Quadratic Games. First version: February 24, 2017 This version: August 3, Abstract

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1 Quadratc Games Ncolas S. Lambert Gorgo Martn Mchael Ostrovsky Frst verson: February 24, 2017 Ths verson: August 3, 2018 Abstract We study general quadratc games wth multdmensonal actons, stochastc payoff nteractons, and rch nformaton structures. We frst consder games wth arbtrary fnte nformaton structures. In such games, we show that there genercally exsts a unque equlbrum. We then extend the result to games wth nfnte nformaton structures, under an addtonal assumpton of lnearty of certan condtonal expectatons. In that case, there genercally exsts a unque lnear equlbrum. In both cases, the equlbra can be explctly characterzed n compact closed form. We llustrate our results by studyng nformaton aggregaton n large asymmetrc Cournot markets and the effects of stochastc payoff nteractons n beauty contests. Our results apply to general games wth lnear best responses, and also allow us to characterze the effects of small perturbatons n arbtrary Bayesan games wth fnte nformaton structures and smooth payoffs. We thank Kostas Bmpks, Ben Golub, Tbor Heumann, Johannes Hörner, Matt Jackson, Davd Myatt, Alessandro Pavan, Andy Skrzypacz, Xaver Vves, Bob Wlson, Anthony Lee Zhang, and semnar and conference partcpants at Stanford, UIUC, the 22nd Coalton Theory Network Workshop, the 2017 Workshop on Markets wth Informaton Asymmetres at Collego Carlo Alberto, the 2018 ASSA Annual Meetng, and the 2018 North Amercan Summer Meetng of the Econometrc Socety for helpful comments and suggestons. Lambert s grateful to Mcrosoft Research New York and the Cowles Foundaton at Yale Unversty for ther hosptalty and fnancal support. Stanford Graduate School of Busness; nlambert@stanford.edu. Stanford Graduate School of Busness; gmartn@stanford.edu. Stanford Graduate School of Busness and NBER; ostrovsky@stanford.edu. 1

2 1 Introducton Games of ncomplete nformaton wth quadratc payoff structures and lnear best responses play an mportant role n the analyss of a dverse varety of questons n economcs, such as the value of macroeconomc nformaton (Morrs and Shn, 2002; Angeletos and Pavan, 2007), the structure of markets under Cournot and Bertrand competton (Vves, 1984; Gal-Or, 1986), the propertes of games on networks (De Martí and Zenou, 2015; Lester, 2016; Myatt and Wallace, 2017), team producton and coordnaton n organzatons (Radner, 1962; Dessen and Santos, 2006), and the behavor of leaders n poltcal communcaton (Dewan and Myatt, 2008, 2012), among many others. In ths paper, we provde a convenent unfed framework for the analyss of rch versons of such games, allowng for a varety of features: multdmensonal actons, stochastc payoff nteractons, and general nformaton structures. Among many other possbltes, our framework accommodates Cournot competton wth asymmetrc nformaton about both the ntercept and the slope of the demand functon, beauty contests n whch the ncentves to conform are heterogenous and stochastc, and games on networks n whch the knowledge of the network structure s partal and dspersed. We consder two types of nformaton structures. In the frst, any fnte nformaton structure s allowed. In partcular, the sgnals of dfferent players may be arbtrarly correlated, may overlap, may contan nformaton about the sgnals of other players, and so on we allow for the same level of generalty as the framework of Aumann (1976), subject only to the fnteness of the sgnal space. We show that genercally there exsts a unque Bayesan Nash equlbrum, and gve a closed-form characterzaton of equlbrum strateges. We then consder nformaton structures that combne fnte and (potentally) nfnte sgnals. The jont dstrbuton of the fnte components of the sgnals s arbtrary, as n the prevous case. The nfnte components have the restrcton that certan condtonal expectatons must be lnear. Ths setup ncludes, as a specal case, the Lnear-Quadratc-Gaussan setup, where (nfnte) sgnals are dstrbuted accordng to a multvarate normal dstrbuton. By also allowng for arbtrary fnte sgnal components, we can capture varous stochastc nteractons and heterogeneous nformaton about the nfnte sgnals (for example, about ther correlatons and observablty across players). We show that genercally there exsts a unque lnear Bayesan Nash equlbrum, and gve a closed-form characterzaton of equlbrum strateges. To llustrate our framework and results, we present two sets of applcatons. Frst, we consder Cournot competton wth quadratc producton costs and lnear demand functons, both of whch may be only partally known by frms. Ths model fts our framework wth fnte sgnals, and as n that framework, we allow for arbtrary nformaton structures about the costs and about the demand functon. In partcular, we allow not just the ntercept but also the slope of the demand functon to be only partally known to the frms. We use our general results to show that n ths settng, an equlbrum of any Cournot competton game always exsts and s unque, and characterze t n closed form. We then prove two results about decentralzed effcency n large asymmetrc markets. Frst, as the market grows large, equlbrum becomes ex-ante effcent. Second, when costs are lnear, equlbrum forces act as an nvsble statstcan : the equlbrum 2

3 quantty converges to the best lnear estmate of the ex-post socally effcent quantty, gven the nformaton avalable to the frms. Second, we analyze beauty contests n whch players may target dfferent state varables and are allowed to have only ncomplete nformaton about the payoff structure of others. Ths generalzaton makes t possble to consder rch versons of beauty contests and coordnaton games: e.g., one can consder a coordnaton game on a network n whch each player knows who her frends are, but has only partal nformaton about the rest of the network, the strengths of the connectons there, etc. In ths framework, we show that there always exsts a unque lnear equlbrum, and characterze t n closed form. We llustrate the framework wth an example that shows some of the novel effects that arse when the strengths of payoff nteractons n a beauty contest are stochastc. We also present several addtonal fndngs. Frst, we dscuss how our closed-form characterzatons (both for the fnte and nfnte nformaton cases) further smplfy f payoff nteractons between players actons are determnstc rather than stochastc (but there s stll uncertanty about the random varables that drectly affect the players payoffs, as n, e.g., the conventonal beauty contests or n Cournot competton games n whch the slope of the demand functon s known). Second, we dscuss the applcablty of our results to general games wth lnear best responses. Fnally, we show how our framework can be used to characterze the effects of small perturbatons n arbtrary Bayesan games wth fnte nformaton structures and smooth payoffs. For the last applcaton, t s essental that the framework allows for stochastc payoff nteracton terms; wthout ths feature, such characterzaton would not be possble. The paper s organzed as follows. In Secton 1.1, we brefly revew the related lterature on ncomplete-nformaton games wth quadratc payoff structures (the lteratures on Cournot competton and beauty contests are brefly revewed n the correspondng sectons). We present the model for arbtrary fnte nformaton structures n Secton 2, and prove the exstence and unqueness of Bayesan Nash equlbrum. We apply ths framework to the questons of equlbrum characterzaton and nformaton aggregaton n Cournot markets n Secton 3. In Secton 4, we present the rcher model wth both fnte and potentally nfnte sgnals. Applcatons to beauty contests are presented n Secton 5. In Secton 6, we present the smplfed closed-form characterzaton for the case n whch payoff nteracton coeffcents are determnstc. Fnally, n Secton 7, we dscuss the applcatons of our framework to general games wth lnear best responses and to the characterzaton of the effects of small perturbatons n general Bayesan games. 1.1 Related Lterature A large lterature spannng multple felds has nvestgated ncomplete-nformaton games wth quadratc payoff structures. In ths secton, we brefly revew some key papers on the general propertes of such games, hghlghtng the relatve contrbutons of our work. We revew the lteratures more drectly related to our applcatons to Cournot competton and to beauty contests n Sectons 3 and 5, respectvely. The semnal paper of Radner (1962) formalzes team problems as mult-agent decson problems 3

