Correlated Equilibrium in Games with Incomplete Information

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1 Correlated Equlbrum n Games wth Incomplete Informaton Drk Bergemann y Stephen Morrs z Frst Verson: October Current Verson: May 4, Abstract We de ne a noton of correlated equlbrum for games wth ncomplete nformaton n a general settng wth nte players, nte actons, and nte states. We refer to ths soluton concept as Bayes correlated equlbrum. For a gven common pror over the payo relevant states and types, we show that the set of Bayes correlated equlbrum probablty dstrbutons equals the set of probablty dstrbutons over actons, states and types that mght arse n any Bayes Nash equlbrum consstent wth the gven common pror over states and types. We de ne a game of ncomplete nformaton n terms of a payo envronment, or the basc game, and a belef envronment, or the nformaton structure. We show how the nformaton structure a ects the set of predctons that can be made about the Bayes correlated equlbrum dstrbuton. We show that a more nformed nformaton structure reduces the set of Bayes equlbrum dstrbutons as t mposes addtonal ncentve constrants. Keywords: Correlated equlbrum, ncomplete nformaton, robust predctons, nformaton structure. JEL Classfcaton: C7, D8, D8. We acknowledge nancal support through NSF Grant SES 85. Forges, Olver Gossner, Eran Shmaya and semnar partcpants at the Pars School of Economcs. y Department of Economcs, Yale Unversty, New Haven, U.S.A., drk.bergemann@yale.edu. z Department of Economcs, Prnceton Unversty, Prnceton, U.S.A., smorrs@prnceton.edu. We have bene ted from comments from Francose

2 Correlated Equlbrum May 4, Introducton We present a noton of correlated equlbrum n games wth ncomplete nformaton. Aumann (974), (987) ntroduced the noton of correlated equlbrum n games wth complete nformaton. A number of de ntons of correlated equlbrum n games wth ncomplete nformaton have been suggested, notably n Forges (99). Our de nton s drven by a d erent motvaton from the earler lterature; we seek the soluton concept whch characterzes the set of Bayes Nash equlbra whch can be sustaned by some nformaton structure n a xed economc settng. Ths leads us to suggest an equlbrum noton that we shall call Bayes correlated equlbrum, whch s a (weaker) of the weakest de nton of ncomplete nformaton correlated equlbrum (Bayesan soluton) n Forges (99). We dstngush the "payo envronment" and the "belef envronment" n the de nton of the game. By payo envronment, we refer to the set of actons, the set of payo relevant states, the utlty functons of the agents, and the common pror over the payo relevant states. By belef envronment, we refer to the nformaton structure, the type space of the game, whch s generated by a mappng from the payo relevant states to a probablty dstrbuton over types. The separaton between payo and belef envronment enables us to ask how changes n the belef envronment a ect the equlbrum set for a gven and xed payo envronment. By contrast wth the earler lterature, we allow nformaton not known to any of the players to be re ected n the equlbrum dstrbuton. In games of complete nformaton, the noton of correlated equlbrum was meant to descrbe the set of possble equlbrum outcomes whch can be acheved when the agents may have access to some, unobserved and unmodelled correlaton opportuntes. A correlated equlbrum was smply de ned by jont dstrbuton over the actons of the agents. In a game wth ncomplete nformaton, the unobserved and unmodelled correlaton opportuntes stll exst, but the exstence of the prvate nformaton of the agents means that there are some constrants on the correlaton opportuntes as the actons have to be consstent wth the prvate nformaton of the agents. From ths perspectve, the prvate nformaton of the agents mposes restrctons on the correlaton that can arse n the Bayes correlated equlbrum. An mportant specal case s then gven by the null nformaton system n whch the type space of each agent conssts of a sngleton type for every agent. Ths null nformaton system then mposes null restrctons on the jont correlaton of the agents actons and payo state over and above the common pror of the payo relevant states. The set of Bayes correlated equlbra for a gven payo envronment s then largest under the null nformaton system as the lack of prvate nformaton means that there are no constrants mposed beyond a consstency requrement whch asks that the margnal of the equlbrum dstrbuton over the states equals the common pror of the states. Subsequently, we ask how the presence of prvate nformaton restrcts the correlaton opportuntes of the agents actons. In partcular, we compare

3 Correlated Equlbrum May 4, nformaton structures and ask whch nformaton structure contans more nformaton for the agents, and hence mposes more restrctons on the set Bayes correlated equlbra. We present a de nton as to when one nformaton structure s more nformed than another nformaton structure. The crteron of more nformed represents an extenson of the garblng condton by Blackwell (95) to an envronment wth many agents. We establsh that an nformaton structure s more nformed than another nformaton structure f and only f t supports a smaller set of Bayes correlated equlbra. The present de nton of Bayes correlated equlbrum s used promnently n the analyss of our companon paper, "Robust Predctons n Games wth Incomplete Informaton", (Bergemann and Morrs ()). In the companon paper, we analyze how much can be sad about the jont dstrbuton of actons and states on the bass of the knowledge of the payo envronment alone. There we refer to robust predctons as those predctons whch can be made wth the knowledge of the payo envronment alone, and wthout any assumpton about the belef envronment. In the companon paper, the analyss was con ned to an envronment wth quadratc and symmetrc payo functons, a contnuum of agents and normally dstrbuted uncertanty about the common payo relevant state. But ths tractable class of models enabled us to o er robust predctons n terms of restrctons on the rst and second moments of the jont dstrbuton over actons and state. By contrast, here we present the de nton of the Bayes correlated equlbrum n a canoncal game theoretc framework wth a nte number of agents, a nte set of pure acton and a nte set of payo relevant states. After we ntroduce the relevant notons, we show towards the end of ths paper how the present results translate nto the settng wth a contnuum of anonymous agents that we consdered n (Bergemann and Morrs ()). A number of papers have consdered alternatve de ntons of correlated equlbrum n games wth ncomplete nformaton, most notably Forges (99) and Forges (6). In ths paper, we document the relatonshp between our verson of correlated equlbrum and the varous de ntons n the lterature. In the dscusson of the varous de ntons of correlated equlbrum, we wll nd t s useful to dvde restrctons that the varous soluton concepts mpose on the jont dstrbuton over actons, states and types nto two classes: feasblty condtons on the dstrbutons of acton type state pro les, whch are requred to hold ndependently of the payo functons, and ncentve compatblty condtons whch are ratonalty constrants on players acton choces. The only feasblty condton that we mpose n de nng the Bayes correlated equlbrum s a consstency requrement that demands that the acton type state dstrbuton of the equlbrum mples the dstrbuton on the exogenous varables, namely the common pror on the payo relevant states and types. In contrast, n many of the exstng soluton concepts, the feasblty condtons are ntended to capture the outcome of some form of communcaton among the agents wth an unnformed medator. It s then natural to mpose addtonal restrcton on the acton state type

