Higher Order Beliefs in Dynamic Environments

Transcription

1 University of Pennsylvania Department of Economics June 22, 2008

2 Introduction: Higher Order Beliefs Global Games (Carlsson and Van Damme, 1993): A B A 0, 0 0, θ 2 B θ 2, 0 θ, θ Dominance Regions: A if θ < 0; B if θ > 2 Multiplicity: θ (0, 2)

3 Introduction: Higher Order Beliefs Global Games (Carlsson and Van Damme, 1993): A B A 0, 0 0, θ 2 B θ 2, 0 θ, θ Dominance Regions: A if θ < 0; B if θ > 2 Multiplicity: θ (0, 2) θ 1 = 3 θ 2 = 1.5 θ 3 = 1.5 Player 1: Player 2:

4 Introduction: Higher Order Beliefs Global Games (Carlsson and Van Damme, 1993): A B A 0, 0 0, θ 2 B θ 2, 0 θ, θ Dominance Regions: A if θ < 0; B if θ > 2 Multiplicity: θ (0, 2) θ 1 = 3 θ 2 = 1.5 θ 3 = 1.5 Player 1: Player 2: Discontinuity between Common Knowledge and k-mutual Belief (k finite) (cf. Rubinstein, 1989): the fact that Dominance Regions are NOT ruled out by Higher Order Beliefs determines the result.

5 Introduction: Higher Order Beliefs Static Games: multiplicity of rationalizable outcomes is not robust to perturbation of higher order beliefs

6 Introduction: Higher Order Beliefs Static Games: multiplicity of rationalizable outcomes is not robust to perturbation of higher order beliefs Weinstein and Yildiz (2007, WY): If the underlying space of uncertainty is rich enough, whenever a game has multiple Rationalizable outcomes, any of these is uniquely rationalizable in an arbitrarily close game (perturbing higher order beliefs only).

7 Introduction: Higher Order Beliefs Static Games: multiplicity of rationalizable outcomes is not robust to perturbation of higher order beliefs Weinstein and Yildiz (2007, WY): If the underlying space of uncertainty is rich enough, whenever a game has multiple Rationalizable outcomes, any of these is uniquely rationalizable in an arbitrarily close game (perturbing higher order beliefs only). (Non-)Robustness of Refinements: refinements of rationalizability (e.g. Nash Eq.) are not robust to perturbations of higher order beliefs if and only if they agree with those of rationalizability. (Robustness = upper hemicontinuity)

8 Introduction: What is the Role of Higher Order Beliefs in Dynamic Games? Dynamic Environments: little is known

9 Introduction: What is the Role of Higher Order Beliefs in Dynamic Games? Dynamic Environments: little is known Dynamic Global Games: no systematic analysis contrasting results (in terms of impact of perturbations of higher order beliefs on multiplicity)

10 Introduction: What is the Role of Higher Order Beliefs in Dynamic Games? Dynamic Environments: little is known WY s analysis cannot be applied to normal forms of dynamic games: WY s Richness Condition: the underlying space of uncertainty Θ must be rich enough that for each player strategy s i there is a payoff state θ s i that makes s i strictly dominant.

11 Introduction: What is the Role of Higher Order Beliefs in Dynamic Games? Dynamic Environments: little is known WY s analysis cannot be applied to normal forms of dynamic games: WY s Richness Condition: the underlying space of uncertainty Θ must be rich enough that for each player strategy s i there is a payoff state θ s i that makes s i strictly dominant. 1 a 1 ( 1 1 ) a 2 a 3 2 b 1 b 2 ( 2 )( b 1 b 2 )( 0 )( )

12 Dynamic Environments Do WY s results (uniqueness /(non-)robustness) extend to dynamic environments? How?

13 Dynamic Environments Do WY s results (uniqueness /(non-)robustness) extend to dynamic environments? How? one possibility is to maintain the same richness condition for the (reduced) normal form of the game and modify (normal fom)-rationalizability introducing trembles (see WY)

14 Dynamic Environments Do WY s results (uniqueness /(non-)robustness) extend to dynamic environments? How? one possibility is to maintain the same richness condition for the (reduced) normal form of the game and modify (normal fom)-rationalizability introducing trembles (see WY) PROBLEM: are tremble-based solution concepts still refinements of such modified rationalizability?

15 Dynamic Environments Do WY s results (uniqueness /(non-)robustness) extend to dynamic environments? How? one possibility is to maintain the same richness condition for the (reduced) normal form of the game and modify (normal fom)-rationalizability introducing trembles (see WY) PROBLEM: are tremble-based solution concepts still refinements of such modified rationalizability? alternatively: weaken the richness condition, and exploit the extensive form:

16 Dynamic Environments Do WY s results (uniqueness /(non-)robustness) extend to dynamic environments? How? one possibility is to maintain the same richness condition for the (reduced) normal form of the game and modify (normal fom)-rationalizability introducing trembles (see WY) PROBLEM: are tremble-based solution concepts still refinements of such modified rationalizability? alternatively: weaken the richness condition, and exploit the extensive form: Sequential Rationalizability (SR) (similar to (but weaker than) Pearce s (1984) Extensive Form Rationalizability)

17 Dynamic Environments Do WY s results (uniqueness /(non-)robustness) extend to dynamic environments? How? one possibility is to maintain the same richness condition for the (reduced) normal form of the game and modify (normal fom)-rationalizability introducing trembles (see WY) PROBLEM: are tremble-based solution concepts still refinements of such modified rationalizability? alternatively: weaken the richness condition, and exploit the extensive form: Sequential Rationalizability (SR) (similar to (but weaker than) Pearce s (1984) Extensive Form Rationalizability) MAIN RESULT: SR is the strongest solution concept that is robust to perturbations of higher order beliefs.