4 n whch agents share a common goal but nformaton s decentralzed, and shows that under the quadratc payoff specfcaton, the person-by-person optmal decson polces are lnear n sgnals. U (2009) uses Bayesan potental games n order to apply Radner s methods to games n whch the common goal assumpton does not hold. A lmtaton of ths approach s that varous symmetry assumptons on the payoffs are requred for the games to admt a potental functon. In contrast, our paper allows for general asymmetrc payoff structures. Basar (1978a) allows for asymmetres n the payoff structure, as well as for multdmensonal actons and sgnals. He employs a contracton-mappng approach that gves suffcent condtons for equlbrum unqueness n general. However, ths approach does not gve necessary and suffcent condtons under whch the lnear equlbrum exsts and s unque. Furthermore, n contrast to Basar (1978a), we provde closed-form solutons for the unque lnear equlbrum, and allow for stochastc payoff nteracton terms. Basar (1978b) studes 2-player quadratc games n whch payoff nteracton terms are allowed to be stochastc. Lke Basar (1978a), Basar (1978b) only gves suffcent condtons for equlbrum exstence, and does not provde closed-form solutons. A recent paper that also studes general quadratc games wth rch asymmetrc nformaton structures s Bergemann, Heumann, and Morrs (2017). The focus of that paper, however, s dfferent from ours. Bergemann, Heumann, and Morrs (2017) focus on games wth sngle-dmensonal actons, fxed payoff nteracton terms, and Gaussan nformaton structures, and explore such topcs as dentfcaton n games, the range of outcomes across nformaton structures, and connectons to Bayes Correlated Equlbrum (Bergemann and Morrs, 2013, 2016). By contrast, we provde exstence, unqueness, and closed-form characterzaton results for games wth multdmensonal actons, stochastc payoff nteracton terms, and general nformaton structures wth lnear condtonal expectatons. Ths level of generalty allows us to explore such applcatons as Cournot competton games wth uncertan slope of demand, beauty contests n whch the weghts that players put on matchng others actons are stochastc, and the characterzaton of the effects of small perturbatons n arbtrary Bayesan games wth fnte nformaton structures and smooth payoffs. 2 Games wth Fnte Informaton Structures In ths secton, we state and prove our frst man result: generc equlbrum exstence and unqueness for quadratc games wth arbtrary fnte nformaton structures. We mpose no restrctons on the nformaton structure (beyond fnteness), whch enables us to represent arbtrary nformatonal asymmetres and nterdependences. 1 In Secton 4, we present parallel results for games wth potentally nfnte state and sgnal spaces, under some addtonal assumptons. 1 Our nformatonal framework s equvalent to the canoncal parttonal nformaton structure of Aumann (1976) wth a fnte underlyng state space, although ts mathematcal descrpton s slghtly dfferent. 4

5 2.1 Model There are n players, = 1,..., n. Pror to takng an acton, each player prvately observes a sgnal s whose set S of possble realzatons s fnte and contans k 1 elements. We denote by s = (s 1 ; s 2 ;... ; s n ) the random vector summarzng the sgnals of all n players. Players share a common pror belef P about the jont probablty dstrbuton over sgnal vectors s. Wthout loss of generalty, we assume that for every player and realzaton s S, the probablty of player observng sgnal s S s postve. We do not mpose any other restrctons on dstrbuton P. As n Aumann (1976), ths lack of restrctons s the key feature that allows our framework to accommodate arbtrary nformaton asymmetres and nterdependences, by approprately choosng the jont dstrbuton P over sgnal vectors s. 2.2 Actons and Payoffs After observng hs sgnal s, each player chooses acton a R m, where m 1 s the dmenson of player s acton. The payoff π of player depends on hs own acton a, the actons of other players (denoted by vector a ), and the vector of sgnals s. Specfcally, π (a, a, s) = 1 2 at Γ (s)a + a T Γ, (s)a + a T g (s) + h (a, s), (1) where Γ (s) s a matrx of sze m m, Γ, (s) s a matrx of sze m ( j m j), g (s) s a vector of sze m, and h s an arbtrary functon of a and s. Note that the presence of functon h does not affect the ncentves of player (and thus the equlbra of the game), but does n general affect the effcency and welfare propertes of varous strategy profles. The only assumpton we mpose on the parameters of payoff functons π s that for each and each sgnal realzaton s S, the condtonal expectaton E[Γ (s) s = s s a symmetrc and negatve defnte matrx. The symmetry assumpton s made for notatonal convenence and s wthout loss of generalty. The assumpton that the matrx s negatve defnte s substantve: t ensures that player s optmzaton problem s well-defned and has a fnte soluton. In the case of sngle-dmensonal actons, ths assumpton reduces to the condtonal expectaton of the quadratc coeffcent n equaton (1) beng negatve. We mpose no other assumptons on the parameters of payoff functons π. In partcular, the dmensons of dfferent players acton spaces can be dfferent; the nteracton terms Γ, between dfferent players actons can be dfferent and asymmetrc, and can vary arbtrarly as functons of the entre vector of sgnals s; and functons g, whch determne the drect payoff nteractons between the vector of sgnals s and the players actons, can be arbtrary as well. 2,3 2 Because of ths generalty, the quadratc games we consder are n general not potental games. Applyng Theorem 4.5 n Monderer and Shapley (1996), a quadratc game wth one-dmensonal actons s a potental game f and only f 2 π a a j = 2 π j a a j. We do not mpose any such symmetry restrctons between Γ,j and Γ j,. 3 Usng standard technques, any game of ncomplete nformaton wth fnte sgnals and multdmensonal actons can be expressed as a game of complete nformaton wth sngle-dmensonal actons. However, n our settng, we fnd t more convenent and transparent to work drectly wth ncomplete nformaton and multdmensonal actons. 5

