 2012, Gentry, L and Lu 2015 and Prokopovych and Yannes

7 2.1 Mode A Bayesan game Γ can be descrbed as foows: The set of payers: I = {1, 2,..., n}. The prvate nformaton space of payer s T for each I. Each T s endowed wth the σ-agebra T. Let T be the product space n =1 T and T the product σ- agebra n =1 T. For each payer I, s a fnte set of actons uness otherwse noted avaabe to payer. Let = 1 n. Payer s sad to be dummy f contans ony one acton. 15 The nformaton structure: the common pror of a the payers s λ, a probabty measure on the measurabe space T, T. For each I, λ s the margna probabty of λ on T. Suppose that T, T, λ s atomess for each non-dummy payer I. 16 Let λ be absoutey contnuous 17 wth respect to the product measure 1 n λ, wth qt 1,..., t n beng the correspondng densty.e., Radon-Nkodym dervatve. 18 For each I, gven the acton profe x and the type profe t T, the payoff of payer s u x, t. We assume that each u s an ntegraby bounded mappng from T to R such that u x, s T -measurabe for each x. 19 Hereafter, the notaton denotes the set of a the payers n I except payer. Let λ = λ. Let M be the space of probabty measures on. For each payer I, a behavora strategy resp. pure strategy s a measurabe functon from T to M resp.. Snce a pure acton n can be regarded as a Drac measure n M concentrated at the pont n, a pure strategy can aso be vewed as a behavora strategy. Let L T the set of a behavora strateges for payer I, and L T = I LT. Gven a strategy profe f = f 1,..., f n, payer s expected payoff s U f = T u x, t I f t, dx λdt. 15 The types of a dummy payer can be vewed as some common states nfuencng other payers types and payoffs. We aways assume that there are at east two non-dummy payers to avod trvaty. 16 A measurabe set D n a probabty space Ω, F, P s sad to be an atom f P D > 0, and for any F- measurabe subset D 1 of D, P D 1 = 0 or P D. A probabty space s atomess f t has no atoms. When Ω s a Posh space wth the correspondng Bore σ-agebra F on Ω, Ω, F, P s atomess f and ony f every snge pont n Ω has measure zero. Smar to the remark n Footnote 10, t s aso not expected to have a pure-strategy equbrum n a Bayesan game wth atomc types n genera. 17 Let T, T, λ be a probabty space. A fnte measure ν s sad to be absoutey contnuous wth respect to λ f for any D T, λd = 0 mpes νd = 0. In ths case, there exsts a λ-amost unque λ-ntegrabe functon q such that νd = qtλdt for any D T. Such a functon q s caed the Radon-Nkodym dervatve of ν D wth respect to λ, see Loeb 2016, p As noted n Footnote 4, such an assumpton s standard n the terature. 19 A mappng φ: T R s sad to be ntegraby bounded f there s a rea-vaued ntegrabe functon h on T, T, λ such that φx, t ht for a x, t T. If u x, ony depends on payer s type t but not on other payers types, then she s sad to have prvate vaue. be 7

8 A behavora resp. pure strategy equbrum s a behavora resp. pure strategy profe f = f1, f 2,..., f n such that f maxmzes U f, f for each payer I. 2.2 Coarser nter-payer nformaton Before statng the condton of coarser nter-payer nformaton, we frst defne the noton of nowhere equvaence. Let Ω, F, P be an atomess fnte postve measure space, and G a sub-σ-agebra of F. For a set D F wth P D > 0, et G D resp. F D be the restrcted σ-agebra {D D : D G} resp. {D D : D F} on D. The σ-agebra F s sad to be nowhere equvaent to G under P f the strong competon of G D n F D under P s not equa to F D for any D F of postve measure. 20 Fx a non-dummy payer I. We frst consder her densty weghted payoff by ettng w x, t = u x, t qt for each x and t T. For a gven strategy profe f = f 1,..., f n, t s cear that payer s expected payoff can be rewrtten as U f = T w x, t I f t, dx I λ dt. Defnton 1. Let G be a countaby generated sub-σ-agebra of T. Payer s sad to have smpe coarser nter-payer nformaton f T s nowhere equvaent to G under λ, and w x,, t s G -measurabe for a x, t and. Payer s sad to have coarser nter-payer nformaton f T s nowhere equvaent to G under λ, and for some postve nteger J and each, w x, t = 1 j J w j x, t s I ρ j st s where for j = 1,..., J, 1 w j x, s 1 s nλ s -ntegrabe and w j x,, t