4 Correlated Equlbrum May 4, 4 dstrbuton n equlbrum whch have hold condtonal on the agents types. For example, the Bayesan soluton, the weakest of Forges ve de ntons, mposes the restrcton, referred to here as jon feasblty, that the dstrbuton over states condtonal on agents types s not changed condtonal on the medator s recommendatons. Our noton of Bayes correlated equlbrum s closest to the Bayesan soluton but s strctly weaker than the Bayesan soluton, because we do not nsst on jon feasblty. A number of papers - notably Gossner (), Lehrer, Rosenberg, and Shmaya () and Lehrer, Rosenberg, and Shmaya () - have examned comparatve statcs of how changng the nformaton structure e ects the set of predctons that can be made about players actons, under Bayes Nash equlbrum or alternatve soluton concepts. We revew these results and report a new result. We dscussed above that as the agents become more nformed, where nformaton s encoded n ther type, the set of possble predctons must be reduced. As the agents have more prvate nformaton, the ncentve constrants, here referred to as obedence constrants, wll become tghter. The role of the prvate nformaton n re nng the equlbrum predcton s mportant n our "Robust Predcton" agenda. We wll formalze ths result ths result here n the general framework of the current paper rather than n the spec c envronment of quadratc payo functons and normally dstrbuted uncertanty of (Bergemann and Morrs ()). We llustrate the noton of Bayes correlated equlbrum and the resultng robust predctons n varety of examples, among them a rst prce aucton wth prvate values and a sender-recever game, whch s closely related to a problem studed by Gentzkow and Kamnca (), where senders are allowed to commt to a communcaton strategy. In Bergemann and Morrs (5) and later work, we studed a mechansm desgn envronments and de ned the noton of robust mechansm. In ths earler settng, the agents knew ther own "payo types", and whle there was common knowledge of how utltes depended on the pro le of payo types, the agents were allowed to have any belefs and hgher order belefs about others payo types. We then de ned a mechansm to be robust f the socal choce functon or correspondence could be truthfully mplemented n the drect mechansm as a Bayes Nash equlbrum for any belefs and hgher order belefs about others payo types. In Bergemann and Morrs (7), we dscussed the game theoretc framework underlyng the analyss n the mechansm desgn envronment. The noton of Bayes correlated equlbrum s motvated by the same concern for robustness but t encodes a less demandng noton of robustness. The Bayes correlated equlbrum nssts that the common pror over the state and type dstrbuton s preserved, and n the case of the null nformaton structure that the common pror over the state alone s preserved, but all addtonal correlaton due to unobserved communcaton or nformaton among the agents s permtted. We proceed as follows. In Secton, we descrbe a general ncomplete nformaton game and compare Bayes Nash equlbrum wth a soluton concept whch we call Bayes correlated equlbrum. In Secton,

5 Correlated Equlbrum May 4, 5 we descrbe our robust predctons approach and explan the key role played an "epstemc" result: the set of Bayes correlated equlbrum probablty dstrbutons over actons, types and payo -relevant varables equals the set of probablty dstrbutons of actons, types and payo -relevant varables that mght arse n a Bayes Nash equlbrum f players were able to observe addtonal nformaton sgnals beyond ther orgnal types. In Secton 4, we explan how the soluton concept we dub "Bayes Correlated Equlbrum" relates to the lterature, n partcular Forges (99) and Forges (6). In Secton 5, we report results on comparng nformaton structures. In Secton 6, we revew specal cases n order to llustrate the robust predctons agenda more broadly. In Secton 7, we descrbe analogues of our results for contnuum anonymous player games, whch apply to our work n Robust Predctons n Games wth Incomplete Informaton. Secton 8 concludes and contans a dscusson of the relaton to the sgned covarance result of Chwe (6) and the "payo types" envronments of Bergemann and Morrs (7). Bayes Nash and Bayes Correlated Equlbrum Throughout the paper, we wll x a nte set of players and a nte set of payo relevant states of the world. There are I players, ; ; ::; I, and we wrte for a typcal player. We wrte for the payo relevant states of the world and for a typcal element of. A "basc game" G conssts of () for each player, a nte set of acton A and a utlty functon u : A! R; and () a full support pror (), where we wrte A = A :: A I. Thus G = (A ; u ) I = ;. An "nformaton structure" S conssts of () for each player, a nte set of types or "sgnals" T ; and () a sgnal dstrbuton :! (T ), where we wrte T = T :: T I. S = (T ) I = ;. Together, the "payo envronment" or "basc game" G and the "belef envronment" or "nformaton structure" S de ne a standard "ncomplete nformaton game". Thus Whle we use d erent notaton, ths dvson of an ncomplete nformaton game nto the "basc game" and the "nformaton structure" s a standard one n the lterature, see, for example, Lehrer, Rosenberg, and Shmaya (). A (behavoral) strategy for player n the ncomplete nformaton game game (G; S) s b : T! (A ). Wrte B for the set of strateges of player n the game (G; S). The followng s the standard de nton of Bayes Nash Equlbrum n ths settng. De nton A strategy pro le b s a Bayes Nash Equlbrum (BNE) of (G; S) f for each = ; ; ::; I,

6 Correlated Equlbrum May 4, 6 t T and a A wth b (a jt ) >, we have u ((a ; b (t )) ; ) () ((t ; t ) j) for each a A. t T ; t T ; u a ; b (t ) ; () ((t ; t ) j). The relevant space of uncertanty n the ncomplete nformaton game (G; S) s A T, and we wll wrte for a typcal element of (A T ). There are two knds of constrants mposed n de nng alternatve versons of ncomplete nformaton correlated equlbrum: "feasblty" constrants and "ncentve compatblty" condtons. Our preferred de nton wll mpose one feasblty condton: De nton Dstrbuton (A T ) s consstent for (G; S) f, for all t T and, we have (a; t; ) = () (tj) () aa Ths smply says the margnal of dstrbuton on the exogenous varables T and s consstent wth the descrpton of the game (G; S). We wll also mpose the weakest natural ncentve compatblty condton, "obedence", that says that a player who knows hs type t, hs recommended acton a and the dstrbuton only has an ncentve to follow that recommendaton. De nton Dstrbuton (A T ) s obedent for (G; S) f, for each = ; ::; I, t T and a A, we have u ((a ; a ) ; ) ((a ; a ) ; (t ; t ) ; ) () for all a A. a A ;t T ; a A ;t T ; u a ; a ; ((a ; a ) ; (t ; t ) ; ) ; Now our leadng de nton of correlated equlbrum for ncomplete nformaton games wll be: De nton 4 A probablty dstrbuton (A T ) s a Bayes Correlated Equlbrum (BCE) of (G; S) f t s consstent and obedent.