6 2.3 Bayesan Nash Equlbrum A profle of pure strateges a ( ) of all players s a Bayesan Nash equlbrum f for every player, for every sgnal realzaton s, acton a ( s ) maxmzes player s expected payoff, condtonal on the realzaton of the sgnal, gven the prmtves of the game and the strateges of other players Equlbrum Exstence, Unqueness, and Characterzaton We can now state and prove the frst man result of the paper. Theorem 1 Genercally, there exsts a unque Bayesan Nash equlbrum. The remander of ths secton presents the proof of Theorem 1 and the closed-form characterzaton of the unque Bayesan Nash equlbrum. The proof conssts of two parts. Lemma 1 provdes a condton that guarantees equlbrum exstence and unqueness. It also provdes closed-form expressons for the equlbrum strateges when the condton s satsfed. Lemma 2 shows that the condton for exstence and unqueness requred n Lemma 1 holds genercally. To state Lemma 1, we need to ntroduce addtonal notaton. Let Γ j (s) be the matrx of sze m m j correspondng to the terms multplyng actons a T a j of players and j n equaton (1), for a partcular vector of sgnals s. Let Φ be a block matrx defned as follows. Enumerate the k possble sgnal realzatons of player as 1, 2,..., k, and let K = k k n. Matrx Φ conssts of K K blocks: each block row corresponds to one player and one possble sgnal realzaton of that player, s, 5 and analogously, each block column corresponds to one player j and one possble sgnal realzaton of that player, s j. The block of matrx Φ n block row (, s ) and block column (j, s j ) s gven by Φ (, s ),(j, s j ) = E[Γ j (s) s = s, s j = s j P[s j = s j s = s. Note that the sze of block Φ (, s ),(j, s j ) s the same as the sze of matrx Γ j (s): m m j. Also, when = j, then Φ (, s ),(, s ) = E[Γ (s) s = s, and when s ŝ, then Φ (, s ),(,ŝ ) = 0. Condton 1 Matrx Φ s nvertble. Lemma 1 The game has a unque Bayesan Nash equlbrum f and only f Condton 1 s satsfed. The proof of Lemma 1 s n Appendx A.1. In addton to showng the result, the proof also shows that when Condton 1 s not satsfed, then ether the game has no Bayesan Nash equlbrum, or t has nfntely many. Also, the proof provdes a convenent closed-form soluton for the equlbrum 4 In general, Bayesan games may of course have mxed-strategy equlbra n addton to pure-strategy ones, but as we wll note n the proof of Theorem 1, n our settng best responses are always unque. Thus, there cannot be any mxed-strategy equlbra, and so to smplfy notaton we only talk about pure strategy profles and equlbra. 5 I.e., block row 1 corresponds to the frst sgnal realzaton of player 1, block row 2 corresponds to the second sgnal realzaton of player 1,..., block row k 1 corresponds to the last sgnal realzaton of player 1, block row (k 1 + 1) corresponds to the frst sgnal realzaton of player 2, and so on. and 6

7 when the condton s satsfed. Specfcally, slghtly abusng notaton, let a R m k denote a stacked vector of player s actons, one for each possble realzaton of sgnal s (.e., the frst m elements of vector a are player s acton after observng s = 1, the next m elements of a are the acton after observng s = 2, and so on). Let vector a = (a 1 ; a 2 ;... ; a n ) denote the combned profle of all players actons. Smlarly, let g R m k denote a stacked vector of player s condtonal expectatons of g (s), one for each realzaton of sgnal s (.e., the frst m elements of vector g are equal to E[g (s) s = 1, the next m elements of g are equal to E[g (s) s = 2, and so on). Let g = (g 1 ; g 2 ;... ; g n ) denote the vector combnng the vectors g of all ndvdual players. When Condton 1 s satsfed, the unque Bayesan Nash Equlbrum s gven by a = Φ 1 g. (2) The second step of the proof of Theorem 1 shows that Condton 1 holds genercally. Formally, consder the followng one-dmensonal collecton of quadratc games, parameterzed by γ R. The nformaton structure s the same for all the games n the collecton, and payoffs are gven by π (a, a, s) = 1 2 at Γ (s)a + a T (γγ, (s)) a + a T g (s) + h (a, s). (3) Lemma 2 Condton 1 holds for all γ R, except for at most fntely many values. The proof of Lemma 2 s n Appendx A.2. Together wth Lemma 1, ths concludes the proof of Theorem 1. 3 Applcaton: Cournot Competton In ths secton, we apply the results of Secton 2 to Cournot markets wth ncomplete nformaton. Our model uses the classcal lnear demand functons and lnear-quadratc costs that are used n Palfrey (1985) and Vves (1988), also n the context of ncomplete nformaton. Whle these two papers use symmetrc nformaton structures n whch frms receve..d. sgnals (condtonally on the state of the world), we leverage the framework of the prevous secton to derve mplcatons for the case of arbtrary fnte nformaton structures. Rch nformaton structures n Cournot markets are also consdered by Lambert, Ostrovsky, and Panov (2018a), who study nformaton aggregaton propertes of large asymmetrc markets. They assume that the frms nformaton s jontly normally dstrbuted, and show that as the market grows large, nformaton gets aggregated. As our results n Secton 3.3 demonstrate, ths result depends on the property of jontly normal dstrbutons that one random varable can be decomposed as a lnear combnaton of other jontly normal random varables and dosyncratc nose. Wthout ths property, under general nformaton structures, full nformaton aggregaton does not occur. Instead, the decentralzed market serves, n effect, as an nvsble statstcan, wth equlbrum market quanttes and prces provdng optmal lnear estmators of socal welfare-maxmzng quanttes and prces. 7