s G -measurabe for a x and t, 2 ρ j s s nonnegatve and ntegrabe on T s, T s, λ s. A Bayesan game s sad to have smpe coarser nter-payer nformaton f each nondummy payer has smpe coarser nter-payer nformaton. Assume that payer has smpe coarser nter-payer nformaton n terms of G. Then, her tota nformaton s aways more than her nter-payer nformaton, gven on any nontrva event n T. By Lemma 3 beow, there s a countaby generated σ-agebra G such that G G T, T, G, λ s atomess and T s nowhere equvaent to G under λ. It s obvous that the G -measurabty condton n the above defnton s st satsfed when G s repaced by G, whch means that payer aso has smpe coarser nter-payer nformaton n terms of G. Thus, wthout oss of generaty, we assume that T, G, λ s atomess hereafter. 20 The strong competon of G D n F D under P s the coecton of a sets n the form E E 0, where E G D and E 0 s a P -nu set n F D, and E E 0 denotes the symmetrc dfference E \ E 0 E 0 \ E., 8

9 For Bayesan games wth coarser nter-payer nformaton, payers may have nterdependent payoffs and correated types. In partcuar, t s nessenta whether types are ndependent or correated, snce the densty q can be absorbed nto the densty weghted payoff. Note that the condton of smpe coarser nter-payer nformaton s a speca case of the condton of coarser nter-payer nformaton for J = 1. nterpret G as payer s nter-payer nformaton. In ths case, we can That s, G s payer s nformaton fow to a other payers, whch descrbes the nfuence of payer s prvate nformaton n other payers densty weghted payoffs. On the one hand, the measurabty of payer s payoff n terms of her own type s descrbed by the σ-agebra T ; namey, a the functons n the coecton {w x,, t : x, t } are measurabe wth respect to T. On the other hand, the measurabty of any other payer s densty weghted payoff n terms of payer s type s represented by the σ-agebra G. The condton of smpe coarser nter-payer nformaton goes beyond the case of ndependent atomess types and prvate vaues 21 n the sense that each payer s prvate nformaton can nfuence hersef fuy and other payers partay. However, Exampe 1 beow shows that a Bayesan game wth prvate vaues and ndependent types condtoned on bnary common states may fa to satsfy the condton of smpe coarser nter-payer nformaton. On the other hand, the condton of coarser nter-payer nformaton aows each payer s prvate nformaton to nfuence hersef fuy and other payers partay for each common state, whch means that the condton s satsfed n Exampe 1. Exampe 1. Suppose that there are two payers, I = {1, 2}. Payer s prvate nformaton space s T = [0, 1] endowed wth the Bore σ-agebra T = B[0, 1]. Both payers have fnte actons. For each I, payer s payoff functon u s bounded and ony depends on the acton profe x and her own prvate type t. The nformaton structure s descrbed as foows. Let T 0 = {H, L}, whch s the set of common states unobservabe to both payers. We assume that H and L are drawn wth equa probabty. If t 0 = H, then a par t 1, t 2 w be drawn from T 1 T 2 under the unform dstrbuton λ = η η, where η s the unform dstrbuton on [0, 1]. If t 0 = L, then a par t 1, t 2 T 1 T 2 w be drawn under the dstrbuton ˆλ, whch has the densty 4t 1 t 2 wth respect to η η. It s cear that condtoned on the unobservabe common state t 0 T 0, t 1 and t 2 are drawn ndependenty. There are two natura ways for representng ths exampe nto a Bayesan game. 21 In ths case, each payer s payoff u ony depends on the acton profe x and her own type t, and λ = I λ. Then, a the functons n the coecton {w j x,, t : x, t, j } do not depend on t, and hence are measurabe wth respect to the trva σ-agebra {, T }. By Lemma 3 beow, one can fnd a countaby generated sub-σ-agebra G T such that T, G, λ s atomess and T s nowhere equvaent to G. 9