7 Correlated Equlbrum May 4, 7 As wll dscuss n detal below n Secton 4, ths s essentally the de nton of "Bayesan soluton" n Forges (99), wth the d erence that we work wth a ncomplete nformaton game descrpton that does not ntegrate out payo relevant states and thus allows the medator to make acton recommendatons that depend on a payo -relevant state that s observed by nobody. We wll dscuss n the next secton why ths de nton s nterestng for our robust predctons agenda. A Bayes Nash Equlbrum b s a strategy pro le n B. A Bayes Correlated Equlbrum s an element of (A T ) and thus a dstrbuton over acton type state pro les. To compare the two soluton concepts, we would lke to dscuss the dstrbuton of acton type state pro les generated by a BNE. De nton 5 Dstrbuton (A T ) s nduced by strategy pro le b B f, for each a A, t T and, we have! IY (a; t; ) = () (tj) b (a jt ). Dstrbuton (A T ) s Bayes Nash acton type state dstrbuton of (G; S) f there exsts a Bayesan Nash Equlbrum b of (G; S) that nduces t. = We also have the followng straghtforward observaton: Lemma Every Bayes Nash acton type state dstrbuton of (G; S) s a Bayes Correlated Equlbrum of (G; S). We wll also sometmes be nterested n the nduced acton state dstrbutons,.e., what we can say f types are not observed. De nton 6 Acton state dstrbuton (A ) s nduced by (A T ) f t s the margnal of on A. Acton state dstrbuton (A ) s a BNE acton state dstrbuton of (G; S) f t s nduced by a Bayes Nash acton type state dstrbuton of (G; S). Acton state dstrbuton (A ) s a BCE acton state dstrbuton of (G; S) f t s nduced by a Bayes Correlated Equlbrum of (G; S). An mportant specal case s when the nformaton system s "null" wth the players knowng nothng t about the states. Formally, the null nformaton system S = I, = ; where t s the sngleton type of player and t j = for each. We wll abbrevate the (degenerate) ncomplete nformaton game (G; S ) to G. Observe that n the specal case of a null nformaton system, the space A T reduces to A and the consstency condton () on (A ) becomes (a; ) = () () aa

8 Correlated Equlbrum May 4, 8 for all ; and the obedence constrant () reduces to u ((a ; a ) ; ) ((a ; a ) ; ) (4) a A ; a A ; for each = ; ::; I, a A and a A. u a ; a ; ((a ; a ) ; ) ; Now we have: Lemma If (A ) s nduced by a BCE acton type state dstrbuton (A T ), then s a BCE of G. As we wll dscuss n detal below, ths result s n the sprt of Proposton 4 of Forges (99), whch shows that "any" correlated equlbrum soluton concept of (G; S) generates an equlbrum of the basc game G. Robust Predctons Consder an analyst who knows that. G descrbes actons, payo functons dependng on fundamental states, and a pror dstrbuton on fundamental states.. Players have observed at least nformaton system S.. The full, common pror, nformaton system s common certanty among the players. 4. The players actons follow a Bayes Nash Equlbrum. What can she deduce about the jont dstrbuton of actons, types n the "nformaton structure" S and states? In ths secton, we wll formalze ths queston and show that all she can deduce s that the dstrbuton wll be a BCE dstrbuton of (G; S). To formalze ths, let S e = (Z ) I = ; be a supplementary nformaton system, over and above S, and suppose each agent observes a supplementary sgnal z Z, where : T! Z descrbes the dstrbuton of supplementary sgnals. Now G; S; S e s an "augmented ncomplete nformaton game". Wrte : T Z! (A ) for a behavor strategy of player n the augmented ncomplete nformaton game.

9 Correlated Equlbrum May 4, 9 De nton 7 A strategy pro le s a Bayes Nash Equlbrum of the augmented game each = ; ; ::; I, t T, z Z and a A wth (a j (t ; z )) >, we have for each a A. a A ;t T ;z Z ; a A ;t T ;z Z ; G; S; S e f, for u ((a ; (t ; z )) ; ) () ((t ; t ) j) ((z ; z ) j (t ; t ) ; ) u a ; (t ; z ) ; () ((t ; t ) j) ((z ; z ) j (t ; t ) ; ). Wrte for the probablty dstrbuton over A T generated by strategy pro le, so (a; t; ) = () (tj)! IY (zjt; ) (a jt ; z ) zz De nton 8 A probablty dstrbuton (A T ) s a BNE acton type state dstrbuton of G; S; S e f there exsts a BNE of G; S; S e such that =. = Proposton A probablty dstrbuton (A T ) s a Bayes Correlated Equlbrum of (G; S) f and only f t s a BNE acton type dstrbuton dstrbuton of u; ; S; S e for some augmented nformaton system e S. Proof. Suppose that s a correlated equlbrum of (u; ; S). a A ;t T ; a A ;t T ; for each, t T, a A and a A ; and Thus u ((a ; a ) ; ) ((a ; a ) ; (t ; t ) ; ) u a ; a ; ((a ; a ) ; (t ; t ) ; ) ; (a; t; ) = () (tj) aa for all t T and. Construct an augmented nformaton system S e = (Z ) I = ; wth each Z = A and (ajt; ) = (aj; t).

10 Correlated Equlbrum May 4, Now n the augmented ncomplete nformaton game wth (a jt ; a ) = for all, t and a. Clearly, we have =. Now = t T ;z Z ; t T ;z Z ; G; S; S e, consder the "truthful" strategy pro le u a ; (t ; z ) ; () ((t ; t ) j) ((z ; z ) j (t ; t ) ; ) u a ; a ; ((a ; a ) ; (t ; t ) ; ) and thus Nash equlbrum condtons are mpled by the correlated equlbrum condtons on. Conversely, suppose that s a Nash equlbrum of G; S; S e. Now (a j (t ; z )) > mples t T ;z Z ; t T ;z Z ; u ((a ; (t ; z )) ; ) () ((t ; t ) j) ((z ; z ) j (t ; t ) ; ) u a ; (t ; z ) ; () ((t ; t ) j) ((z ; z ) j (t ; t ) ; ). for each a A. Thus But = (a j (t ; z )) z Z (a j (t ; z )) z Z u ((a ; (t ; z )) ; ) () ((t ; t ) j) ((z ; z ) j (t ; t ) ; ) t T ;z Z ; t T ;z Z ; u a ; (t ; z ) ; () ((t ; t ) j) ((z ; z ) j (t ; t ) ; ). (a j (t ; z )) u a ; (t ; z ) ; () ((t ; t ) j) ((z ; z ) j (t ; t ) ; ) z Z t T ;z Z ; u a ; a ; ((a ; a ) ; (t ; t ) ; ) a A ;t T ; and thus BNE condtons mply that s a BCE. An alternatve formulaton of ths result would be to say that BCE captures the mplcatons of common certanty of ratonalty (and the common pror assumpton) n the game (G; S), snce requrng BNE n some game wth augmented nformaton s equvalent to descrbng a belef closed subset where the game (G; S) s beng played and there s common certanty of ratonalty. Thus ths s an ncomplete nformaton analogue of the Aumann (987) characterzaton of correlated equlbrum for complete nformaton games and thus - as descrbed n more detal n the next secton - corresponds to the "partal Bayesan approach" of Forges (99), wth the d erence that she works wth the reduced game - ntegratng out the payo states.

