Common Belief Foundations of Global Games

Transcription

1 of Global Games Stephen Morris and Hyun Song Shin Conference on Complementarities and Information IESE Business School June 2007

2 Introduction COMPLEMENTARITIES lead to multiple equilibria Multiplicity sensitive to INFORMATIONAL assumptions: relaxing informational assumptions gives rise to "uniqueness" Carlsson & van Damme "Global Games and Equilibrium Selection" Econometrica

3 "Regime Change" Game Continuum of players "invest" or "not invest" Payos { Cost of investing: c 2 (0; 1) { Return to investing 1 if proportion investing is at least 1 0 otherwise 2

4 Equilibrium 1. Common knowledge of Multiple equilibria if 2 (0; 1) 3

5 Equilibrium 1. Common knowledge of Multiple equilibria if 2 (0; 1) 2. Lack of common knowledge of N y; 1, xi N ; 1 unique equilibrium if and only if 2 2 each player invests if x i > bx (y), does not invest if x i < bx (y) as! 1, bx! 1 c. 4

6 "Regime Change" example references Morris & Shin "Coordination Risk and Price of Debt" EER 2004 Metz "Private and Public Information in Self-Fullling Currency Crises" JE 2002 Hellwig "Public Information, Private Information and the Multiplicity of Equilibria in Coordination Games" JET 2002 Rochet & Vives "Coordination Failures and Lender of Last Resort" JEEA 2004 Dasgupta "Coordination and Delay in Global Games" JET forthcoming Morris & Shin "Liquidity Black Holes" RF

7 Morris & Shin "Catalytic Finance" JIE 2006 Angeletos, Hellwig & Pavan "Signalling in a Global Game" JPE 2006 Angeletos, Hellwig & Pavan "Policy in a Global Coordination Game" Angeletos, Hellwig & Pavan "Dynamic Global Games of Regime Change" Econometrica 2006 Angeletos & Werning "Crises and Prices" AER

8 Usefulness of Methodology? Catchy name, cute model but is it relevant? 1. Robustness to Endogenous Public Information (e.g., Angeletos & Werning 2005 etc...) 2. Relevance of Private Information (e.g., Svensson 2005, Sims 2006 etc...) 3. What about other ways of relaxing common knowledge assumptions (Weinstein & Yildiz 2007)? "" may help address these questions... 7

9 1. Carlsson & van Damme Morris & Shin "Informational Events that Trigger Currency Attacks" Philly Fed Working Paper Morris & Shin "Global Games: Theory and Applications" Econometric Society World Congress Hellwig

10 This Paper Can we express necessary and sucient conditions for a unique rationalizable outcome in natural language of higher order beliefs? 9

11 I: Common Knowledge of Rank Beliefs In normal information structure, what does a player believe about the rank of his private signal? Consider player observing x if at percentile c, then p (x ) = c or = x 1 (c) p 2. thus at percentile c or lower if x 1 (c) p 3. believes N y+x + ;

12 4. probability at cth percentile or lower is p (x y) 1 p 1 (c) which tends to c as 2! 1. Thus (in limit) common knowledge that players have uniform beliefs over own rank in population. "Monotonicity" and common knowledge of (any) rank beliefs implies unique rationalizable outcome. 11

13 II: Common Knowledge of Beliefs in Dierences x i N ; 1 proportion optimists believe N x i + ; 1 proportion 1 pessimists believe N x i ; 1 equilibrium characterized by ; x o and x p 12

14 equilibrium conditions: p (1 ) 1 x p p + 1 (x o ) 1 ( p ( x o )) = c 1 p x p + = c = 1 So = (1 ) p 1 (1 c) p p + 1 (1 c) p Common knowledge of "beliefs in dierences". "Montonicity" and common knowledge of beliefs in dierences implies unique rationalizable outcome. 13

15 Model 1. Background Players I = f1; :::; Ig Finite "payo states" 14

16 Model 1. Background Players I = f1; :::; Ig Finite "payo states" 2. Type Space T = (T i ; i ) I i=1 i's types: T i i's belief: i : T i! (T i ) 15

17 Model 1. Background 2. Type Space T = (T i ; i ) I i=1 3. Binary Action Game with Strategic Complementarities = ( i ) I i=1 i chooses a i 2 f0; 1g i (Z; ) is payo gain to action 1 over 0 in state if Z is the set of opponents choosing 1, i.e. u i (1; a i ; ) u i (0; a i ; ) = i (fj 6= ija j = 1g ; ) i : 2 I=fig! R, increasing in Z 16

18 Question What joint restriction on higher order beliefs (T ) and payos () gives unique rationalizable outcomes? 17

19 Generalized Belief Operators I "Simple" event { F = F i i=1;i { each F i T i { F is an event in T = T i i=1;i 18

20 Generalized Belief Operators II Belief Operator 8 9 < B i i (F ) = : t X = i 2 F i i (t i ; ) i (fj 6= i jt j 2 F j g ; ) 0 ; t i ; { t i 2 B i i j 6= i (F ) means that type t i thinks it likely that t j 2 F j for many { monotonic: F F 0 ) B i i { B (F ) = i=1;i B i i (F ) (F ) B i i (F 0 ) 19