8 We are not the frst to step away from the assumpton of jont normalty of sgnals and parameters n Cournot competton wth ncomplete nformaton. L et al. (1987) pont out that to obtan tractable results n a Cournot settng, t s suffcent to consder dstrbutons for whch certan condtonal expectatons are lnear. They also note the advantages of certan non-gaussan dstrbutons, whch make t possble to crcumvent the ssues wth negatve quanttes and prces that can arse under the Gaussan specfcaton (Hurkens, 2014). However, to the best of our knowledge, ours s the frst specfcaton that allows tractable analyss of lnear-demand settngs n whch frms are uncertan not only about the ntercept of the demand functon, but also about ts slope. Note also that analogously to the current settng, the framework of Secton 2 can also be used to study models of dfferentated Bertrand and dfferental Cournot competton wth mult-product frms and rch asymmetrc nformaton structures about varous demand and cost parameters. 3.1 Model of Cournot Competton There s a market wth n frms, = 1,..., n. The frms produce and sell a good to a populaton of consumers. Each frm observes a fnte sgnal s S, usng the notaton of Secton 2.1. In addton, there s a state of the world ω that takes values n a fnte set Ω. For each frm, the margnal probablty dstrbuton over sgnals has full support. The jont dstrbuton of sgnals and state of the world s otherwse arbtrary. After each frm has observed ts sgnal s, t decdes on a quantty q R to produce; ths s the player s acton n our general framework. Each frm ncurs cost of producton c (ω)q + d (ω)q 2, wth d ( ) 0 and c ( ) 0. Consumer demand s lnear, and nverse demand s P (Q, ω) = α(ω) β(ω)q, where α( ) > 0 and β( ) > 0. For a gven profle of quanttes q 1,..., q n suppled by the frms, and a gven state of the world, the realzed proft of frm s π = (α(ω) β(ω)q) q c (ω)q d (ω)q 2 where Q = j q j s the total quantty suppled by the frms. Ths model of Cournot competton s therefore a specal case of our framework of Secton 2. 6 We refer to the game just descrbed as a Cournot game. The followng proposton shows that there always exsts a unque equlbrum of the Cournot game. Note that ths statement s stronger than the statement of Theorem 1, because t does not rely on genercty. 6 Strctly speakng, our Cournot competton model does not ft the framework of Secton 2.1, as t nvolves a state of the world that s not part of the fnte sgnals of the frms. Ths ssue, however, s purely notatonal. To make the model ft the framework of Secton 2.1, one can ntroduce an addtonal player, Nature, whose sgnal space s Ω and whose payoff s gven by, for example, π N = a 2 N, so that t always plays a N = 0. Alternatvely, wthout ntroducng an artfcal player, one can replace the payoff functons π (a, a, s, ω) wth ther expectatons condtonal on the vector of frms sgnals s: π (a, a, s) = E[π (a, a, s, ω) s. Ether approach brngs the model nto the formal framework of Secton

9 Proposton 1 There exsts a unque Bayesan Nash equlbrum of the Cournot game. The proof of Proposton 1 s n Appendx A.3. The dea of the proof s as follows. Frst, we show that n the specal case α( ) = c ( ) = 0 for every, all frms producng zero (for all sgnal realzatons) s the unque Bayesan Nash equlbrum. Thus, by Lemma 1, Condton 1 s satsfed. Second, we observe that n the general case, wth no condtons on α( ) and c ( ), matrx Φ (defned n Secton 2.4), whch determnes exstence and unqueness of equlbrum va Condton 1, does not depend on α( ) and c ( ). Hence, snce Condton 1 s satsfed n the specal case α( ) = c ( ) = 0 for every, t s also satsfed n the general case. We conclude by Lemma 1 that there exsts a unque Bayesan Nash equlbrum n the general case. 3.2 On the Effcency of Large Markets We now show that a property of nvsble hand of compettve markets holds under ncomplete nformaton for arbtrary fnte nformaton structures, extendng the work of Vves (1988) to arbtrary fnte nformaton structures. Our result also relates to Palfrey (1985), who gves condtons on the nformaton structures for the frst-best market outcome to be acheved n large Cournot markets for frms wth constant margnal costs. To establsh the results of ths subsecton, we consder a sequence of markets of the type just descrbed. The markets of the sequence are ndexed k = 1, 2,... Each market k corresponds to a market of Secton 3.1 n whch both the number of frms and consumer demand are scaled by a factor k. Specfcally, market k has N groups of frms, and each group ncludes k frms. There contnues to be a fnte state of the world ω takng values n Ω. Every frm of group observes the same fnte sgnal s S, and ncurs cost c (ω)q + d (ω)q 2 when producng q unts. Inverse demand n market k s P = α(ω) β(ω) Q k. By Proposton 1, there exsts a unque Bayesan Nash equlbrum n every market k. In a Cournot envronment wth ncomplete nformaton, one must dstngush between two relevant noton of effcency, whch we refer to as ex-ante versus ex-post effcency. The noton of ex-post effcency assumes all uncertanty has resolved. Specfcally, fxng a gven state of the world ω, a vector of producton quanttes s ex-post effcent when the total surplus s maxmzed among all possble vectors of producton quanttes. Thus, ex-post effcency corresponds to a socal planner who has full knowledge of the state of the world, and recommends to each frm an output level so as to maxmze total surplus. In contrast, ex-ante effcency captures the dea that the state of the world s not known by the frms at the tme they must decde on producton quanttes. Furthermore, nformaton s decentralzed: each frm s decson can only depend on ts own sgnal. Formally, we say that a strategy profle s ex-ante effcent when the expected total surplus s maxmzed among all frm strategy profles. Hence, ex-ante effcency corresponds to a socal planner who can dctate the strateges to be followed by each frm, but does not have any partcular nformaton on the state of the world. When frms observe ω drectly then both ex-ante and ex-post effcency reduce to the 9