11 Correlated Equlbrum May 4, 4 A Number of Legtmate De ntons of Correlated Equlbrum n Incomplete Informaton Games Forges (99) s ttled and dent es " ve legtmate de ntons of correlated equlbrum n games wth ncomplete nformaton." Forges (6) descrbes a mstake n Forges (99) that leads to a sxth de nton. Our purpose n ths secton s to revew these sx de ntons and understand ther relaton to the soluton concept we dub "Bayes correlated equlbrum." Let us hghlght a few d erences between our formulaton of games and soluton concepts to bear n mnd as we descrbe the relaton:. Whle we drectly de ne soluton concepts for (G; S) as subsets of acton type state dstrbutons (A T ), she characterzes the set of equlbrum payo s satsfyng a set of restrctons whch mplctly de ne the soluton concept.. Whle we work wth a "basc game", G = (A ; u ) I = ;, descrbng pror and payo s and an "nformaton structure" S = (T ) I = ;, she dstngushes between the "decson problem wth ncomplete nformaton," (A ; u ) I = and ncludes the pror on payo relevant states n her descrpton of the "nformaton scheme".. Whle we nclude the dstrbuton of payo relevant states n our soluton concept, she ntegrates out payo relevant states. 4. Whle we and Forges (6) allow for any nte number of players, Forges (99) focussed on the two player case for smplcty. We start wth ve de ntons of correlated equlbrum for a ncomplete nformaton game (G; S). s useful to dvde restrctons nto two classes: feasblty condtons on the dstrbutons of acton type state pro les, whch are requred to hold ndependent of the payo functons, and ncentve compatblty condtons whch are ratonalty constrants on players acton choces. It The closest solutons rely only on addtonal feasblty constrants, mantanng obedence as the only ncentve compatblty constrant. Recall that the only feasblty condton we mposed n de nng Bayes Correlated Equlbrum was the consstency requrement (De nton ) that the acton type state dstrbuton mpled the dstrbuton on exogenous varables (types and states) was that of the game (G; S). If the soluton concept s ntended to capture the outcome of communcaton among the players perhaps allowng for an unnformed medator, t s natural to mpose the addtonal restrcton that the dstrbuton over states condtonal on agents types s not changed condtonal on the medator s recommendatons:

12 Correlated Equlbrum May 4, De nton 9 Dstrbuton (A T ) s jon feasble for (G; S) f, for all a A and t T such that we have for all. (a; t; ) > ; (a; t; ) () (tj) a; t; = tj (5) Ths assumpton s (mplctly) mantaned n all Forges soluton concepts for (G; S) and s made explct n Lehrer, Rosenberg, and Shmaya () and Lehrer, Rosenberg, and Shmaya () (e.g., condton 4 on page 676 n Lehrer, Rosenberg, and Shmaya ()). De nton A probablty dstrbuton (A T ) s a Bayesan soluton of (G; S) f t s consstent, jon feasble and obedent. Ths s the soluton concept dscussed n Secton 4.4 of Forges (99) and one of the two dscussed n secton.5 of Forges (6). Lehrer, Rosenberg, and Shmaya () refer to ths as a "global equlbrum." It also corresponds to the set of jontly coherent outcomes n Nau (99), just ed from no arbtrage condtons. Forges and Koessler (5) provde a just caton f players are able to certfy ther types to the medator. Ths soluton concept allows players to learn about other players types from the medator s recommendaton. The followng condton removes ths possblty: De nton Dstrbuton (A T ) s belef nvarant for (G; S) f, for all t T and a A such that ((a ; a ) ; (t ; t ) ; ) > ; a A ;t T ; we have ((a ; a ) ; (t ; t ) ; ) () ((t ; t ) j) for each t T. a A ; = (6) (a ; a ) ; t ; t ; () t ; t j a A ;t T ; t T ; Ths s condton on page 676 n Lehrer, Rosenberg, and Shmaya (). As Forges (6) puts t, "the omnscent medator can use hs knowledge of the types to make hs recommendatons but the players

13 Correlated Equlbrum May 4, should not be able to nfer anythng on the others types from ther recommendatons." Ths restrcton s added to gve the second soluton concept: De nton A probablty dstrbuton (A T ) s a belef nvarant Bayesan soluton of (G; S) f t s consstent, jon feasble, belef nvarant and obedent. Ths s the second soluton concept dscussed n Secton.5 of Forges (6); t was dscussed nformally n Secton 4.4 of Forges (99) but t was then mstakenly clamed that t was equvalent to agent normal form correlated equlbrum. Ths soluton concept s also used n Lehrer, Rosenberg, and Shmaya () and Lehrer, Rosenberg, and Shmaya (). Because they do not work wth the reduced game,.e., they explctly dscussed payo relevant states lke, and they must explctly mpose a jon feasblty restrcton. The belef nvarant Bayesan soluton allows the medator to use nformaton about players types to make a recommendaton to players. Suppose that the medator has no nformaton about the players types when decdng what strategy to recommend as a functon of the players types. Ths s re ected n the next feasblty restrcton. A pure strategy n the ncomplete nformaton game s functon : T! A. Wrte for the set of pure strateges of agent and for the set of pure strategy pro les, = ::: I. De nton Dstrbuton (A T ) s agent normal form feasble for (G; S) f there exsts q () such that (a; t; ) = () (tj) q () (7) fj(t)=ag for each a A, t T and. One can show that agent normal form feasblty mples belef nvarance. Ths restrcton s added to gve the thrd soluton concept: De nton 4 A probablty dstrbuton (A T ) s an agent normal form correlated equlbrum of (G; S) f t s consstent, unnformed medator feasble, agent normal form feasble (and thus belef nvarant) and obedent. Ths s the soluton concept dscussed n Secton 4. of Forges (99) and Secton. of Forges (6). It corresponds to applyng the complete nformaton de nton of correlated equlbrum to the agent normal form of the reduced ncomplete nformaton game. It was also studed by Samuelson and Zhang (989) and Cotter (994). The soluton concept only makes sense on the understandng that the players receve a recommendaton for each type but do not learn what recommendaton they would have receved f they had been d erent types. If they dd learn the whole strategy that the medator choose for them n the strategc form game, then an extra ncentve compatblty condton would be requred:

