The Canonical Reputation Model and Reputation Effects

Transcription

1 The Canonical Reputation Model and Reputation Effects George J. Mailath University of Pennsylvania Lecture at the Summer School in Economic Theory Jerusalem, Israel June 25,

2 The Canonical Reputation Model A long-lived player 1 faces a sequence of short-lived players, in the role of player 2 of the stage game. A i, finite action set for each player. Y, finite set of public signals of player 1 s actions, a 1. ρ(y a 1 ), prob of signal y Y, given a 1 A 1. Player 2 s ex post stage game payoff is u 2 (a 1, a 2, y), and 2 s ex ante payoff is u 2 (a 1, a 2 ) := y Y u 2 (a 1, a 2, y)ρ(y a 1 ). Each player 2 maximizes her stage game payoff u 2. 2

3 The player 2 s are uncertain about the characteristics of player 1: Player 1 s characteristics are described by his type, ξ Ξ. All the player 2 s have a common prior μ on Ξ. Type space is partitioned into two sets, Ξ = Ξ 1 Ξ 2, where Ξ 1 is the set of payoff types and Ξ 2 is the set of behavioral (or commitment or action) types. For ξ Ξ 1, player 1 s ex post stage game payoff is u 1 (a 1, a 2, y, ξ), and 1 s ex ante payoff is u 1 (a 1, a 2, ξ) := y Y u 1 (a 1, a 2, y, ξ)ρ(y a 1 ). Each type ξ Ξ 1 of player 1 maximizes (1 δ) t 0 δt u 1 (a 1, a 2, ξ). 3

4 Player 1 knows his type and observes all past actions and signals, while each player 2 only the history of past signals. A strategy for player 1: σ 1 : t=0 (A 1 A 2 Y ) t Ξ Δ(A 1 ). If ˆξ Ξ 2 is a simple action type, then!ˆα 1 Δ(A 1 ) such that σ 1 (h t 1, ˆξ) = ˆα 1 for all h t 1. A strategy for player 2: σ 2 : t=0 Y t Δ(A 2 ). 4

5 Space of outcomes: Ω := Ξ (A 1 A 2 Y ). A profile (σ 1, σ 2 ) with prior μ induces the unconditional distribution P Δ(Ω). For a fixed simple type ˆξ = ξ(ˆα 1 ), the probability measure on Ω conditioning on ˆξ (and so induced by ˆα 1 in every period and σ 2 ), is denoted P Δ(Ω). Denoting by P the measure induced by (σ 1, σ 2 ) and conditioning on ξ ˆξ, we have P = μ(ˆξ) P + (1 μ(ˆξ)) P. (1) Given a strategy profile σ, U 1 (σ, ξ) denotes the type-ξ long-lived player s payoff in the repeated game, ] U 1 (σ, ξ) := E P [(1 δ) δ t u 1 (a t, y t, ξ) ξ. t=0 5

6 Denote by Γ(μ, δ) the game of incomplete information. Definition A strategy profile (σ 1, σ 2 ) is a Nash equilibrium of the game Γ(μ, δ) if, for all ξ Ξ 1, σ 1 maximizes U 1(σ 1, σ 2, ξ) over player 1 s repeated game strategies, and if for all t and all h2 t H 2 that have positive probability under (σ 1, σ 2 ) and μ (i.e., P(ht 2 ) > 0), [ E P u2 (σ 1 (ht 1, ξ), σ 2 (ht 2 )) [ 2] ht = max E P u2 (σ 1 a 2 A (ht 1, ξ), a 2) h t ]

7 Denote by Γ(μ, δ) the game of incomplete information. Definition A strategy profile (σ 1, σ 2 ) is a Nash equilibrium of the game Γ(μ, δ) if, for all ξ Ξ 1, σ 1 maximizes U 1(σ 1, σ 2, ξ) over player 1 s repeated game strategies, and if for all t and all h2 t H 2 that have positive probability under (σ 1, σ 2 ) and μ (i.e., P(ht 2 ) > 0), [ E P u2 (σ 1 (ht 1, ξ), σ 2 (ht 2 )) [ 2] ht = max E P u2 (σ 1 a 2 A (ht 1, ξ), a 2) h t ] 2. 2 Our goal: Reputation Bound (Fudenberg & Levine 89 92) Fix a payoff type, ξ Ξ 1. What is a good lower bound, uniform across Nash equilibria σ and Ξ, for U 1 (σ, ξ)? Our tool (Gossner 2011): relative entropy. 7