10 classcal noton of effcency under complete nformaton. Our noton of ex-ante effcency concdes wth the noton of second-best decentralzed effcency of Vves (1988), also for Cournot games, and wth the effcency noton of Angeletos and Pavan (2007) for a broader class of quadratc games. We let TS a k be the maxmum expected total surplus that can be acheved n market k over all possble strategy profles. Ether TS a k s nfnte for all k, or TSa k s proportonal to the number of frms. 7 We restrct attenton to the nterestng case of fnte maxmum total surplus, and we let TS a be the normalzed maxmum expected total surplus per frm that can be acheved n any market k. Analogously, we let TS k be the expected total surplus of the unque equlbrum n market k, and TS k the expected total surplus per frm. Under ncomplete nformaton, t s well-known that t s generally not possble for the market equlbrum to be ex-post effcent 8 (see, for example, Proposton 1 of Vves (1988)). Ex-ante effcency s thus the natural benchmark for models of Cournot competton wth ncomplete nformaton, as a second-best welfare property. Our next result shows that as markets grow large and many frms share the same nformaton, the market equlbrum becomes ex-ante effcent. Proposton 2 In the lmt as the market sze grows, the market equlbrum becomes ex-ante effcent: lm k TS k = TS a. The decentralzed effcency of large Cournot markets was observed by Vves (1988) n the case of..d. sgnals. Our result mples that ths effcency result does not rely on the symmetry of nformaton. The proof of Proposton 2 s n Appendx A.4. It reles on the observaton that the strateges whch obtan n the Bayesan Nash equlbrum correspond to the strateges that a planner would want to enforce to maxmze a based expected total surplus, where the bas s addtve and nversely proportonal to market sze. As the market sze grows, the objectve of the planner becomes close to the unbased expected total surplus, and the equlbrum strateges yeld a expected total surplus that, n turn, becomes close to the surplus obtaned under an ex-ante effcent strategy profle. 3.3 The Invsble Statstcan Because nformaton s ncomplete, n general as the market grows large the equlbrum market quantty cannot converge to the ex-post effcent quantty. However, these two quanttes are related n a statstcal sense. In ths secton, we show that the equlbrum market quantty converges to the best lnear estmate of the ex-post effcent quantty. Throughout ths secton, margnal costs are constant and dentcal across frms: d ( ) = 0 and c (ω) = c(ω) for every. Let Q denote the set of all real-valued mappngs wth vectors of frms sgnals and states of the world as nput. Elements of Q are market quanttes determned by the realzaton 7 The possblty of unbounded total surplus s an artfact of the general quadratc framework from whch our model of Cournot competton derves, and whch does not restrct frms to nonnegatve output levels. Such stuaton may occur when the costs are lnear and the margnal costs vary across frms. 8 Palfrey (1985) shows that f margnal costs are constant and dentcal across frms, and f frms receve..d. sgnals on the state of the world that determnes the frms costs and the ntercept of the demand functon, ex-post effcency s acheved n large markets under certan general condtons on the dstrbutons of sgnals and states. 10

11 of the frms sgnals and the state of the world; they can be nterpreted as random varables. For the cost structures under consderaton, both the ex-ante and ex-post effcent market quanttes exst n every market k. Let us denote them by Q a k and Qp k, respectvely. Note that Qa k, Qp k Q. We contnue to make use of the bar notaton to denote per-frm averages, that s, Q p = Q p k /(Nk) and Q a = Q a k /(Nk). We omt subscrpt k because per-frm average effcent quanttes do not depend on the market k. To state our result, we ntroduce the collecton C Q of addtvely separable mappngs: C = {Q Q wth Q(s 1,..., s n, ω) = q 1 (s 1 ) + + q N (s N )}. Note that Q a C, but that n general, Q p C. In the followng, elements of C can be nterpreted as possble statstcal estmates of Q p. These estmates are addtvely separable n the sgnal values, and so correspond to classcal statstcal lnear estmates n whch each frm sgnal s a dummy varable. We refer to the elements of C as lnear estmates. Consder the statstcal queston of fndng a lnear estmate Q that mnmzes the β-weghted quadratc error E[β(ω)(Q Q p ) 2. When such a lnear estmate exsts, we refer to t as best lnear estmate of Q p wth respect to the β-weghted squared error. The followng proposton shows how ths statstcal queston relates to Cournot markets. It conssts of two parts. The frst part of the proposton shows that the ex-ante effcent producton s the unque best lnear estmate of the ex-post effcent producton for the β-weghted quadratc error. The second part of the proposton shows that, n large markets, the equlbrum producton per frm approxmates the best lnear estmate of the ex-post effcent producton level. In the followng, Q k denotes the equlbrum quantty n market k, and let Q k denote the output per frm. Both are a functon of the frms sgnals. Proposton 3 There exsts a unque best lnear estmate Q of Q p wth respect to the β-weghted squared error, and the followng obtans: 1. Q a = Q. 2. For every realzaton of jont sgnals, lm k Q k = Q. As opposed to the prevous result of Secton 3.2 on the ex-ante effcency of large markets, Proposton 3 does not apply when the costs are quadratc. One drect mplcaton of the above result s that, f (and only f) Q p C, then at the lmt, the large Cournot market becomes ex-post effcent. For the case of..d. sgnals, and assumng constant and dentcal margnal costs, Palfrey (1985) shows that under farly general condtons on dstrbutons, ex-post effcency of large market obtans. Our result suggests that, wth asymmetrc nformaton, ex-post effcency requres stronger condtons: ex-post effcency only obtans under addtve separablty of the frst-best output. 11

12 The proof of Proposton 3 s n Appendx A.5. Unqueness of the best lnear estmate owes to the convexty of the squared error, and to the convexty of the set of lnear estmates. The key to the frst statement of the proposton s the observaton that, when there are no quadratc costs, the problem of mnmzng the β-weghted error s dentcal to the problem of maxmzng the expected surplus, so that f the problems have a unque maxmzer, the maxmzers must be dentcal. The second statement of the proposton uses an argument related to the proof of Proposton 2. The equlbrum market quantty s the unque soluton of an optmzaton problem that can be nterpreted as the mnmzaton of the β-weghted squared error wth a bas. The magntude of the bas vanshes as the market sze grows, consequently the problem of fndng the equlbrum market quantty becomes very smlar to the problem of fndng the quantty that mnmzes the β-weghted squared error, and the two maxmzers become arbtrarly close. 4 Games wth Infnte Informaton Structures In some applcatons, usng nfnte sgnal spaces may be more convenent than restrctng attenton to fnte sgnal spaces. For example, t s common to assume that sgnals of players are jontly normally dstrbuted. In ths secton, we consder a settng wth potentally nfnte sgnal spaces, wth some addtonal assumptons on the dstrbuton of sgnals and the payoff structure. We restrct attenton to lnear equlbra (n whch the actons of players are lnear functons of some of the sgnals). Our second man result shows that genercally, there exsts a unque lnear equlbrum. Ths equlbrum can be characterzed n closed form. 4.1 Model There are n players, = 1,..., n. Pror to takng an acton, each player prvately observes a par of sgnals: s S (where, as n Secton 2, set S s fnte and contans k elements) and θ R l (where l 1 s the dmenson of sgnal θ ). We denote by s = (s 1 ; s 2 ;... ; s n ) and θ = (θ 1 ; θ 2 ;... ; θ n ) the vectors summarzng the sgnals of all n players. For convenence, we refer to s s as fnte sgnals of the players and θ s as nfnte sgnals (although the latter are not requred to have an nfnte number of possble realzatons). We make the followng assumptons about the jont dstrbuton of sgnals (s, θ). Frst, for every player and realzaton s S, the probablty of player observng s s postve. As n Secton 2, ths s wthout loss of generalty. Also as n Secton 2, we do not mpose any other restrctons on the jont dstrbuton of fnte sgnals s. Second, for every player and realzaton s S, the condtonal dstrbuton of θ has fnte frst and second moments, and matrx Var(θ s = s ) has full rank. The latter assumpton s also wthout loss of generalty, as redundant sgnals can always be replaced wth ndependent nose. Thrd, condtonal on the fnte sgnals of a subset of players, ther nfnte sgnals do not contan any addtonal nformaton about other players fnte sgnals. Formally, take any subset T of players, wth T 1 and T < n. Take any profle of fnte and nfnte sgnal realzatons ( s T, θ T ) of players n 12