18 The Model: Finite Multistage Games with Observable Actions Dynamic Bayesian Games Type Space: Γ T = N, H, Z, Θ, (A i, T i, u i ) i N T = (T i, τ i ) i N s.t.τ i : T i (Θ T i )(each type in a type space represents an infinite hierarchy of beliefs) Universal Type Space (Mertens and Zamir, 1985): T = (Ti, τi ) i N where T is the set of all (collectively coherent) profiles of hierarchies generated by Θ payoffs: u i : Z Θ T R interim (reduced form) strategies: s i S i s.t. s i : H A i and s i (h) A i (h) outcome function: O : S Z

19 Sequential Rationality Beliefs: Conditional Probability Systems [CPS] (Myerson, 1986) µ i H (Θ T i S i ) such that: 1)for each h H, µ i ( h) (Θ T i S i ) 2) Θ T i S i (h) dµi ( h) = 1 3) Consistent with Bayesian Updating whenever possible

20 Sequential Rationality Beliefs: Conditional Probability Systems [CPS] (Myerson, 1986) µ i H (Θ T i S i ) such that: 1)for each h H, µ i ( h) (Θ T i S i ) 2) Θ T i S i (h) dµi ( h) = 1 3) Consistent with Bayesian Updating whenever possible Sequential Best Responses s i r i ( µ i t i ) if and only if h H (si ) s i maximizes over s i S i (h) Θ 0 T i S i u i (O (s i, s i ), θ, t i, t i ) dµ i ( h)

21 Sequential Rationality Beliefs: Conditional Probability Systems [CPS] (Myerson, 1986) µ i H (Θ T i S i ) such that: 1)for each h H, µ i ( h) (Θ T i S i ) 2) Θ T i S i (h) dµi ( h) = 1 3) Consistent with Bayesian Updating whenever possible Sequential Best Responses s i r i ( µ i t i ) if and only if h H (si ) s i is a best response for t i to µ i ( h)

22 Sequential Rationality Beliefs: Conditional Probability Systems [CPS] (Myerson, 1986) µ i H (Θ T i S i ) such that: 1)for each h H, µ i ( h) (Θ T i S i ) 2) Θ T i S i (h) dµi ( h) = 1 3) Consistent with Bayesian Updating whenever possible Sequential Best Responses s i r i ( µ i t i ) if and only if h H (si ) s i is a best response for t i to µ i ( h) s i S i is sequentially rational for t i, if there exist a conjecture µ i H (Θ T i S i ) s.t. s i r i ( µ i t i ) and mrg Θ T i µ i ( φ) = τ ti.

23 Sequential Rationalizability Definition (SR) For each i and t i T i, let SR T,0 i (t i ) = S i. Define recursively, for k = 1, 2,..., and t i T i SR T,k i (t i ) = ŝ i SR T,k 1 i (t i ) : (1) ŝ i is sequentially rational for t i w.r.t. some CPS µ i (2) µ i (t i, s i φ) = 0 if s i / SR T,k 1 i (t i ) Sequential Rationalizability is defined as the set of pairs (t, s) SR T = SR T,k k 0

24 Sequential Rationalizability ( 1 1 a 1 ) 1 a 2 a 3 2 SR = {a 1, a 2 } {b 1, b 2 } EFR = {a 2, b 1 } b 1 b 2 ( 2 )( b 1 b 2 )( 0-1 )( 0-2 )

25 Sensitivity to Higher Order Beliefs: The Richness Condition a 1 1 ( 1 1 ) a 2 a 3 2 b 1 b 2 ( 2 )( b 1 b 2 )( 0 )( )

26 Sensitivity to Higher Order Beliefs: The Richness Condition a 1 1 ( 1 1 ) a 2 a 3 2 ( b 1 b 2 b 1 b 2 )( )( )( 0-1 )

27 Sensitivity to Higher Order Beliefs: The Richness Condition a 1 1 ( 1 1 ) a 2 a 3 2 ( b 1 b 2 b 1 b 2 )( )( )( 0-1 ) Richness Condition: for each player strategy s i there exists a payoff state θ s i s.t. for all s i S i, s i is the unique strategy s.t.: for all histories that can be reached by s i, s i h is best response to s i h. (s i is sequentially dominant)

28 Sensitivity to Higher Order Beliefs Richness Condition: for each player strategy s i there exists a payoff state θ s i s.t. s i is sequentially dominant. Theorem Under the richness condition, for any finite types profile ˆt T and any s SR (ˆt ), there exists a sequence of finite types profiles ˆt m ˆt and s.t. SR (ˆt m) = {s} for each m (the convergence ˆt m ˆt is in the product topology)