10 1. One can add a dummy payer 0 wth a sngeton acton space and the prvate type space T 0. Then ths s a 3-payer game, n whch the space of type profes s T = T 0 T 1 T 2 and the common pror s λ I = 1 2 δ H λ δ L ˆλ, where δ t0 s the Drac measure concentrated at the state t 0 {H, L}. 2. One can aso vew ths exampe as a 2-payer game, n whch the space of type profes s T = T 1 T 2 and the common pror s λ II = 1 2 λ + ˆλ. Cam 1. In both cases, the Bayesan game above does not satsfy the condton of smpe coarser nter-payer nformaton for payers 1 and 2, whe the condton of coarser nterpayer nformaton does hod. The proof of the above cam s eft n Appendx B. 3 Man Resuts In ths secton, we state the man resuts of ths paper, whch characterze severa fundamenta propertes of pure-strategy equbra n Bayesan games, such as exstence, purfcaton from behavora strateges, and convergence for a sequence of games, by the condton of coarser nter-payer nformaton. The proofs are gven n Appendx A. Fx n 2 and the payer space I = {1, 2,..., n}. For each 1 n, payer has prvate nformaton space T, T, λ and nter-payer nformaton G. Let H n be the coecton of a Bayesan games wth the payer space I and the nformaton spaces {T, T /G, λ } I. 3.1 Exstence of pure-strategy equbra Any Bayesan game consdered n ths subsecton s n H n. The foowng theorem shows that the condton of coarser nter-payer nformaton s not ony suffcent, but aso necessary for the exstence of a pure-strategy equbrum. Theorem 1. Every Bayesan game has a pure-strategy equbrum f and ony f every non-dummy payer has coarser nter-payer nformaton. The proposton beow consders the case that some payers may or may not have coarser nter-payer nformaton. Proposton 1. If some payers have coarser nter-payer nformaton, then those payers can st pay pure strateges n some equbrum. Remark 1. In the proof of the necessty part of Theorem 1, we ndeed show that t s suffcent for a subcass of Bayesan games to have pure-strategy equbra. In partcuar, every payer has coarser nter-payer nformaton f ether of the foowng condtons hods: 1. every Bayesan game wth type-rreevant payoffs has a pure-strategy equbrum; 2. every Bayesan game wth ndependent types has a pure-strategy equbrum. 10

11 3.2 Condtona purfcaton By purfcaton n a fxed game, one usuay means a method to reate behavora-strateges to pure-strateges preservng the equbrum property n the game. In ths subsecton, we consder the noton of condtona purfcaton n terms of any behavora-strategy profe and any payer. Any Bayesan game consdered n ths subsecton s n H n. Defnton 2. Let f = f 1, f 2,..., f n and g = g 1, g 2,..., g n be two behavora-strategy profes. Fx payer I. 1. The strategy profes f and g are sad to be payoff equvaent for payer f U f = U g, and U h, f = U h, g for any gven behavora strategy h of payer. 2. The strategy profes f and g are sad to be condtona dstrbuton equvaent for payer f for any D G and B, D f B t λ dt = D g B t λ dt. 3. When g s a pure-strategy for payer, f and g are sad to be beef consstent for payer f g t supp f t for λ -amost a t T, where supp f t s the support of the probabty measure f t. When g s a pure-strategy for payer, g s sad to have a condtona purfcaton of f for payer f f and g are payoff equvaent, condtona dstrbuton equvaent, and beef consstent for payer. When g s a pure-strategy profe, g s sad to be a condtona purfcaton of f f f and g are payoff equvaent, condtona dstrbuton equvaent, and beef consstent for every payer. If f s an equbrum and g s a condtona purfcaton of f, then t s cear that g s an equbrum. The foowng theorem shows that the condton of coarser nter-payer nformaton s both necessary and suffcent for the exstence of a condtona purfcaton of a genera behavora strategy. Theorem 2. Every behavora-strategy profe n any Bayesan game possesses a condtona purfcaton f and ony f every non-dummy payer has coarser nter-payer nformaton. The foowng proposton shows that f some payers have coarser nter-payer nformaton, then those payers can st have a condtona purfcaton. Proposton 2. If payer has coarser nter-payer nformaton, then every behavorastrategy profe f has a condtona purfcaton for payer. 3.3 Cosed graph property In ths subsecton, we consder the cosed graph property for pure-strategy equbra n Bayesan games. It means that any sequence of pure-strategy equbra, for a sequence of Bayesan games convergng to a mt game, shoud possess a subsequence convergng to a pure-strategy equbrum of the mt game. 11