13 set T. Then the probablty of any realzaton s T of fnte sgnals of players outsde of T, condtonal on the entre profle of sgnals ( s T, θ T ), s equal to the probablty of realzaton s T condtonal only on the profle of fnte sgnal realzatons s T. One mmedate case of dstrbutons satsfyng ths condton s vector θ beng ndependent of vector s. However, more nterestng dependences are also allowed. For example, n 1 players may observe ndependent dentcally dstrbuted sgnals θ, and the n-th player may observe a subset of those sgnals, wth hs sgnal s n determnng whch subset of the n 1 ndependent sgnals he observes. 9 Another example s two players observng sgnals θ drawn from two potentally correlated dstrbutons (wth several possble correlaton coeffcents), and also observng partally nformatve sgnals s about the correlaton coeffcent. Next, for convenence, we assume that for every player, for every realzaton s of hs fnte sgnal, the expected value of hs nfnte sgnal s zero: E[θ s = s = 0. Ths s wthout loss of generalty. Note, however, that n conjuncton wth the prevous assumpton, ths renormalzaton mples a stronger property: for every subset T of players, wth T 0 and T n, for any profle of ther fnte sgnal realzatons s T that has a postve probablty, for any player (who may or may not be a part of T ), we have E[θ s T = s T = Fnally, we assume that for every player and every player j, the condtonal expectaton E[θ j s = s, s j = s j, θ = θ s a lnear functon of θ. Ths s a substantve assumpton, whch we dscuss n more detal n Secton 4.5. Note that whle the expectaton has to be a lnear functon of θ, the coeffcents n ths lnear functon are allowed to depend on s and s j (although of course the case when they do not depend on one or both of these fnte sgnals s also allowed). In partcular, these assumptons allow for the case of jontly normal sgnals θ whose covarance matrx may depend on the profle of fnte sgnals s. 4.2 Actons and Payoffs After prvately observng hs sgnal, each player chooses acton a R m, where m 1 s the dmenson of player s acton. Note that there s n general no relaton between the sze k of player s fnte sgnal space S, the dmenson l of hs nfnte sgnal θ, and the dmenson m of hs acton a. The payoff of player depends on hs own acton a, the actons of other players (denoted by vector a ), and the entre vectors of sgnals s and θ (whch nclude both the sgnals of player and the sgnals of other players): π (a, a, s, θ) = 1 2 at Γ (s)a + a T Γ, (s)a + a T g (s, θ) + h (a, s, θ), (4) where Γ (s) s a matrx of sze m m, Γ, (s) s a matrx of sze m ( j m j), g (s, θ) s a vector of sze m, and h s an arbtrary functon of a, s, and θ. 11 As before, the presence of 9 Formally, for the frst n 1 players, S = 1, and θ R k. For the n-th player, S n = {0, 1} n 1 and θ n R (n 1)k. When the j-th element of s n s equal to 1, components (k(j 1) + 1,..., kj) of θ n are equal to θ j. When the j-th element of s n s equal to 0, these components are random nose. 10 The proof of ths statement s gven n Appendx A The only assumpton we need to mpose on functon h s that the expected value of h (a, s, θ) s fnte for every profle of lnear strateges of players other than, condtonal on every possble realzaton ( s, θ ) of player s sgnals. 13

14 functon h does not affect the ncentves of player (and thus the equlbra of the game), but does n general affect the effcency and welfare propertes of varous strategy profles. We assume that for each and each sgnal realzaton s S, the condtonal expectaton E[Γ (s) s = s s a symmetrc negatve defnte matrx. In addton, we assume that for every player, the expectaton E[g (s, θ) s = s, θ = θ s a lnear functon of θ. Note that whle the expectaton has to be a lnear functon of θ, the coeffcents n ths lnear functon are allowed to depend on s. We mpose no other restrctons on payoff functons π. 4.3 Lnear Equlbrum We say that a strategy of player s lnear f t can be represented as a (s, θ ) = κ (s ) + Λ (s )θ, where for every realzaton s of sgnal s, κ ( s ) s a vector n R m and Λ ( s ) s a matrx n R m l. Note, n partcular, that the senstvty of player s acton to hs nfnte sgnal θ may depend on the realzaton of hs fnte sgnal s. We say that a profle of strateges of all players s a lnear equlbrum f (1) every strategy a ( ) n the profle s lnear, and (2) the profle of strateges s a Bayesan Nash equlbrum,.e., for every player, for every realzaton ( s, θ ) of sgnals (s, θ ), acton a ( s, θ ) maxmzes player s expected payoff, gven the prmtves of the game and the strateges of other players. 4.4 Equlbrum Exstence and Unqueness We can now state and prove the second man result of the paper. Theorem 2 Genercally, there exsts a unque lnear equlbrum. The remander of ths secton presents the proof of Theorem 2 and the closed-form characterzaton of the unque lnear equlbrum. As n the proof of Theorem 1, the argument conssts of two parts. Lemma 3 provdes condtons that guarantee the exstence and unqueness of lnear equlbrum. It also provdes closed-form expressons for the equlbrum strateges when these condtons are satsfed. Lemma 4 shows that the condtons for exstence and unqueness requred n Lemma 3 hold genercally. The frst condton n Lemma 3 s the same as Condton 1 stated earler (n Secton 2). To gve the second condton, we need to ntroduce some addtonal notaton. By assumpton, the expectaton E[θ j s = s, s j = s j, θ = θ s a lnear functon of θ. Denote ths lnear functon by E[θ j s = s, s j = s j, θ = θ = Q T j ( s, s j ) θ. 12 Let Ψ be a block matrx defned as follows. As n Secton 2, enumerate the k possble sgnal realzatons s of player as 1, 2,..., k, and let K = k k n. Matrx Ψ conssts of K K blocks: each block row corresponds to one player and one possble sgnal realzaton of that player, s, and analogously, each block column corresponds to one player j and one possble sgnal realzaton 12 Note that ths lnear functon does not have a constant term. As we show n Appendx A.6, ths lack of the constant term s mpled by our normalzaton of the sgnal dstrbutons. 14