29 Example θ = (θ 1, θ 1, θ 2 ) Θ = {θ, θ a3, θ a 3, θ a2b1 } θ = (0, 0, 0) θ a3 = (0, 3, 0) θ a 3 = (0, 4, 0) θ a2b1 = (2, 0, 2) θ = (0, 0, 0) let t denote CK of θ SR(t ) = {a 1, a 2 } {b 1, b 2 } ( 1 1 a 1 ) 1 a 2 a 3 b 1 b 2 ( 2 + θ1 )( θ1 1 + θ b 1 b 2 )( θ 1 θ 2 2 )( θ 1 1 )

30 Example t m t s.t. SR(t m ) = {a 1 b 2 } for each m θ a3 d suppose that 1 observes the true state θ Θ = {θ, θ a3, θ a3 } with prior ( 1 2ɛ 2, ɛ, 1 2 ). Fix ɛ (0, 1 6 ). If 1 observes θ or θ a3, the game starts, with messages delivered with ɛ prob p (0, 1 2ɛ ) at each round. 1 1+p θ θ a3 θ 1 1 ɛ ɛ p 1+p 1 1+p 2 θ a3 p 1+p θ θ a ( 1 1 ) a 1 1 a 2 a 3 b 1 b 2 2 b 1 b 2 ( ) ( ) ( ) ( ) 2 + θ1 θ1 θ 1 θ θ 2 2 θ 2 2 1

31 Example t m t s.t. SR(t m ) = {a 2 b 1 } for each m game: suppose that 1 observes the true state θ Θ = {θ, θ a2b1 } s.t. θ = (0, 0, 0) and θ a2b1 = (2, 0, 2) If he observes θ, the game starts, with messages delivered with prob 1 2 at each round. d θ... θ a2b1 θ θ ( 1 1 a 1 1 ) a 2 a 3 2 b 1 b 2 b 1 b ( ) ( ) ( ) ( ) 2 + θ1 θ1 θ 1 θ θ 2 2 θ 2 2 1

32 Structure of SR Theorem (Structure of SR) Under the richness assumption, the set U = {t T : # (SR (t)) = 1} is open and dense in T. Moreover, the unique outcome is locally constant. If # (SR i (t i )) = n, then t i is on the boundary of n open sets in which each of the n strategies is uniquely SR. Corollary Generic Uniqueness of any Perfect-Bayesian Equilibrium outcome. Corollary Non-Robustness of PBE-predictions to perturbations of higher order beliefs.

33 Other Results Upper Hemicontinuity: the SR-correspondance is u.h.c in t.

34 Other Results Upper Hemicontinuity: the SR-correspondance is u.h.c in t. Type Space-Invariance: if two types t i and t i in two (possibly different) type-spaces induce the same hierarchy of beliefs over Θ, SR i (t i ) = SR i (t i ) (only hierarchies matter)

35 Other Results Upper Hemicontinuity: the SR-correspondance is u.h.c in t. Type Space-Invariance: if two types t i and t i in two (possibly different) type-spaces induce the same hierarchy of beliefs over Θ, SR i (t i ) = SR i (t i ) (only hierarchies matter) Alternative Characterization: (for generic games) SR is equivalent to the Dekel-Fudenberg (1991) procedure(*) applied to the interim normal form of the game. ((*): one round of deletion of weakly dominated strategies followed by iterated deletion of (strictly) dominated strategies)

36 Conclusion In writing down a game (a model), we generally make simplifying assumptions about higher order beliefs. It is important then that the implications of the analysis are robust to possible misspecifications of these. (Robustness = u.h.c.)

37 Conclusion In writing down a game (a model), we generally make simplifying assumptions about higher order beliefs. It is important then that the implications of the analysis are robust to possible misspecifications of these. (Robustness = u.h.c.) Robustness: suppose that uncertainty is rich (i.e.: richness condition). What solution concept to use, if you re concerned with this problem of robustness? (ANSWER: Sequential Rationalizability: it is the strongest u.h.c. solution concept)

38 Conclusion Perturbations of beliefs: Do higher order beliefs play the same role in dynamic environments that they play in static ones? what kind of perturbations to beliefs deliver the familiar results in dynamic environments?

39 Conclusion Perturbations of beliefs: Do higher order beliefs play the same role in dynamic environments that they play in static ones? what kind of perturbations to beliefs deliver the familiar results in dynamic environments? in the literature on Dynamic Global Games (DGG), beliefs are perturbed at each stage (more like a repeated global game). Here, perturbations are once and for all (more similar to reputation models).

40 Conclusion Perturbations of beliefs: Do higher order beliefs play the same role in dynamic environments that they play in static ones? what kind of perturbations to beliefs deliver the familiar results in dynamic environments? The results of this paper suggest that nothing inherent to dynamic environments prevents higher order beliefs from having the familiar impact on multiplicity. The unusual results from DGG are rather due to the nature of the perturbations that have been used.