12 Let T be a Posh space.e., a compete separabe metrc space for each I. Suppose that there s a sequence of Bayesan game {Γ k } k 0 wth the payer space I = {1,..., n}, the type space T and the acton space for each I. In the k-th game, the common pror s a Bore probabty measure λ k wth margna λ k assumptons for each k 0. for payer. We make the foowng 1. For each I, λ k s absoutey contnuous wth respect to λ0 wth the Radon-Nkodym dervatve p k. Denote pk t = I pk t and p 0 t 1 for t = t 1,..., t n T. 2. The probabty λ k s absoutey contnuous wth respect to I λ k wth the Radon- Nkodym dervatve q k. The payoff functon of payer n Γ k s u k, whch s a bounded measurabe mappng from T to R. Denote w k x, t = u k x, t q k t for each x and t T. For k 1, gven a pure-strategy profe g1 k,..., gk n, we denote µ k = λk gk as the dstrbuton nduced by g k on MT : µ k B A = λ k {t B : g kt A } for any set B T and A. Defnton 3. A Bayesan game Γ 0 s sad to have the cosed graph property f gven any sequence of pure-strategy equbra {g k 1,..., gk n} k 1 for any sequence of Bayesan games {Γ k } k 1 such that 1. µ k weaky converges to some µ 0, 2. {q k p k } k 0 and {p k } k 0 are ntegraby bounded by some ψ 1 : T R + and ψ 2 : T R + respectvey based on the measure j I λ 0 j, 3. for each x, w k x, t converges to w 0 x, t and p k t converges to p 0 t for j I λ 0 j - amost a t T, 22 there exsts a pure-strategy equbrum g 0 1,..., g0 n n game Γ 0 wth µ 0 = λ0 g0 I. for each A Bayesan game Γ n H n s sad to be contnuous f q s contnuous I λ -amost everywhere, and for each I and x, u x, s contnuous I λ -amost everywhere. The foowng theorem shows that the condton of coarser nter-payer nformaton characterzes the cosed graph property for contnuous Bayesan games. Theorem 3. Every contnuous Bayesan game has the cosed graph property f and ony f every non-dummy payer I has coarser nter-payer nformaton. 22 Mgrom and Weber 1985 consdered the cosed graph property for behavora-strategy equbra, where the respectve sequences of payoff functons and denstes are assumed to be unformy convergent. 12

13 4 Appcatons Auctons wth externates We frst consder a genera aucton game wth externates. We sha show that the condton of coarser nter-payer nformaton s satsfed by ths cass of auctons, and hence a pure-strategy equbrum exsts. Let I = {1,..., n} be the set of bdders wth n 2. There are K aternatves. The state of the word s drawn from the set T 0 = {t 01,..., t 0J }, whch s unknown to the bdders. The state t 0j occurs wth probabty τ j > 0, and 1 j J τ j = 1. Each bdder has a prvate sgna t that s drawn from a compact rectange T R n K, endowed wth the Bore σ-agebra T = BT. Condtoned on the common state t 0j, bdder s sgna s drawn accordng to a Bore probabty measure λ j wth a densty functon q j > 0 wth respect to the Lebesgue measure on T, ndependent of other bdders sgnas. The nterpretaton s that the coordnate t k of t nfuences the vaue of bdder n aternatve k. Bdder s vauaton s hence v t 0j, k, t k 1,..., tk n when the common state s t 0j and the aternatve k s chosen. 23 R +. After recevng her sgna t, bdder submts a seaed bd b from a fnte set B n An aocaton rue Q: I B M{1,..., K} then determnes the aternatve that s to be chosen, and bdder pays a transfer x b 1,..., b n R for the bddng profe b 1,..., b n I B. Defne a transton probabty ν from T = 1 n T to MT 0 such τ that for 1 j J, ν{t 0j } t = j 1 n qj t 1 r J τ r I qr t, whch s the condtona probabty of the common state t 0j, gven the reazed type profe t T. When the type profe s t 1,..., t n and the bddng profe s b 1,..., b n, bdder s payoff s 1 k K 1 j J u t 1,..., t n, b 1,..., b n = v t 0j, k, t k 1,..., t k n ν{t 0j } t Qk b 1,..., b n x b 1,..., b n. Cam 2. In the above aucton, the condton of coarser nter-payer nformaton s satsfed, and hence a pure-strategy equbrum exsts. Bertrand ogopoy games Next, we consder a Bertrand prcng game n whch frms face demand shocks, and have prvate costs and nterdependent payoffs. The shock s unobservabe to the frms, whe the frms receve prvate sgnas ndependenty condtoned on the reazed shock. There are n 2 frms competng n the market. The set of unobservabe demand shocks n the market s T 0 = {t 01,..., t 0J }, where t 0j occurs wth probabty τ j. Condtoned on t 0j, 1 each frm receves a prvate sgna y from a compact nterva Y R endowed wth the Bore σ-agebra Y and a Bore probabty measure µ j on Y ; 2 each frm 23 Such knd of nterdependent payoff structure was consdered n Jehe and Modovanu 2001 for the case J = 1. 13