14 Correlated Equlbrum May 4, 4 De nton 5 Dstrbuton (A T ) s strategc form ncentve compatble for (G; S) f there exsts q () such that (a; t; ) = () (tj) q () (8) fj(t)=ag for each a A, t T and ; and, for each = ; ::; I, t T, a A and such that (t ) = a, we have () q ( ; ) A u ((a ; a ) ; ) (9) for all a A. a A ;t T ; () f j (t )=a g a A ;t T ; f j (t )=a g q ( ; ) A u a ; a ; Note that ths condton mples both agent normal form feasblty and obedence. gves the fourth soluton concept: Ths restrcton De nton 6 A probablty dstrbuton (A T ) s a strategc form correlated equlbrum of (G; S) f t s consstent, unnformed medator feasble and strategc form ncentve compatble (and thus agent normal form feasble, belef nvarant and obedent). Ths s the soluton concept dscussed n Secton 4. of Forges (99) and Secton. of Forges (6). Ths soluton concept was studed by Cotter (99). Thus far we have smply been addng restrctons, so that the soluton concept have become stronger as we go from Bayesan soluton, to belef nvarant Bayesan soluton, to agent normal form correlated equlbrum, to strategc form correlated equlbrum. For the Bayesan soluton, an omnscent medator who observes players types for free s assumed. For agent normal form and strategc form correlated equlbrum, the players types cannot play a role n the selecton of recommendatons to the players. An ntermedate assumpton s that the players can report ther types to the medator, but wll do so truthfully only f t s ncentve compatble to do so. Wrte : T! A for the medator s recommendaton strategy mpled by (A T ), so that, for each t T and wth (a ; t; ) >, a A (ajt; ) = (a; t; ) (a ; t; ) a A for each a A.

15 Correlated Equlbrum May 4, 5 De nton 7 Dstrbuton (A T ) s truth tellng for (G; S) f, for each = ; ::; I and t T, we have () ((t ; t ) j) ((a ; a ) j (t ; t ) ; ) () aa;t T ; aa;t T ; for all t T and : A! A. () ((t ; t ) j) ( (a ) ; a ) j t ; t ; ; Note that ths condton mples obedence (De nton ). One can show that ths condton s mpled by strategc form ncentve compatblty. Now we have the fth soluton concept: De nton 8 A probablty dstrbuton (A T ) s a communcaton equlbrum of (G; S) f t s consstent, jon feasble and ncentve compatble (and thus obedent). Ths s the soluton concept dscussed n Secton 4. of Forges (99) and Secton.4 of Forges (6), and developed earler n the work of Myerson (98) and Forges (986). Thus we have Forges ve soluton concepts for the ncomplete nformaton game (G; S):. Bayesan soluton (De nton );. Belef nvarant Bayesan soluton (De nton );. Agent normal form correlated equlbrum (De nton 4); 4. Strategc form correlated equlbrum (De nton 6); and 5. Communcaton equlbrum (De nton 8). As documented by Forges (99) and Forges (6) and mpled by the above de ntons, we have that the Bayesan soluton [] s weaker than the belef nvarant Bayesan equlbrum soluton [], whch s weaker than the agent normal form correlated equlbrum [], whch s weaker than the strategc form correlated equlbrum [4]; and also the Bayesan soluton [] s weaker than communcaton equlbrum [5] whch s weaker than strategc form correlated equlbrum [4]. Examples reported n Forges (99) and Forges (6) that each weak ncluson s strct and that the belef nvarant Bayesan soluton [] and agent normal form correlated equlbrum [] cannot be ranked relatve to communcaton equlbrum [5]. Our de nton of Bayes Correlated Equlbrum s weaker than the Bayesan soluton, the weakest of Forges ve, because we do not mantan jon feasblty.

16 Correlated Equlbrum May 4, 6 The followng s a trval (one player) example showng that Bayes Correlated Equlbrum s a more permssve soluton concept than any of Forges ve soluton concepts for (G; S). Suppose there s one player, I =, and two states, = ;. Let the basc game G = (A ; u ; ) be de ned by A = fa ; a g, u (a ; ) =, u a ; = and u (a ; ) = u a ; =, and () = =. And consder the null nformaton system S. Consstency (), obedence (4) and jon feasblty (5) together mply that (a ; ) = a ; = and a ; = a ; =. Ths s thus the unque Bayesan soluton, belef nvarant Bayesan soluton, agent normal form correlated equlbrum, strategc form correlated equlbrum and communcaton equlbrum. However, consstency () mples only that (a ; ) + a ; = a ; + a ; = and obedence (4) mples only that (a ; ) a ; a ; a ;. There are many Bayesan Correlated Equlbra satsfyng the above constrants. The one maxmzng the player s utlty has (a ; ) = a ; = and a ; = a ; =. In Secton 4.5, Forges (99) dscusses how more soluton concepts are concevable, ncludng by droppng jon feasblty, and gves an example lke the above llustratng ths pont. In Secton 6, Forges (99) consders a "unversal Bayesan approach" n whch a pror "nformaton scheme" (n our language, pror on and nformaton system) s not taken as gven. Thus her "unversal Bayesan soluton" s de ned for (A ; u ) I =. Expressng her deas n the language of acton state dstrbutons, she studes the followng soluton concept. De nton 9 A probablty dstrbuton (A ) s a unversal Bayesan soluton of (A ; u ) I = f t sats es (4). Thus a probablty dstrbuton (A ) s Bayes Correlated Equlbrum of G = (A ; u ) I = ; f and only f t s a unversal Bayesan soluton and sats es (). Note that applyng Proposton to the specal case of the null nformaton system, we have that (A ) s a Bayes Correlated

