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Yale Lecture on Correlated Equilibrium and Incomplete Information Yale lecture on Correlated Equilibrium and Incomplete Information Stephen Morris March 2012 Introduction Game Theoretic Predictions are very sensitive to beliefs/expectations and higher order beliefs/expectations or (equivalently) information structure (Higher order) beliefs/expectations are rarely observed What predictions can we make and analysis can we do if we do not observe (higher order) beliefs/expectations? Robust Predictions Agenda: Basic Question Fix "payo¤ relevant environment" = action sets, payo¤-relevant variables ("states"), payo¤ functions, distribution over states = incomplete information game without higher order beliefs about states Assume payo¤ relevant environment is observed by the analyst/econometrician Analyze what could happen for all possible higher order beliefs (maintaining common prior assumption and equilibrium assumptions) Make set valued predictions about joint distribution of actions and states Robust Predictions Agenda: Minimal Information More generally, the analyst might know something about players’higher order beliefs but cannot rule out the possibility that they know more What predictions can he then make? Narrower set of predictions (more incentive constraints) More knowledge of agents’information thus allows tighter identi…cation of underlying game Partial Identi…cation (not this lecture) Incomplete Information Correlated Equilibrium Robust predictions = set of outcomes consistent with equilibrium given any additional information the players may observe set of outcomes that could arise if a mediator who knew the payo¤ state could privately make action recommendations set of incomplete information correlated equilibrium outcomes very permissive de…nition of incomplete information correlated equilibrium: we dub it "Bayes correlated equilibrium" This Talk 1 De…nition of Bayes correlated equilibrium 2 "Epistemic result": Bayes correlated equilibria equal set of distributions that could arise if players observed additional information 3 One player, two action, two state example (c.f. Kamenica Gentzkow 11) 4 Other de…nitions of incomplete information correlated equilibrium drop "join feasibility" from Forges’93 de…nition of Bayesian solution drop "belief invariance" from (implicit) de…nition in Liu 05/11 5 Comparative statics of information more information reduces the set of Bayes correlated equilibria many player garblings; ordering many player information Setting players i = 1; :::; I (payo¤ relevant) states Payo¤ Relevant Environment actions (A )I i i=1 I utility functions (ui ) , each ui : A R i=1 ! state distribution () 2 G = (A ; u )I ; i i i=1 ("basic game", "pre-game") Information Environment signals (types) (T )I i i=1 signal distribution : (T1 T2 ::: TI ) ! S = (T )I ; i i=1 ("higher order beliefs", "type space," "signal space") Games with Incomplete Information The pair (G; S) is a standard game of incomplete information A (behavioral) strategy for player i is a mapping bi : Ti (Ai ) ! DEFINITION. A strategy pro…le b is a Bayes Nash Equilibrium (BNE) of (G; S) if, for all i, ti and ai with bi (ai ti ) > 0, j bj (aj tj ) ui ((ai ; a i ) ; ) () (t ) 0 j 1 j a i A i ;t i T i ; j=i 2 X 2 2 Y6 @ A bj (aj tj ) ui ai0; a i ; () (t ) 0 j 1 j a i A i ;t i T i ; j=i 2 X 2 2 Y6 @ A for all a Ai . i0 2 Bayes Correlated Equilibrium DEFINITION. An action type state distribution (A T ) is a Bayes Correlated Equilibrium (BCE) of 2 (G; S) it is obedient, i.e., for each i, ti , ai and ai0, ui ((ai ; a i ) ; ) ((ai ; a i ) ; (ti ; t i ) ; ) a i A i ;t i T i ; 2 X 2 2 ui ai0; a i ; ((ai ; a i ) ; (ti ; t i ) ; ) a i A i ;t i T i ; 2 X 2 2 and consistent, i.e., (a; t; ) = () (t ) . j a A X2 Augmented Information Structure We know players observe S but we don’tknow what additional information they observe. Augmented information structure S = (Z )I ; , where i i=1 : T (Z) ! e Augmented information information game G; S; S Player i’sbehavioral strategy bi : Ti Zi (Ai ) ! e Augmented Information Structure DEFINITION. A strategy pro…le b is a Bayes Nash Equilibrium (BNE) of (G; S; S ) if, for all i, ti ; zi and ai with bi (ai ti ; zi ) > 0, 0 j bj (aj tj ; zj ) ui ((ai ; a i ) ; ) (z; t; ) 0 j 1 a i ;t i ;z i ; j=i X Y6 @ A bj (aj tj ; zj ) ui ai0; a i ; (z; t; ) 0 j 1 a i ;t i ;z i ; j=i X Y6 @ A for all a Ai , where i0 2 (z; t; ) = () (t ) (z t; ) j j BNE Action Type State Distributions DEFINITION. An action type state distribution (A T ) is a BNE action type state distribution of 2 (G; S; S0) if there exists a BNE strategy pro…le b such that I (a; t; ) = () (t ) bi (ai ti ; zi ) (z t; ) . j j j z Z i=1 ! X2 Y Result THEOREM. Action state distribution is a BNE action type state distribution of G; S; S for some S if and only if it is a BCE of (G; S). e e c.f. Aumann 1987, Forges 