15 of that player, s j. The block of matrx Ψ n block row (, s ) and block column (j, s j ) s gven by Ψ (, s ),(j, s j ) = ( ) Q j ( s, s j ) E [Γ j (s) s = s, s j = s j P[s j = s j s = s where denotes the Kronecker product of two matrces. 13 When actons a or nfnte sgnals θ of all players are sngle-dmensonal, the Kronecker product reduces to regular multplcaton by a scalar. When both actons a and nfnte sgnals θ of all players are sngle-dmensonal, the expressons smplfy further and each block Ψ (, s ),(j, s j ) s just a real number. In the general case, the sze of block Ψ (, s ),(j, s j ) s m l m j l. Condton 2 Matrx Ψ s nvertble. Lemma 3 The game has a unque lnear equlbrum f and only f Condtons 1 and 2 are satsfed. The proof of Lemma 3 s n Appendx A.7. In addton to showng the result, the proof also shows that when Condton 1 or Condton 2 are not satsfed, then ether the game has no lnear equlbrum, or t has nfntely many. Also, the proof provdes a closed-form soluton for the equlbrum when both condtons are satsfed. Specfcally, recall our representaton of lnear strateges as a (s, θ ) = κ (s ) + Λ (s )θ. Slghtly abusng notaton, let κ R m k be the stacked vector of player s constant terms κ (s ), one for each possble realzaton of sgnal s (.e., the frst m elements of κ are κ (1), the next m elements are κ (2), and so on). Let vector κ = (κ 1 ;... ; κ n ) denote the combned profle of all players constant terms. Next, consder the vectorzaton vec Λ ( s ),.e., the column vector of sze m l n whch the columns of matrx Λ ( s ) are stacked on top of each other. 14 Let Λ R m l k be the stacked vector of these vectorzatons vec Λ ( s ), one for each realzaton s. Fnally, let vector Λ = (Λ 1 ;... ; Λ n ) combne the vectors Λ of ndvdual players. Next, by assumpton, the expectaton E[g (s, θ) s = s, θ = θ s a lnear functon of θ. Denote ths functon by E[g (s, θ) s = s, θ = θ = G ( s ) + F ( s ) θ. Let g R m k be the stacked vector of ndvdual vectors G ( s ), one for each possble realzaton of s (.e., g = (G (1);... ; G (k )), and let g = (g 1 ;... ; g n ) denote the vector combnng these terms for all players. Fnally, let f = (vec F (1);... ; vec F (k )) R m l k, and let f = (f 1 ; f 2 ;... ; f n ). When Condtons 1 and 2 are satsfed, the unque lnear equlbrum s gven by κ = Φ 1 g (5) Λ = Ψ 1 f. (6) The second step of the proof of Theorem 2 shows that Condtons 1 and 2 hold genercally. Formally, consder the followng one-dmensonal collecton of quadratc games, parameterzed by 13 If matrx A s of sze k m wth elements a j and matrx B s another matrx, of any sze, then A B s defned as a block matrx wth k m blocks, n whch the sze of each block s equal to the sze of B and each block (, j) s equal to a jb. See for detals. 14 See for detals. 15

16 γ R. The nformaton structure s the same for all the games n the collecton, and payoffs are gven by π (a, a, s, θ) = 1 2 at Γ (s)a + a T (γγ, (s)) a + a T g (s, θ) + h (a, s, θ) (7) Lemma 4 Condtons 1 and 2 hold for all γ R, except for at most fntely many values. The proof of Lemma 4 s n Appendx A.8. Together wth Lemma 3, ths concludes the proof of Theorem Lnear Condtonal Expectatons As mentoned n Secton 4.1, the lnearty of condtonal expectatons assumpton (.e., the assumpton that for every player and every player j, the condtonal expectaton E[θ j s = s, s j = s j, θ = θ s a lnear functon of θ ) s substantve. However, there are many nterestng and common examples of jont dstrbutons that satsfy ths restrcton. The most common example s the multvarate normal dstrbuton. For example, f vector θ s dstrbuted normally, wth any varance-covarance matrx, and s ndependent of vector s, then E[θ j s = s, s j = s j, θ = θ s a lnear functon of θ, for any and j. More complex examples bult from normal dstrbutons are also possble. For example, fnte sgnals s may determne correlatons between the nfnte sgnals of dfferent players. For a very dfferent example of jont dstrbutons satsfyng the condtonal lnearty assumpton, suppose random varables α and β are ndependent (and other than that, come from arbtrary dstrbutons), and suppose player observes both of them (.e., θ = (α; β)), whle player j observes only one (e.g., θ j = (α)). Then both E[θ j θ and E[θ θ j are lnear functons. More generally, f there are several ndependent random varables, and each agent observes a subset of them, the resultng nformaton structure satsfes the lnearty condton. For example, suppose there are no fnte sgnals s (formally, a fxed vector s s observed wth probablty 1), and consder a network of agents n whch the nfnte sgnal (type) of each agent s a random varable drawn from some dstrbuton (possbly a dfferent one for each agent). The types of agents are ndependent of each other. Each agent observes ther own type, and the types of ther neghbors n the network. Then, for all and j, E[θ j θ = θ s a lnear functon of θ. For yet another example, suppose random varables α and β are ndependent and dentcally dstrbuted, and player observes each of them separately, whle player j only observes ther sum. Agan, both E[θ j θ and E[θ θ j are lnear functons. As before, we can use ths observaton to explore rcher models n whch some players observe specfc varables whle others only observe ther aggregates (and n whch who observes what may depend on fnte sgnals s). Fnally, note that f a player s sgnal θ s bnary, then for any jont dstrbuton of θ and θ j, the expectaton E[θ j θ s a lnear functon of θ. We return to ths observaton n Secton 6.2, where t serves as a bass for a connecton between two seemngly dstnct models. 16