14 bears a prvate margna cost of producton z from a compact nterva Z R + endowed wth the Bore σ-agebra Z and a Bore probabty measure ν j on Z ; 3 the jont sgnas y 1, z 1,..., y n, z n are drawn based on the probabty measure I µ j νj. For each I, the densty of µ j νj wth respect to the product Lebesgue measure on T = Y Z s q j > 0. After observng the prvate sgna y, z, frm proposes a prce n the fnte set P. 24 Frm s payoff from servng the market s gven by p z Dp, t 0, y, where p P s the sae prce, and D s the market demand functon non-ncreasng n p for each t 0 T 0 and y = y 1,..., y n I Y. Frm serves the market f she has the owest prce, and has zero producton when her prce s strcty hgher than the owest prce. If there are more than one frms who propose the owest prce, then a those frms share the market wth equa probabty. The mode can aso be apped to procurement aucton settngs, where the bd s a per-unt prce, and the buyer s demand decreases wth the wnnng prce. Defne a transton probabty ν from 1 n T to MT 0 such that ν{t 0j } t = τ j 1 n qj t 1 r J τ r I qr t, 1 j J. When the sgna profe s t = t 1,..., t n and the prce profe s p = p 1,..., p n, frm s payoff s 1 j J u t, p = p z Dp,t 0j,y K ν{t 0j } t, p p, and #{ I : p = p } = K, 0, otherwse, where #{ I : p = p } s the number of eements n the set { I : p = p }, and t = y, z for I. Cam 3. In the Bertrand game wth ncompete nformaton, the condton of coarser nter-payer nformaton s satsfed, and hence a pure-strategy equbrum exsts. Monotone equbra n auctons wth CARA preferences Beow, we consder an a-pay aucton G wth nterdependent vaues, mutdmensona types and CARA preferences. There are n 2 potentay asymmetrc bdders. Each bdder s utty functon exhbts constant absoute rsk averson CARA. 25 We use ths exampe to ustrate that Theorem 1 can be usefu for provng equbrum exstence resut n some settngs when the acton spaces are not necessary fnte. Let I = {1,..., n} be the set of bdders. Each bdder has a prvate sgna t that s drawn from T = 1 n [0, t ] R n +, endowed wth the Bore σ-agebra T = BT. Bdder draws an affated sgna t accordng to a Bore probabty measure λ wth a densty functon q > 0 wth respect to the Lebesgue measure on T, ndependent of other bdders sgnas. After recevng her sgna t, bdder submts a seaed bd b from 24 It means that frms cannot set prces n unts smaer than, say, a penny. 25 Reny 2011 and Gentry, L and Lu 2015 recenty address the exstence ssue on the monotone equbrum n envronments wth CARA preferences and a contnuum of bds. These two papers focus on the case wth prvate vaues. Our exampe aows for the case wth nterdependent payoffs. 14

15 B = {o} [b, b ] wth b > b 0, where o < 0 represents the outsde opton, and [b, b ] s the set of feasbe bds wth the reserve prce b. Any bdder submttng a bd above o pays her bd. The mappng Q : I B [0, 1] determnes the probabty that bdder wns the good. The bdder wth the hghest bd above o wns the object. The bdder submttng o wns wth probabty 0, and does not need to pay. If there are more than one bdders submttng the hghest bd above o, then a these bdders wn wth equa probabty. The constant rsk averson eve of bdder s α > 0. Gven the fna vaue r, the payoff of bdder s u r = 1 e α r α. Foowng the payoff structure as used n Jehe and Modovanu 2001 and the frst exampe of ths secton, we assume that the vaue of bdder at the sgna profe t 1,..., t n s gven by v t, t = t + κ t + c, where κ 0 for 1 n and c 0. That s, 1 the coordnate t of t nfuences the vaue of bdder ; 2 c can be vewed as the nta weath of bdder. If bdder wns the good by bddng b o, then her payoff s u v t, t b. If bdder oses by bddng b 0, then she gets u c b. If bdder chooses o, then she gets u c. We sha prove that a monotone pure-strategy equbrum exsts n the aucton G. We frst construct a sequence of auctons {G k } k=1 wth dscretzed bddng sets. For k suffcenty arge, we verfy that the condton of coarser nter-payer nformaton s satsfed n G k, and hence a pure-strategy equbrum exsts by Theorem 1. We then prove the exstence of a monotone pure-strategy equbrum n the aucton G by foowng some standard mt arguments. Cam 4. The aucton G has a pure-strategy monotone equbrum. 