1993 Proof is quite mechanical and analogous to standard arguments Idea of Proof Suppose that is a BCE. Consider the augmented I information system S = (Zi )i=1 ; with e Zi = Ai (a t; ) = (a t; ) j j Conversely, if is a BNE action type space distribution then consistency holds by construction and the obedience constraints are average of BNE best response conditions Predictions with No Information Null information structure S0 = (Z )I ; with each i i=1 0 0 Zi = t and t = 1. i j Write G for the game with null information (G; S0) Important special case Bayes Correlated Equilibrium (with Null Information) DEFINITION. An action state distribution (A ) is a Bayes Correlated Equilibrium (BCE) of G if is obedient2 , i.e., for each i, ai and ai0, ui ((ai ; a i ) ; ) ((ai ; a i ) ; ) X2 ui ai0; a i ; ((ai ; a i ) ; ) X2 and consistent, i.e., for each (a; ) = () . a A X2 BNE Action State Distributions DEFINITION. An action state distribution (A ) is a BNE action state distribution of G if there exists2 a BNE strategy pro…le b of G such that I (a; ) = () (t ) bi (ai ti ) . j j t T i=1 ! X2 Y Result COROLLARY. Action state distribution is a BNE action state distribution of (G; S) for some S if and only if it is a BCE of G. If is a singleton, this is exactly Aumann 87 Simplest Example: Setting with One Player and Two States Kamenica and Gentzkow (2011) I = 1 = 0; 1 ; innocent, guilty f g f g Simplest Example: Basic Game A = a0; a1 = acquit, convict f g f g Payo¤s u given by 0 1 a0 0 a1 0 1 Prior has (0) = and (1) = 1 . Simplest Example: Information Structure S = (T ; ) Finite T Write k (t) = (t k ) j No Information Say > 1 Some Extremal Bayes Correlated Equilibria: e¢ cient 0 1 a0 0 a1 0 1 most acquitals 0 1 a0 1 a1 0 0 most convictions 0 1 1 a0 (1 ) 0 1 a1 (1 ) 1 Robust Predictions Maximum Utility in BCE of (G; S) is + (1 )(1 ) Minimum Utility in BCE of (G; S) is max 0 (t) ; (1 )(1 ) 1 (t) t f g X Other Examples 1 Continuum Symmetric Player Linear Normal Games ("Robust Predictions in Incomplete Information Games") no identi…cation of sign or magnitude of interaction e¤ect without informational assumptions exact identi…cation with known information structure smooth partial identi…cation in between 2 First Price Auction (two player symmetric uniform [0; 1]) (Wednesday theory seminar) revenue = 1 3 minimum BCE revenue = 0:14? maximum BCE revenue = 1 2 "Legitimate De…nitions" Forges (1993): "Five Legitimate De…nitions of Correlated Equilibrium in Games with Incomplete Information" Forges (2006) gives #6 Liu (2005, 2011) gives one more relevant one More Feasibility Restrictions DEFINITION. Action type state distribution (A T ) is join feasible for (G; S) if there exists f : T 2 (A) such that ! (a; t; ) = () (t ) f (a t) j j for each a; t; . DEFINITION. Action type state distribution is a Bayesian solution of (G; S) if it is a BCE (i.e., is obedient and consistent) and also satis…es join feasibility. This is Forges’weakest solution concept If there is a dummy player who observes , then join feasibility is "free" and BCE = Bayesian solution In a private values environment (as in Wednesday’sseminar), join feasibility is also "free" More Feasibility Restrictions DEFINITION. Action type state distribution (A T ) is belief invariant for (G; S) if 2 ((ti ; t i ) ) () (t i ; ti ; ai ) = j j ti ; t0 i 0 0 j t0 i ;0 X Liu (2005) showed that (subjective) belief invariant BCE correspond to interim correlated rationalizability in the sense of Dekel, Fudenberg and Morris (2007) and thus capture the implications of common certainty of rationality for …xed Mertens-Zamir hierarchies. One Player Example I = 1 = 0; 1 f g ( ) = ( ) = 1 0 1 2 Payo¤s u 0 1 2 a0 3 0 1 a1 0 3 unique Bayesian solution and unique belief invariant BCE: 1 1 (a0; 0) = (a0; 1) = 2 ; ex ante expected utility 3 a BCE: (a ; ) = (a ; ) = 1 ; ex ante expected utility 1 0 0 1 1 2 2 Primer on Other De…nitions 1 Bayesian solution (BCE + join feasibility) 2 Belief invariant BCE (BCE + belief invariance) 3 Belief invariant Bayesian solution (BCE + join feasibility + belief invariance) 4 Agent normal form correlated equilibrium (BCE + uninformed mediator) (4) not equal to (3) as claimed in Forges (1993) 5 Strategic form correlated equilibrium (BCE + uninformed mediator + implied extra info revelation) 6 Communication equilibrium (BCE + truthtelling constraints) Relationships One ordering from biggest to smallest: 1 Bayes correlated equilibrium 2 Bayesian solution 3 belief invariant Bayesian solution 4 agent normal form CE 5 strategic form CE Belief invariant BCE (Liu) is contained in (1) and contains (3) Communication equilibrium is contained in (5) and contains (2) No other rankings possible Main Result Write BCE (G; S) for the set of action state distributions induced by Bayes Correlated Equilibria of BCE (G; S), i.e., (A T ) s.t.
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