17 The lst above s by no means exhaustve there are many other types of jont dstrbutons of sgnals for whch our assumpton of lnear condtonal expectatons s satsfed. 15 So whle the assumpton s certanly substantve, t nevertheless allows for a wde varety of nterestng cases. 5 Applcaton: Beauty Contests In ths secton, we llustrate the framework and results of Secton 4 wth an applcaton: beauty contests wth potentally uncertan relatve weghts that players put on coordnatng wth others. In beauty contests (Morrs and Shn, 2002), players receve sgnals about an uncertan varable, and ther optmal acton s a weghted average of ther estmate of the value of the varable and ther estmate of the average acton of other players. The standard BC framework assumes that all players are ex ante dentcal, and n partcular, receve dentcally dstrbuted sgnals and put the same weghts on matchng the actons of all other players. In ther analyses of beauty contests, Bergemann et al. (2017) and Lambert et al. (2018b) allow for rch asymmetrc nformaton structures, but mantan the assumpton that each agent cares equally about matchng all other agents actons (.e., each player tres to match a weghted average of the true state of the world and the average acton of other players). Golub and Morrs (2017), Lester (2016), and Myatt and Wallace (2017) relax ths assumpton, and allow players to put more weght on matchng some agents actons and less (or none) on matchng other agents actons, examnng the nterplay between beauty contests and network settngs. In such envronments, Golub and Morrs (2017) consder common and heterogeneous prors, and focus on the characterzaton of hgher-order expectatons and ther lmt as the weght of the coordnaton component n payoff functons goes to 1. Lester (2016) and Myatt and Wallace (2017) explore endogenous nformaton acquston. 16 The contrbuton of the current secton of our paper s to not only allow players to put dfferent weghts on matchng dfferent agents actons, but to also allow these weghts to be stochastc, wth players potentally havng rch nformaton both about these weghts and about the states of the world (and about other players knowledge of those parameters). We also allow players to target dfferent state varables, allowng for many other nterpretatons of the framework (e.g., the framework can be vewed as a model of a coordnaton game, n whch each player has some (ndvdual) blss acton that he would pck n the absence of nteractons wth other players, but also puts some weght on pckng an acton close to the actons of some other players), and to receve multdmensonal nformaton (so that, for example, one dmenson of the sgnal s nformatve about a player s own optmal blss acton and another dmenson s nformatve about another player s blss acton; or one dmenson s a player s own sgnal, another dmenson s a sgnal shared wth some other players, 15 See, e.g., L et al. (1987). 16 Wth the excepton of Golub and Morrs (2017), all papers on beauty contests mentoned above consder games wth jontly normal nformaton structures. Golub and Morrs (2017) consder beauty contests wth fnte nformaton structures, whch (n the case of common prors) ft the model of Secton 6.1 below. As we explan n Secton 6.2, all these models are specal cases of a unfed model of quadratc games wth fxed nteracton coeffcents and nformaton structures that satsfy the lnear condtonal expectatons property. 17

18 and the thrd, fnte dmenson, contans nformaton about other players payoff functons). In Secton 5.1, we descrbe our general model of beauty contests. In Secton 5.2, we show that n ths model, there s always a unque lnear equlbrum (unlke Theorem 2, ths statement s always true, not just genercally). In Secton 5.3, we consder a partcular example of a beauty contest wth uncertan nteractons and show that allowng for uncertan nteracton terms n players payoff functons leads to economcally novel predctons. 5.1 Model of Beauty Contests There are n players, = 1,..., n. Each player observes a fnte sgnal s S and an nfnte sgnal θ R l. Each player also has a blss pont b R, whch s a random varable. Denote by b = (b 1 ;... ; b n) the random vector of blss ponts. As n Secton 4, we do not mpose any restrctons on the dstrbuton of fnte sgnals s, apart from assumng that for each player and each sgnal realzaton s S, the probablty of player observng s s postve. For the profle of nfnte sgnals θ and the vector of blss ponts b, we assume that they are jontly normally dstrbuted, wth the mean of θ beng equal to zero, the mean of b beng equal to b, and the varance-covarance matrx of (θ; b) beng arbtrary, subject only to the constrant that for each, matrx Var(θ ) has full rank (as before, ths assumpton s wthout loss of generalty). Note that dfferent blss ponts b can be assumed to be dentcal, but are not requred to. We assume that s s ndependent of θ and b. 17 Each player s acton s a real number: a R. The payoff of each player s a functon of hs blss pont, hs acton, the actons of other players, and the profle of fnte sgnals s, and s equal to π (a, a, b, s) = γ (s)(a b ) 2 γ j (s)(a a j ) 2, (8) j where for every s, γ (s) > 0 and for each j, γ j (s) In the classcal models of beauty contests, players have an ncentve to choose an acton that s close to a common random blss pont, and at the same tme that s close to the average acton of the other players. The weghts that players put on each component can vary across players, but are fxed and commonly known. Our model of beauty contests allows for more flexblty along three dmensons. Frst, dfferent players may have dfferent blss ponts. Second, a player may put dfferent weghts on matchng the actons of dfferent other players. Fnally, the weghts that players put on matchng the blss ponts and on matchng varous other players actons may be stochastc, wth complex nformaton structures about these weghts. E.g., on a network, a player may know the preferences of her neghbors, but only have probablstc nformaton about the preferences of agents who are further away. Smlarly, 17 The results of ths secton can be extended to more general cases n whch the dstrbuton of θ s not necessarly normal, and n whch s and θ are not ndependent, whle stll remanng wthn the framework of Secton 4. We restrct attenton to the current set of assumptons for expostonal convenence. 18 As n the Cournot competton model of Secton 3, our beauty contests model does not formally ft the framework of Secton 4, because the blss ponts are not drectly parts of ether fnte sgnals s or nfnte sgnals θ. Smlarly to the Cournot competton model, to ft formally the framework of Secton 4.1, one can ntroduce a dummy player Nature whose fnte sgnal space conssts of only one element and whose nfnte sgnal s equal to vector b and whose payoff π N s gven by π N = a 2 N, so that t always plays a N = 0. 18