17 Correlated Equlbrum May 4, 7 Equlbrum of G f and only f there exsts an nformaton system S and a Bayes Correlated Equlbrum (A T ) of (G; S) whch nduces (A ). Ths then corresponds to Forges Proposton 4 when appled to the soluton concept of Nash equlbrum (although she states the results n terms of equlbrum payo s rather than dstrbutons). As she notes, her Proposton 4 s a natural ncomplete nformaton generalzaton of Aumann (987) and our Proposton s an example of such a generalzaton stated n a d erent language. 5 Comparng Informaton Systems An mportant result for our robust predctons agenda s that as players become more nformed, the set of possble predctons must be reduced, snce obedence constrants wll become tghter. We wll formalze ths result n the next sub-secton. Frst, we revew some exstng results on comparng nformaton systems. 5. The Exstng Lterature The followng useful termnology s used n Lehrer, Rosenberg, and Shmaya () and Lehrer, Rosenberg, and Shmaya (). De nton Informaton system S s a garblng of S f there exsts : T! (T ) and satsfyng t j = tt (tj) t jt for each t T and. The map s called a garblng that transforms S to S. Ths says that the jon of the nformaton n S s a garblng n the sense of Blackwell (95) of the jon of the nformaton n S. Garblng s non-communcatng f, for each = ; :::; I, t T, t T, t ; t j (t ; t ) = t ; t j t ; et t T t T for all t ; et T. De nton Informaton system S s a non-communcatng garblng of S f there exsts a non-communcatng garblng that transforms S nto S. Ths condton requres that each player s nformaton n S s a Blackwell garblng of hs nformaton n S. If garblng s a non-communcatng garblng, we wrte (t jt ) for the (t ndependent) probablty

18 Correlated Equlbrum May 4, 8 of t condtonal on t,.e., t jt t ; t j (t ; t ) t T Garblng s coordnated f there exst (f; :::; Kg) and, for each, : T f; :::; Kg! (T ) such that t jt = for each t T and t T. K (k) k= IY = t jt ; k De nton Informaton system S s a coordnated garblng of S f there exsts a coordnated garblng that transforms S nto S. A garblng s ndependent f t s coordnated wth K =, so that there exsts, for each, : T! (T ) such that t jt = for each t T and t T. K (k) k= IY = t jt ; k De nton Informaton system S s an ndependent garblng of S f there exsts a ndependent garblng that transforms S nto S Lehrer, Rosenberg, and Shmaya () and Lehrer, Rosenberg, and Shmaya () note that, by de - nton, an ndependent garblng s a coordnated garblng, a coordnated garblng s a non-communcatng garblng and a non-communcatng garblng s a garblng, and present elegant examples showng that none of the reverse mplcatons s true. Say that an nformaton system S s larger that S under a gven equlbrum concept f, for every game G, every acton state dstrbuton nduced by an equlbrum of (G; S ) s also nduced by an equlbrum of (G; S). Informaton system S s equvalent to S under a gven equlbrum concept f S s larger than S and S s larger than S under that equlbrum. Lehrer, Rosenberg, and Shmaya () show that (n Theorem.8) that. Two nformaton systems are equvalent under Bayes Nash Equlbrum f and only f they are ndependent garblngs of each other.. Two nformaton systems are equvalent under Agent Normal Form Correlated Equlbrum (De nton 4) f and only f they are coordnated garblngs of each other.

19 Correlated Equlbrum May 4, 9. Two nformaton systems are equvalent under the Belef Invarant Bayesan Soluton (De nton ) f and only f they are non-communcatng garblngs of each other. They do not report an analogous result for Bayes Correlated Equlbrum. for the "larger than" relaton. They do not report results Lehrer, Rosenberg, and Shmaya () consder common nterest games. Say that nformaton system s S better than S under a gven soluton concept f, for every common nterest game G, the maxmum (common) equlbrum payo s hgher n (G; S) than (G; S ). They show. (Theorem.5) Informaton system S s better than S under Bayes Nash Equlbrum f and only f S s a coordnated garblng of S.. (Theorem 4.) Informaton system S s better than S under Agent Normal Form Correlated Equlbrum (De nton 4) f and only f S s a coordnated garblng of S.. (Theorem 4.) Informaton system S s better than S under Strategc Form Correlated Equlbrum (De nton 6) f and only f S s a coordnated garblng of S. 4. (Theorem 4.5) Informaton system S s better than S under the Belef Invarant Bayesan Soluton (De nton ) f and only f S s a non-communcatng garblng of S. 5. (Theorem 4.6) Informaton system S s better than S under Communcaton Equlbrum (De nton 8) f and only f S s a garblng of S. Gossner () studes Bayes Nash equlbrum only as a soluton concept. nformaton games but also reports results for ncomplete nformaton games. Hs focus s on complete The dea of hs results s that more correlaton possbltes are better for the set of BNE that can be supported. To state Gossner s result, wrte BNE (G; S) for the set of BNE acton state dstrbutons of (G; S) (see De nton 6),.e., the set of dstrbutons on A that can be nduced by a BNE of (G; S). De nton 4 Informaton system S s BNE-larger than nformaton system S f BNE (G; S ) BNE (G; S) for all basc games G. have An ndependent garblng s fathful f whenever for each, t T and t T wth (t jt ) >, we t ; t j () e et T ;e t ; = et j e () ((t ; t ) Y j t j j jt A t T j6= e (t ; t ) j e t T ; e

20 Correlated Equlbrum May 4, for all t T and. De nton 5 Informaton system S s a fathful ndependent garblng of S f there exsts a fathful ndependent garblng that transforms S nto S. Intutvely, ths states that nformaton system S allows more correlaton possbltes than S but does does not gve more nformaton about belefs and hgher order belefs about payo relevant states. we have: Proposton Informaton system S s BNE-larger than S f and only f S s a fathful ndependent garblng of S. Ths s Theorem 9 n Gossner (). Now [In the bre y descrbed (Secton 6) statement of Gossner s result, hs de nton of BNE-larger ("rcher" n hs language) refers only to dstrbutons over acton pro les, and not over acton pro les and ; however hs arguments would apply the above result.] An nterestng specal case s when S s unnformatve,.e., contans nether nformaton about nor correlaton opportuntes, so that there exst, for each, (T ) such that t j = IY = t for all t T and. In ths case, BNE (G; S ) s just equal to the ndependent dstrbutons over actons generated by Nash equlbra n the basc game G. Ths S s a fathful ndependent garblng of S for any S whch s not nformatve about : smply set t jt = IY = t for all t T and. Now BNE (G; S) contans BNE (G; S ) because there are weakly more correlaton possbltes n S. 5. More Informaton Reduces the set of Bayes Correlated Equlbra We present a new result showng that more nformaton reduces the set of Bayes Correlated Equlbra. Note that ths result seems to go n the opposte drecton to Gossner s result, as we are seeng that more nformaton rules out more outcomes. The explanaton for ths apparent contradcton s that by usng BCE as a soluton concept, the players can correlate ther behavor for free n equlbra and the only mpact of more nformaton s to mpose more ncentve constrants. The relevant formalzaton of less nformaton s as follows:

21 Correlated Equlbrum May 4, De nton 6 Informaton system S s less nformed than S f there exst : T! (T ) and, for each, : T! (T ), such that t j = tt t jt; (tj) for each t T and, satsfyng also that for each = ; :::; I, t T, t T, for all t T and. t T t ; t j (t ; t ) ; = t jt Note that f S s a non-communcatng garblng of S exactly f the above de nton s sats ed but wth the functon not dependent on. Thus f S s a non-communcatng garblng of S, then S s less nformed than S. But a robust example n the Appendx (Secton 9.) shows that the converse s not true. Now we have: De nton 7 Informaton system S s BCE-larger than nformaton system S f BCE (G; S) BCE (G; S ) for all games G. Theorem S s BCE-larger than S f and only f S s less nformed than S: A sketch of the proof s the Appendx (Secton 9.). The argument nvolves relatng the hgher order belefs about under the two nformaton systems. See Tang () for more dscusson of ths ssue. 6 The Robust Predctons Agenda We wll report a couple of examples to llustrate the logc of our approach. 6. One Player, Bnary State, Bnary Acton Games Suppose there s one player and two states. There are two states, = f ; g. Consder the game G wth A = fa ; a g; u (a ; ) =, u (a ; ) = and u (a ; ) = u (a ; ) = ; and ( ) = and ( ) =. Consder an arbtrary nformaton system S = (T; ) and wrte k (t) for the probablty of sgnal t n state k. We are nterested n Bayes Correlated Equlbra of the game (G; S). In the specal case where S s the null nformaton system, ths reduces to the problem of ndng outcomes of a sender recever game where we can exogenously choose the sender strategy. Ths s thus closely related to the problem studed by Gentzkow and Kamnca (), where senders are allowed to commt to a communcaton strategy.

22 Correlated Equlbrum May 4, Suppose that the medator recommends acton a f the player observes sgnal t n state k probablty k (t) (and thus a wth probablty k (t)). Thus the medator s behavor s gven by ( ; ) wth each k : T! [; ]. Now f the player observes sgnal t and s advsed to take acton a, he attaches probablty to state and thus follows the recommendaton f or (t) (t) (t) (t) + ( ) (t) (t) wth ( ) (t) (t) ( ) (t) (t) () (t) (t) (t) (t). () If the player observes sgnal t and s advsed to take acton a, he attaches probablty to state and thus follows the recommendaton f (t) ( (t)) (t) ( (t)) + ( ) (t) ( (t)) ( ) (t) ( (t)) ( ) (t) ( (t)) or ( ) (t) (t) ( ) (t) (t) + ( ) (t) ( ) (t) () or (t) (t) (t) (t) + (t). (4) (t) Now the two obedence constrants () and (4) can be combned n the constrant that (t) (t) (t) + max ;. (5) (t) Now dstrbuton (A T ) s a Bayes Correlated Equlbrum f and only f 8 ( ) (t) (t), f (a; ) = (a ; ) >< ( ) (t) ( (t)), f (a; ) = (a ; ) (a; t; ) = (t) (t), f (a; ) = (a ; ) >: (t) ( (t)), f (a; ) = (a ; ) for some ( ; ) satsfyng (5). To understand how the set of BCE vary wth d erent nformaton structures, we can consder some extreme ponts. Consder the player s ex ante utlty ( (t) ( (t)) + ( ) ( ) (t) (t)). tt

23 Correlated Equlbrum May 4, Ths s maxmzed by settng (t) = and (t) = for all t T, gvng maxmum ex ante utlty U (S) = + ( ) ( ). We wrte ths as a functon of the nformaton system S, although t turns out to be ndependent of the nformaton system. Now let s nd the BCE mnmzng the player s ex ante utlty. From () and (), we have that ( ) (t) (t) ( ) (t) (t) max f; ( ) (t) ( ) (t) g (6) Thus the utlty mnmzng BCE s attaned by settng (t) = (t) = f (t) (t) and (t) = (t) = f Ths gves mnmum ex ante utlty (t) U (S) = Pr (t) (t) > (t) > +( ) ( ) Pr (t). (t) Notce that more nformaton wll ncrease the mnmum ex ante utlty and not change the maxmum ex ante utlty. Now consder the probablty that acton a s chosen, ( (t) (t) + ( ) (t) (t)). tt Ths s maxmzed by settng (t) = (t) = f (t) (t) and (t) = and (t) solves (t) = (t) (t) (7) otherwse. (S) = Thus the maxmum probablty of acton a n a BCE s (t) Pr > (t) (t) (t) (t) (t) + ( ) (t) Ths s mnmzed by settng (t) = (t) = f (t) > (t)

24 Correlated Equlbrum May 4, 4 and (t) = and (t) solves (t) = (t) (t) (8) otherwse. Thus the mnmum probablty of acton a n a BCE s (t) (S) = Pr (t) 6. Frst Prce Auctons (t) (t) ( ) (t) (t) + ( ) (t) We consder a dscretzed rst prce prvate value aucton wth ndependent unform prors. Suppose there are two players and K states, = K ; K ; :::; K K ; wth typcal element = ( ; ). Consder the game G wth A = A = ; M ; M ; :::; M M ; ; () = K for each ; 8 >< a, f > j u ((a ; a j ) ; ( ; j )) = >: ( a ), f = j., f < j The nformaton structure S has each player observng hs prvate value. Formally, we have S = (T ) =; ; where T = T = K ; K ; :::; K K ; and 8 <, f t = (tj) = :, otherwse. Now a Bayes Correlated Equlbrum s a dstrbuton (A T ) consstency 8 <, f t = K (a; t; ) = :, otherwse aa for all t and ; and obedence ((a ; a j ) ; (t ; t j ) ; ( ; j )) u ((a ; a j ) ; ( ; j )) ((a ; a j ) ; (t ; t j ) ; ( ; j )) u a j ;t j ; j a j ;t j ; j a ; a j ; ( ; j ) for each, ; t, a and a. The followng smple example llustrates the d erence n the bddng behavor n the Bayes Nash Equlbra (BNE) and Bayes Correlated Equlbra (BCE) of the game (G; S) wth a payo type space and a rch type space. Let K = and M = 8. The unque BNE s gven by a symmetrc pure strategy pro le