21 Generalized Common Belief DEFINITION: There is common -belief of F at t if t 2 C (F ) \ k1 B k (F ). DEFINITION: Event F is -evident if F B (F ). PROPOSITION (cf, Aumann 1976, Monderer and Samet 1989): Event F is common -belief at t (t 2 C (F )) if and only if there exists a -evident event F 0 such that t 2 F 0 F. 20

22 Generalized Common Belief Fix X and let X;p i (Z; ) = 1 p, if Z = I= fig and 2 X p, otherwise Now C (T ) is the event that 1. everyone believes X with probability at least p 2. everyone believes with probability at least p that everyone believes X with probability at least p 3. et cetera... 21

23 Rationalizability DEFINITION: Action a i is rationalizable for type t i if a i 2 Ri (; t i), where Ri 0 (; t i ) = f0; 1g 8 there exists i 2 (T i f0; 1g) such that (1) i (t i ; ; a i ) > 0 ) a j 2 Rj >< k (; t j) for all j 6= i R k+1 (2) X i (; t i ) = a i i (t i ; ; a i ) = i (t i ; jt i ) a i X >: (3) a i 2 arg max i (t i ; ; a i ) u i ((a 0 a 0 i ; a i) ; ) i t i ;;a i 9 >= >; R i (; t i ) = \ k1 R k i (; t i ) 22

24 Characterization PROPOSITION: Action 1 is rationalizable for type t i t i 2 B i i C (T ). if and only if Inverse operator: e i (Z; ) = i (I=Z; ) PROPOSITION: Action 0 is rationalizable for type t i t i 2 B e i i C e (T ). if and only if 23

25 Example 1 = f0; 1g Payo to investing is 1 c if in state 1 (the "good state") and the other player invests, c otherwise. The payo to not investing is 0. = 0 (bad) Invest Not Invest Invest c c Not Invest 0 0 = 1 (good) Invest Not Invest Invest 1 c c Not Invest 0 0 i (Z; ) = 1 c, if Z = f3 ig and = 1 c, otherwise 24

26 Example 1 If the state is bad, both players have a dominant strategy to not invest. If the state is good, then the game has multiple Nash equilibria. This game is in the spirit of Rubinstein's (1989) game. Action invest is rationalizable for player 1 if 1. player 1 assigns probability at least c to the good state player 1 assigns probability at least c to both players assigning probability at least c to the good state player 1 assigns probability at least c to both players c-believing that both players p-believe the good state... 25

27 4. and so on... Thus we require common c-belief that the state is good. 26

28 Example 2: Regime Change Game There is a cost of investing: c 2 (0; 1). The return to investing is 1 if proportion investing is at least 1, 0 otherwise i (Z; ) = 1 c, if #Z I 1 1 c, otherwise 27

29 Example 2: Regime Change Game Action 1 if rationalizable for player 1 only if 1. Player 1's probability that 0 is at least c, i.e., Pr 1 ( 0) c 2. Player 1's probability that [the proportion of other players with probability #fj6=1jpr that 0 is at least c] is at least c, Pr j (0)cg 1 I 1 1 c 3. and so on... 28

30 Uniqueness PROPOSITION: Game has unique rationalizable outcome C (T ) \ C e (T ) =?. 29

31 Common Knowledge of Rank Beliefs f i : T i! R Let r i : T i! (f0; :::; I 1g) be an agent's belief about his rank. Thus r i (t i ) [k] = X t i ; i (t i ) [f(t i ; ) j# fj 6= ijf j (t j ) > f i (t i )gg = k] Type prole t has no ties if f i (t i ) 6= f j (t j ) for all i 6= j. There is common knowledge of rank belief if there exists br 2 (f0; :::; I 1g) such that r i (t i ) = br. There is common knowledge of no ties if i (t i ) [(t i ; )] > 0 implies t has no ties. 30

32 PROPOSITION: Assume symmetry, separability, monotonicity and limit dominance. Then common knowledge of rank beliefs and no ties implies unique rationalizable outcomes. 31

33 Common Knowledge of Beliefs about Dierences t i f i (t i ) ; e i (t i ) 2 R i let i be agent i's beliefs about dierences: h i (t i ) j ; j j6=i i hn = i (t i ) o f i (t i ) + j ; j j6=i i PROPOSITION: Assume uniform strict monotonicity and limit dominance. Then common knowledge of beliefs about dierences implies unique rationalizable outcomes. 32

34 Announcement The Stony Brook Game Theory Festival of the Game Theory Society July 9-20, Workshop on Global Games July 19-20, 2007 Organizers: Amil Dasgupta (London School of Economics) Stephen Morris (Princeton University) Alessandro Pavan (Northwestern University) 33