Coordination without Common Knowledge

Transcription

1 Stephen Morris and Hyun Song Shin Arizona State University March 006

2 Introduction Coordination games have multiple equilibria Relaxing common knowledge assumptions gives uniqueness { Carlsson and van Damme (993) { "global games"

3 Investment Example two players, invest or not invest Invest Not Invest Invest ; ; 0 Not Invest 0; 0; 0 N y; x i N ;

4 unique equilibrium if and only if each player invests if x i > bx, does not invest if x i < bx as!, bx!. 3

5 Global Game Approach Uniqueness if not too much "common knowledge" or "public information": nice parameterized normal examples Example generalizes to many players, general payos with strategic complementarities (Frankel, Morris and Pauzner (003)) Comparative statics / Policy Analysis possible 4

6 Applications. Currency Crises (Morris and Shin 998, Guimaeres and Morris 006). Bank Runs (Goldstein and Pauzner 00) 3. Financial Markets Information Structure relaxing common knowledge useful in other contexts (Morris and Shin 00) 5

7 Criticisms? Too easy to apply this methodology?. Endogenous Public Information (e.g., Angeletos and Werning 005). Relevance of Private Information (e.g., Sims 006) 3. What about other ways of relaxing common knowledge assumptions (Yildiz 006))? To address these criticisms and understand global games, must understand higher order belief foundations of results 6

8 Outline of Talk. Review Investment Example and logic of Global Games. Discuss Investment Example with arbitrary beliefs and higher order beliefs 3. Sucient Conditions for global game results: the importance of "approximate common knowledge" and common knowledge of "typicalness" 4. Review endogenous public signals in the light of this discussion 5. Heterogeneous Priors vs Asymmetric Information 7

9 6. Bounded Rationality 7. Learning 8. Summary 8

10 . Investment Example two players, invest or not invest N y; Invest Not Invest Invest ; ; 0 Not Invest 0; 0; 0 x i N ; 9

11 player thinks N yx ; player thinks x N yx ; () suppose player thinks that player invests only if x x { expected payo to investing is u (x ; x ) = E () Pr (x x ) 0s = y ( ) x y x A { unique value of x sets u (x ; x ) = 0 0

12 symmetric cuto equilibrium if u (x ; x ) = 0 0s u (x ; x y x ) ( ) = y x (x y) A ( ) ( ) (x y) A du (x ; x ) dx = s 0s ( ) ( ( ) ( ) (x y) A s p ( ) ( )

13 { thus there is only one symmetric cuto equilibrium if and only if s p 0 ( ) ( ) or game has strategic complementarities, so largest and smallest prole of rationalizable strategies constitute equilibria thus if only one symmetric cuto equilibrium, unique strategy prole surviving iterated deletion of strictly dominated strategies

14 Properties of Equilibrium Strategic Uncertainty Intuition for Uniqueness { Pr (x x jx = x ) if is small. Excess sensitivity to public signals in limit as!, { risk dominant action is played; determinate comparative statics { in many player binary action games, "Laplacian" action is played { uniform devaluation probability in many player many action currency crisis game of Guimaeres and Morris (006) 3

15 . Higher Order Beliefs Analysis Let T i be a nite set types of player i Type t i of player i has beliefs i (t i ) (T j R) over the types of the other player and the state of the world. { write e i (t i ) for type t i 's expectation of { write i (t j jt i ) for the probability that he assigns to player j 6= i being of type t j. 4

16 Fix an event E T T. Let i (Ejt i ) be the probability that type t i assigns to the event E: i (Ejt i ) = X ft j :(t i ;t j )Eg i (t j jt i ) : Let B p i (E) be the set of states where player i assigns probability at least p to the event E: B p i (E) = f(t i; t j ) : i (Ejt i ) pg. 5

17 We also write B p (E) = B p (E) \ Bp (E) C p (E) = \ k [B p ] k (E) Proposition. (Monderer and Samet 989)! C p (E) if and only if there exists F B p (F ) such that! F E For any function f i : T i! R, let B f i i (E) be the set of states where player i assigns probability at least f i to the event E, where f i is a random variable. Thus B f i i (E) = f(t i ; t j ) : i (Ejt i ) f i (t i )g. 6

18 We will also write B f ;f (E) = B f (E) \ Bf (E) C f ;f (E) = \ B f ;f k k (E) Corollary.! F E! C f (E) if and only if there exists F B f (F ) such that Proposition 3. Action I is rationalizable if and only if e i t i X I = B i C e ; e () ; Action N is rationalizable if and only if t i X N = B e i i Ce ; e (). 7

19 Thus the game is dominance solvable if and only if X I \ X N =?. Equivalently, there do not exist E I and E N such that E I \ E N 6=?, E I B e E I and E N B e E N. 8

20 3. Sucient Conditions for Uniqueness Let p i (t i ) = i n(t i ; t j ) : e j (t j ) > e o i (t i ) jt i q i (t i ) = i n(t i ; t j ) : e j (t j ) = e o i (t i ) jt i We will maintain the assumption that there is common knowledge that q i (t i ) = 0 and e i (t i ) 6=, for all i and t i. Now we have: Proposition 4. If there is common knowledge that p = p =, then the game is dominance solvable. 9

21 3. Sucient Conditions for Multiplicity Proposition 5. If C n(t p ; t ) : e o i (t i ) p, then investment is rationalizable; if C n(t p ; t ) : e o i (t i ) p, then not investing is rationalizable. 0

22 4. Endogenous Public Information Atkeson (999) and others: in market situations, prices act as public signals that restore multiplicity Explicit modelling: Tarashev (003), Hellwig, Mukherji and Tsyvinski (005), Angeletos and Werning (005)

23 One dimensional private signals let! 0 (no exogenous public information) let K individuals with private signal accuracy be polled thus endogenous public signal has precision K substituting into uniqueness condition, we get uniqueness if (K) K K

24 or if K K (K ) uniqueness only if private information is suciently inaccurate (Angeletos and Werning (005)) 3

25 two dimensional private signals N y; agent i observes x ik = k " ik, for k = ; each k N 0; k, " ik N 0; k each agent has one public signal (y) and two private signals (x i and x i ) about. 4

26 : higher order beliefs each x ik N thus ; k k k k E i () = y x i x i take large sample K!, we will observe average expectation E () = y 5

27 E () has no necessary relationship to and E () let ; ; ;! at same rate common knowledge of typicalness and unique equilibrium in the limit 6

28 5: dierence in priors Now suppose that player has a prior expectation x of and believes that x is normally distributed with mean x x and variance p p!. 7

29 Suppose that in addition each player observes a conditionally independent public signal y which is normally distributed with mean and precision. Observe that and E (jx ) = y x 0 s Pr (x x B jx ) x x p (y x )! C A Thus if player follows the trigger strategy with cuto x (i.e., investing only if x x ), then player 's expected payo to investing is 8

30 u (x ; x ) = E (j x ) Pr (x x jx ) 0 = y x s x x p (y x )! Now the payo to a player from investing at the other player's cuto is U (x) = u (x; x) = y x 0 s p C (y x) A 9

31 Now du dx = s p s p p p (y x)! Thus we get uniqueness condition 30

32 or 0 C A. Notice that this is independent of, but depends on. Two cases: Pure dierences in priors: = 0, uniqueness for every level of public information The asymmetric information model: = and = p. 3

33 6: bounded rationality Jehiel and Koessler (005) "Revisiting Games of Incomplete Information with Analogy-Based Reasoning" 3

34 7: learning Steiner and Stewart (006) "Learning by Similarity in Global Games" 33

35 8: conclusions 34