25 Correlated Equlbrum May 4, 5 dsplayed n the left matrx below: a n a n :8 8 :56 8 :6 :5 8 :5 :7 4 8 : The entry n each cell s the condtonal probablty, (a j ), that agent wth value submts a bd a. In contrast, the revenue mnmzng BCE gves rse to the condtonal dstrbutons of bds a gven values descrbed the above rght matrx. We observe that the average bd n the revenue mnmzng BCE s strctly below the bd n the BNE. The revenue n the BCE s gven by. whereas n the BNE t s.4. (The examples were computed wth programs wrtten for Matlab.) 7 Anonymous Games In ths secton, we specalze our analyss to the case of anonymous games, where each player s symmetrc n payo s and nformaton, so that players labels are assumed to not matter for ether the descrpton of the game or ther choce of strategy. Once we have an anonymous nte player, nte acton, nte state verson of Bayes Correlated Equlbrum, t s then possble to present analogue results for contnuum player, contnuum acton, contnuum state games. Ths s the foundaton for our quadratc normal modellng n Bergemann and Morrs (): 7. The Fnte Case As before, there are I players and nte state space. A "basc game" G now conssts of () a common acton set A; () a common utlty functon u : A I (A)! R;, where u (a; h; ) s a player s payo f he chooses acton a, the dstrbuton of actons among the I players s h I (A) and the state s. (For any nte set, we wrte I () for the set of probablty dstrbutons on wth support on ; I ; I ; ::; ); and () a full support pror (). Thus a basc game G = (A; u; ). An "nformaton structure" S now conssts of () a common set of types or "sgnals" T ; and () a sgnal dstrbuton :! ( I (T )). Now () ( I (T )) s a probablty dstrbuton over the realzed dstrbuton of sgnals n the populaton. Thus S = (T; ). Now (G; S) descrbes a standard (anonymous) Bayesan game.

26 Correlated Equlbrum May 4, 6 If I (A T ) s a dstrbuton over acton-sgnal pars, wrte marg T I (T ) for the margnal dstrbuton over sgnals, so marg T (t) = (a; t) aa for each t T ; wrte marg A I (A) for the margnal dstrbuton over actons, so marg A (a) = (a; t) ta for each a A. If ( I (A T ) ) s a dstrbuton over acton-sgnal par dstrbutons and states, wrte marg I (T ) ( I (T ) ) for the margnal dstrbuton over realzed dstrbutons of sgnals and states, so marg I (T ) (g; ) = (; ) f I (AT ):marg T =gg for each g I (T ) and. Fnally, wrte for the probablty dstrbuton on I (T ) nduced by () and :! ( I (T )), so (g; ) = () (gj) for each g I (T ) and. De nton 8 (Bayes Correlated Equlbrum ) A probablty dstrbuton ( I (A T ) ) s a Bayes Correlated Equlbrum (BCE) of (G; S) f u (a; marg A ; ) (a; t) (; ) u a ; marg A ; (a; t) (; ) ; (9) I (AT ); I (AT ); for each t T, a A and a A; and marg I (T ) =. () In the specal case of a null nformaton system (so there are no sgnals), then the obedence condton (9) for ( I (A) ) wll be u (a; g; ) g (a) (g; ) u a ; g; g (a) (; ) ; g I (A); g I (A); for each a A and a A whle the consstency condton () wll be marg =. ()

27 Correlated Equlbrum May 4, 7 7. The Contnuum Case There s a contnuum [; ] of players and state space. A "basc game" G now conssts of () a common acton set A; () a common utlty functon u : A (A)! R; where u (a; h; ) s a player s payo f he chooses acton a, the dstrbuton of actons among the contnuum players s h (A) and the state s ; and () a full support pror (). Thus G = (A; u; ). An "nformaton structure" S now conssts of () a common set of types or "sgnals" T ; and () a sgnal dstrbuton :! ( (T )). Now () ( (T )) s a probablty dstrbuton over realzed dstrbutons of sgnals n the populaton. Thus S = (T; ). Now (G; S) descrbes a standard contnuum (anonymous) Bayesan game. Now the de ntons for the contnuum case are as before, except that dstrbutons are over a contnuum populaton and summatons are replaced wth ntegrals. We omt the measurablty condtons that wll be requred n general (they are not an ssue for applcatons usng denstes). As before, f (A T ) s a dstrbuton over acton-sgnal pars, wrte marg T (t) I (T ) and marg A I (A) for the margnal dstrbutons over sgnals and actons respectvely. If ( (A T ) ), wrte marg (T ) ( (T ) ) for the margnal dstrbuton over realzed dstrbutons of sgnals and states. Wrte for the probablty dstrbuton on (T ) nduced by () and :! ( (T )). De nton 9 (Bayes Correlated Equlbrum ) A probablty dstrbuton ( (A T ) ) s a Bayes Correlated Equlbrum (BCE) of (G; S) f Z Z u (a; marg A ; ) (a; t) d u a ; marg A ; (a; t) d; () (AT ); (AT ); for each t T, a A and a A; and marg (T ) =. () In the specal case of a null nformaton system (so there are no sgnals), then the obedence condton () for ( (A) ) wll be Z Z u (a; g; ) g (a) d u a ; g; g (a) d; g(t ); g(t ); for each a A and a A whle the consstency condton () wll be marg = :

28 Correlated Equlbrum May 4, 8 8 Dscusson 8. Payo Type Spaces In Bergemann and Morrs (5) and later work, we studed a robust mechansm envronments n a settng where agents knew ther own "payo types", there was common knowledge of how utltes depended on the pro le of payo types, but agents were allowed to have any belefs and hgher order belefs about others payo types. work. In Bergemann and Morrs (7), we dscussed a game theoretc framework underlyng ths Here we bre y how ths envronment maps nto the settng of ths paper. Suppose that s a product space wth = :::: I. Consder the specal nformaton structure where agent s set of possble sgnals s, and each agent observes the realzaton, so S = ( ) I = ; d, where d s the dentty map d :! wth d () = for all. Now the set of Bayes Correlated Equlbra of a game (G; S) descrbe all the dstrbutons over payo type pro les and actons consstent wth the common pror and common knowledge of ratonalty. Bergemann and Morrs (7) - n the language of ths paper - s an analyss of the structure of Bayes Correlated Equlbra wth the specal nformaton structure S. 8. Sgned Covarance Chwe (6) analyzes statstcal mplcatons of ncentve compatblty n general, and n partcular statstcal mplcatons of correlated equlbrum play. We can state hs man observaton n the language of our paper. Fx any basc game G. Fx any Bayes Correlated Equlbrum (A ) of the basc game (.e., the game wth the null nformaton structure). Fx a player and acton a A. Consder the random varable I a on A that takes value f a s played and otherwse. Fx any other acton a A. Let a ;a be the random varable on A equal to the payo gan to player of choosng acton a rather than a. Then, condtonal on a or a beng played, the random varables I a and a ;a have postve covarance. Ths s the content of the man result n Chwe (6). As he notes, ths s not merely a re-wrtng of the ncentve compatblty constrants, snce these are lnear n probabltes whle the covarance s quadratc n probabltes. Thus hs sgned condtonal covarance result s a necessary property of second order statstcs of a Bayes Correlated Equlbrum. are We sketch a formal statement and proof. The formal de ntons of the random varables I a I a (a; ) = 8 <, f a = a :, otherwse a ;a (a; ) = u ((a ; a ) ; ) u a ; a ; and a ;a