5 Dscussons In ths secton, we revst varous reevant modes of Bayesan games n the terature. 26 We expcty show that the resuts on pure-strategy equbrum exstence and purfcaton n those modes are covered as speca cases of our suffcency resuts n Theorems 1 and 2. Games wth prvate nformaton: mutua ndependence Radner and Rosentha 1982 consdered a cass of Bayesan games n whch the payoffreevant nformaton and strategy-reevant nformaton are expcty separated, and each payer s prvate nformaton s ndependent of the strategy-reevant nformaton of a the other payers. 1. For each payer I, T = Z Y, where Z represents the strategy-reevant nformaton and Y represents the payoff-reevant nformaton. endowed wth the σ-agebra Z and Y respectvey, and T = Z Y. 2. The common pror s λ on T, T, where T = I T and T = I T. Let Z and Y be 26 For smpcty, we cosey foow our notatons for Bayesan games, and ony hghght the changes made to accommodate the partcuar structures of those modes. 15

16 3. Let z resp. ỹ be the projecton from t = z 1, y 1,..., z n, y n to z resp. y. Suppose that the dstrbuton κ nduced by z on Z, Z s atomess. 4. Suppose that the random varabes { z k } k and z, ỹ are mutuay ndependent for each. 5. For each payer, the payoff functon u s an ntegraby bounded mappng from Y R A pure strategy for payer I s a Z -measurabe mappng from Z to. Let v : Z R + be the condtona expectaton of u as foows: v x, z = E u x, ỹ z = z. Consder an auxary game n whch payer I has the acton space, the prvate nformaton space Z, and the payoff functon v x, z. The common pror s κ = I κ. The condton of smpe coarser nter-payer nformaton hods for the auxary game, and hence a pure-strategy equbrum exsts. The foowng cam shows that t s aso a pure-strategy equbrum n the orgna game. Cam 5. The expected payoffs of any payer for any gven strategy profe are the same n the orgna game and n the auxary game. Thus, a pure-strategy equbrum n the auxary game s aso a pure-strategy equbrum n the orgna Bayesan game. Games wth prvate nformaton: condtona ndependence Mgrom and Weber 1985 consdered a cass of Bayesan games wth prvate vaues n whch payers prvate nformaton are ndependent condtoned on fntey many common states. As demonstrated by Exampe 1, such games may fa to satsfy the condton of smpe coarser nter-payer nformaton. We show that the condton of coarser nter-payer nformaton aways hods for ths cass of Bayesan games. For each I, payer s prvate nformaton space s T, T. Let T 0 = {t 01,..., t 0J } be the space of common states that affect the payoffs of a the payers. The state t 0j happens wth probabty τ j > 0 for 1 j J. Gven each t 0j T 0, et λ j be the condtona pror on I T, I T. The margna λ j of λj on T, T s atomess and λ j = I λ j. For each, payer s payoff s an ntegraby bounded functon from T 0 T to R +. A pure strategy of payer s a T -measurabe mappng from T to. As n Exampe 1, we consder two possbe formuatons of the game. 1. One can add a dummy payer 0 wth a sngeton acton space and the prvate type space T 0. Then, ths s a n + 1-payer game, n whch the space of type profes s T = 0 n T and the common pror s λ = 1 j J τ j δ t0j λ j. 16

