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. 9 2 BCE (G; S) = (A ) = m argA 8 2 9 is a BCE < =
DEFINITION. :Information system S0 is BCE-larger than ; information system S if BCE (G; S) BCE (G; S0) for all games G. Main Result
DEFINITION. Information system S0 is BCE-larger than information system S if BCE (G; S) BCE (G; S0) for all games G.
Main Result Preview:
THEOREM. S0 is BCE-larger than S if and only if S is more informed than S0:
Intuition: adding more information adds incentive constraints but also (for some solution concepts) weakens feasibility constraints BCE has no feasibility constraints A Partial Order on Information Structures
DEFINITION. Information structure S is more informed than S0 if there exist : T (T ) such that ! 0
0 t0 = t0 t; (t ) j j j t T X2 for each t T and , and 0 2 0 2
ti0; t0 i (ti ; t i ) ; j t0 i T 0 i X2 is independent of t i and .
Intuition: S0 is a "many player garbling" of S A Partial Order on Information Structures
DEFINITION. Information structure S is informationally equivalent to S0 if S is more informed than S0 and S0 is more informed than S.
Observations:
1 If is a singleton, all information structures are informationally equivalent to each other 2 Two information structures are informationally equivalent if and only if they generate the same probability distribution over Mertens-Zamir belief hierarchies on by Liu (2005/11), each must then be obtainable from the other by adding correlating devices ( belief invariant augmentation) Relating the Partial Order to Augmentations
This ordering on information structures captures the intuition of "augmentations"
LEMMA. Information structure S S is more informed than S for any information structure S and augmentation S. e LEMMA. Information structure S is more informede than S0 if and only if there exists augmentation S such that S S is informationally equivalent to S. e e Relating the Partial Order to Garblings
If we focus on the join of players’information and treat each type pro…le as a signal, we can compare with Blackwell’sclassic ordering:
DEFINITION. (Blackwell) Information system S0 garbling of S if there exists : T (T ) with ! 0
0 t0 = (t ) t0 t j j j t T X2 for each t T and . Map is a garbling that transforms 0 2 0 2 S to S0. Relating the Partial Order to Garblings
Lehrer, Shmaya, Rosenberg (2010/11) have introduced important generalization:
Garbling is non-communicating if, for each i = 1; :::; I , ti Ti , t Ti , 2 i0 2
ti0; t0 i (ti ; t i ) = ti0; t0 i ti ; t i j j t0 i T 0 i t0 i T 0 i X2 X2 e for all t i ; t i T i . 2 Corresponds to player by player belief invariance but cannot depend one
DEFINITION. (Lehrer-Schmaya-Rosenberg 2010/11) Information system S0 is a non-communicating garbling of S if there exists a non-communicating garbling that transforms S into S0. Relating our Partial Order to Garblings
if S is a non-communicating garbling of S then S is more 0 informed than S0 converse is false (example in paper) but if we add a dummy player who knows the true state to both information systems, then S is more informed than S0 if and only if S0 is a non-communicating garbling of S Back to the Main Result
Now we have de…ned and understand the "more informed than" relation, we return to the main result
THEOREM. S0 is BCE-larger than S if and only if S is more informed than S0:
De…nition Two information structures are BCE-equivalent to each other if each is more informed than the other.
COROLLARY. S0 is BCE-equivalent to S if and only if S0 is informationally equivalent to S: Proof of Main Result
Suppose that is a BCE of (G; S) and that S is more informed than S0 Write : T (T ) for the garbling that generates S 0 0 from S ! We will de…ne (A T ) by 0 2 0
0 a; t0; = t0 t; (a; t; ) j t T X2 satis…es consistency by construction 0 obedience constraints for are averages (with weights 0 depending on ) of obedience constraints for belief invariance property of is used Proof of Main Result
Suppose that S is BCE-larger than S 0 Consider a …nite game G (like DFM 2007) where players ";S report their beliefs and higher order beliefs, with an "-grid of possible reports and exact types in S Consider BCE of (G ; S) (for ALL " > 0) where beliefs ";S and higher order beliefs are truthfully reported For each " > 0, there must exist a BCE of (G ; S ) giving " ";S 0 same action state distribution In a sequence of such BCE (as " 0), players’types in S 0 cannot be giving them information! about others’actions (i.e., types in S) and states. So S is more informed than S 0 Existing Comparative Static Results
Lehrer, Shmaya and Rosenberg (2011): Two information structures give rise to the same belief invariant Bayesian solutions in all basic games if and only if they are non-communicating garblings of each other LRS also report analogous results that if you strengthen solution concept (i.e., to agent normal form correlated equilibrium / BNE) using stronger equivalence notions (coordinated / independent garblings) our corollary is what you get if you weaken solution concept (to BCE) and weaken equivalence notion (to informational equivalence) LRS dont have converse, we do because feasibility constraints in LRS solution concepts make more information valuable unlike in BCE Existing Comparative Static Results
Lehrer, Shmaya and Rosenberg (2010) restrict attention to common interest basic games G. Players’highest possible belief invariant Bayesian solution utility in (G; S) is at least as high as their highest possible belief invariant Bayesian solution utility in (G; S0) for all G if and only if S0 is a non-communicating garbling of S. again, analogous results for stronger solution concepts / information structure orderings opposite direction of our main result, i.e., more information increases set of possible distributions intuition: common interest no binding obedience constraints in best equilibrium) our analogous result: the maximum utility in BCE in common interest games is independent of the information structure Existing Comparative Static Results
Gossner (2000) Information structure S gives rise to a larger set of BNE outcomes than information structure S0 in all games if and only if S0 is faithfully reproducible from S. again, opposite direction of our main result, i.e., more information increases set of possible distributions intuitively, S is faithfully reproducible from S if S gives new 0 correlation possibilities but there is no change in beliefs and higher order beliefs Summary
Robust Predictions agenda leads use to a new de…nition of incomplete information correlated equilibrium Interesting connections with the literature Clean comparative static on information structures in many player games Gives rise to new ordering on information structures, tying together in interest ways with existing ones