17 2. The mode s vewed as an n-payer game, n whch the space of type profes s T = I T and the common pror s ˆλ = 1 j J τ j λ j. Cam 6. Both formuatons of the above Bayesan game satsfy the condton of coarser nter-payer nformaton. Games wth prvate and pubc nformaton Fu et a ntroduced a new cass of Bayesan games n whch payers strateges depend on ther strategy-reevant prvate nformaton as we as on some pubcy announced nformaton, and payers payoffs depend on ther own payoff-reevant prvate nformaton and aso payoff-reevant common nformaton. For each payer I, T = Z Y, where Z represents the strategy-reevant nformaton and Y represents the payoff-reevant nformaton. endowed wth the σ-agebras Z and Y respectvey, and T = Z Y. Let Z and Y be A fnte set Z 0 = {z 01,..., z 0J } represents the strategy-reevant states that are pubcy announced to a the payers. Another fnte set Y 0 = {y 01,..., y 0K } represents those payoff-reevant common states that affect the payoffs of a the payers. Denote the power set of Z 0 resp. Y 0 by Z 0 resp. Y 0. Let T 0 = Z 0 Y 0 and T 0 = Z 0 Y 0. The common pror λ s a jont probabty on 0 n T, 0 n T. Gven each z 0j Z 0 and y 0k Y 0, the par z 0j, y 0k happens wth probabty τ jk such that 1 k K τ jk > 0 for each j. Let λ jk be the condtona probabty on I Z Y, I Z Y, gven z 0j Z 0 and y 0k Y 0. Let λ jk be the margna of λ jk on T, T. Let ν jk be the margna of λ jk on Z Z Y, Z Z Y, µ jk be the margna of λ jk on Z Y, Z Y, and κ jk Suppose that ν jk = κ jk µ jk, and κ jk be the margna of λ jk on Z, Z. s atomess. For each payer I, the payoff functon s an ntegraby bounded mappng from Y 0 Y R +. A pure strategy of payer s a Z 0 Z -measurabe mappng from Z 0 Z to. For 1 j J and 1 k K, et v j x, y 0k, z = E u x, y 0k, ỹ z = z, where ỹ resp. z s the projecton from Z Y to Y resp. Z, and the condtona expectaton s taken on the probabty space Z Y, Z Y, µ jk. Consder J auxary games {Γ j } 1 j J such that n the j-th game Γ j, 1 the space of common states s Y 0 = {y 01,..., y 0K }, 2 payer I has the acton space, the prvate nformaton space Z, Z, and the payoff functon v j, 3 the common pror wth a dummy payer 0 s ˆλ j 1 = 1 k K τ jk 1 k K τ jk δ y0k 1 n κ jk. As shown n Cam 6, the condton of coarser nter-payer nformaton hods for each auxary game Γ j, and hence a pure-strategy equbrum f j exsts n Γ j. 17

18 Cam 7. For each I, et g z 0j, z = f j z for z 0j Z 0 and z Z. Then {g } I s a pure-strategy equbrum n the orgna game. Games wth product structure Bare and Duggan 2015 consdered a cass of Bayesan games n whch each payer s prvate nformaton space has a product structure. As shown beow, the game theren satsfes the condton of smpe coarser nter-payer nformaton. 1. For each payer I, T = Z Y, where Z s a Posh space endowed wth the Bore σ-agebra Z, and Y s a Posh space endowed wth the Bore σ-agebra Y, T = Z Y. Let z represent the genera type of payer that may affect every payer s payoff, and y be the prvate type that ony affects payer s payoff. Denote Z = I Z and Z = I Z, Y = I Y and Y = I Y. 2. Let κ be a probabty measure on Z, Z, and κ the margna of κ on Z, Z. Suppose that κ s absoutey contnuous wth respect to I κ wth the Radon- Nkodym dervatve q. 3. Let ν : Z MY be a transton probabty such that ν z s atomess for each z, and ν z = I ν z for z Z. Denote λ = κ ν, whch s the common pror of a the payers. 4. Payer s payoff s an ntegraby bounded mappng from Z Y to R A pure strategy for payer s a T -measurabe mappng from T to. Cam 8. The product Bayesan game above satsfes the condton of smpe coarser nterpayer nformaton. 18

19 References Susan Athey, Snge crossng propertes and the exstence of pure strategy equbra n games of ncompete nformaton, Econometrca , Pauo Bare and John Duggan, Purfcaton of Bayes Nash equbrum wth correated types and nterdependent payoffs, Games and Economc Behavor , Patrck Bngsey, Convergence of Probabty Measures, John Wey & Sons, Adam Brandenburger and Edde Deke, Herarches of beefs and common knowedge, Journa of Economc Theory , Oro Carbone-Ncoau and Rchard P. McLean, On the exstence of Nash equbrum n Bayesan games, Mathematcs of Operatons Research 2017a, pubshed onne. Oro Carbone-Ncoau and Rchard P. McLean, Nash and Bayes-Nash equbra n strategc-form games wth ntranstvtes, workng paper, Rutgers Unversty, 2017b. Aryeh Dvoretsky, Abraham Wad and Jacob Wofowtz, Emnaton of randomzaton n certan statstca decson procedures and zero-sum two-person games, Annas of Mathematca Statstcs , Amanda Fredenberg and Martn Meer, The context of the game, Economc Theory , Hafeng Fu, Yeneng Sun, Nchoas C. Yannes and Zhxang Zhang, Pure strategy equbra n games wth prvate and pubc nformaton, Journa of Mathematca Economcs , Drew Fudenberg and Jean Troe, Game Theory, MIT Press, Cambrdge, Matthew Gentry, Tong L and Jngfeng Lu, Exstence of monotone equbrum n frst prce auctons wth prvate rsk averson and prvate nta weath, Games and Economc Behavor , Peter J. Hammond, Expected utty n non-cooperatve game theory, n Handbook of Utty Theory Savador Barbera, Peter J. Hammond, and Chrstan Sed, eds., vo. 2, ch. 18, , Boston, Kuwer Academc Pubshers, John C. Harsany, Games wth ncompete nformaton payed by Bayesan payers, Parts I III, Management Scence , , , and We He and Yeneng Sun, Condtona expectaton of correspondences and economc appcatons, Economc Theory 2017, pubshed onne. We He, ang Sun and Yeneng Sun, Modeng nfntey many agents, Theoretca Economcs , Zv Heman, A game wth no Bayesan approxmate equbra, Journa of Economc Theory , Phppe Jehe and Benny Modovanu, Effcent desgn wth nterdependent vauatons, Econometrca , M. A Khan, Ka P. Rath and Yeneng Sun, On a prvate nformaton game wthout pure strategy equbra, Journa of Mathematca Economcs , M. A Khan, Ka P. Rath and Yeneng Sun, The Dvoretzky-Wad-Wofowtz theorem and purfcaton n atomess fnte-acton games, Internatona Journa of Game Theory ,

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21 Appendces Appendx A contans the proofs of Theorems 1, 2 and 3. The proofs of Cams 1 8 are eft n Appendx B. A Proofs of Man Resuts The type and strategy of a payer I n a Bayesan game nfuence other payers expected payoffs ony through her condtona acton dstrbuton, gven her nter-payer nformaton G. Gven other payers condtona acton dstrbutons wth respect to ther nter-payer nformaton, payer can choose her optma actons based on her own type. The ssue s that the set of optma strateges taken from such best-responses s usuay nether convex nor compact. Thus, the usua approach for obtanng pure-strategy equbra va a fxed-pont argument or purfcaton fas. However, f payer has coarser nter-payer nformaton.e., T s nowhere equvaent to G, then, by Lemma 1 beow, the set of condtona acton dstrbutons n terms of a vector measure for her optma strateges, 27 gven her nter-payer nformaton G, s convex, compact, and upper hemcontnuous n terms of other payers condtona acton dstrbutons. Based on ths too, the suffcency parts of Theorems 1 and 2 can be proved. The proof for the suffcency part of Theorem 3 needs some decate mtng arguments. For the necessty part of Theorem 1, the subtety s to construct a sequence of Bayesan games so that the exstence of pure-strategy equbra n these games mpes, for each payer and postve nteger m, the exstence of an equa partton {E 1, E 2,..., E m } of T, T, λ ndependent of payer s nter-payer nformaton G. By Lemma 3, T s nowhere equvaent to G. The necessty part of Theorem 2 can be obtaned by the same argument. The proof for the necessty part of Theorem 3 nvoves the constructon of a doube sequence of Bayesan games. For each game correspondng to the postve nteger m n the sequence of Bayesan games for provng the necessty part of Theorem 1, we need to go construct a new sequence of Bayesan games so that the vadty of the cosed graph property for the new sequence mpes for each payer, the exstence of an equa partton {E 1, E 2,..., E m } of T, T, λ ndependent of payer s nter-payer nformaton G. Lemma 3 ndcates that payer has coarser nter-payer nformaton. A.1 Condtona dstrbuton of correspondences va vector measures Let T, T be a measurabe space, and µ k an atomess fnte postve measure on T, T for 1 k K. Denote µ = µ 1,..., µ K. Let µ be a probabty measure such that µ k s absoutey contnuous wth respect to µ for each k. Suppose that G s a countaby generated sub-σ-agebra of T, and s a fnte set. The foowng s the standard defnton of a transton probabty. Defnton 4. A G-measurabe transton probabty from T to s a mappng φ: T M such that φ, B: t φt, B s G-measurabe for every B. Gven the measure µ k, the space of a G-measurabe transton probabtes from T, G, µ k to M s denoted by R T,G,µ k, 1 k K. 27 For a payer s I, the nonnegatve functons ρ j s, j = 1,..., J n Defnton 1 can be naturay used as densty functons for a vector measure. 21