8 Relative Entropy X a finite set of outcomes. The relative entropy or Kullback-Leibler distance between probability distributions p and q over X is d(p q) := x X p(x) log p(x) q(x). By convention, 0 log 0 q = 0 for all q [0, 1] and p log p 0 = for all p (0, 1]. In our applications of relative entropy, the support of q will always contain the support of p. Since relative entropy is not symmetric, often say d(p q) is the relative entropy of q with respect to p. d(p q) 0, and d(p q) = 0 p = q. 8

9 Relative entropy is expected prediction error d(p q) measures observer s expected prediction error on x X using q when true dsn is p: n i.i.d. draws from X under p has probability x p(x)nx, where n x is the number of realization of x in sample. Observer assigns same sample probability x q(x)nx. Log likelihood ratio is L(x 1,..., x n ) = x n x log p(x) q(x), and so 1 n L(x 1,..., x n ) d(p q). 9

10 The chain rule Lemma Suppose P, Q Δ(X Y ), X and Y finite sets. Then d(p Q) = d(p X Q X ) + x P X (x)d(p Y ( x) Q Y ( x)) = d(p X Q X ) + E PX d(p Y ( x) Q Y ( x)). 10

11 Lemma The chain rule Suppose P, Q Δ(X Y ), X and Y finite sets. Then Proof. d(p Q) = d(p X Q X ) + x P X (x)d(p Y ( x) Q Y ( x)) = d(p X Q X ) + E PX d(p Y ( x) Q Y ( x)). d(p Q) = x,y P(x, y) log P X (x) Q X (x) P(x,y) P X (x) Q X (x) Q(x,y) 11

12 Lemma The chain rule Suppose P, Q Δ(X Y ), X and Y finite sets. Then Proof. d(p Q) = d(p X Q X ) + x P X (x)d(p Y ( x) Q Y ( x)) = d(p X Q X ) + E PX d(p Y ( x) Q Y ( x)). d(p Q) = x,y P(x, y) log P X (x) Q X (x) P(x,y) P X (x) Q X (x) Q(x,y) = d(p X Q X ) + x,y P(x, y) log P Y (y x) Q Y (y x) 12

13 Lemma The chain rule Suppose P, Q Δ(X Y ), X and Y finite sets. Then Proof. d(p Q) = d(p X Q X ) + x P X (x)d(p Y ( x) Q Y ( x)) = d(p X Q X ) + E PX d(p Y ( x) Q Y ( x)). d(p Q) = x,y P(x, y) log P X (x) Q X (x) P(x,y) P X (x) Q X (x) Q(x,y) = d(p X Q X ) + x,y P(x, y) log P Y (y x) Q Y (y x) 13

14 Lemma The chain rule Suppose P, Q Δ(X Y ), X and Y finite sets. Then Proof. d(p Q) = d(p X Q X ) + x P X (x)d(p Y ( x) Q Y ( x)) = d(p X Q X ) + E PX d(p Y ( x) Q Y ( x)). d(p Q) = x,y P(x, y) log P X (x) Q X (x) P(x,y) P X (x) Q X (x) Q(x,y) = d(p X Q X ) + x,y P(x, y) log P Y (y x) Q Y (y x) 14

15 Lemma The chain rule Suppose P, Q Δ(X Y ), X and Y finite sets. Then Proof. d(p Q) = d(p X Q X ) + x P X (x)d(p Y ( x) Q Y ( x)) = d(p X Q X ) + E PX d(p Y ( x) Q Y ( x)). d(p Q) = x,y P(x, y) log P X (x) Q X (x) P(x,y) P X (x) Q X (x) Q(x,y) = d(p X Q X ) + x,y P(x, y) log P Y (y x) Q Y (y x) = d(p X Q X ) + x P X (x) y P Y (y x) log P Y (y x) Q Y (y x). 15

16 A grain of truth Lemma Let X be a finite set of outcomes. Suppose p, p Δ(X) and q = εp + (1 ε)p for some ε > 0. Then, d(p q) log ε. 16

17 A grain of truth Lemma Let X be a finite set of outcomes. Suppose p, p Δ(X) and q = εp + (1 ε)p for some ε > 0. Then, d(p q) log ε. Proof. Since q(x)/p(x) ε, we have d(p q) = x q(x) p(x) log p(x) x p(x) log ε = log ε. 17

18 Back to reputations! Fix ˆα 1 Δ(A 1 ) and suppose μ(ξ(ˆα 1 )) > 0. In a Nash eq, at history h2 t, σ 2(h2 t ) is a best response to α 1 (h t 2 ) := E P[σ 1 (h t 1, ξ) ht 2 ] Δ(A 1), that is, σ 2 (h t 2 ) maximizes a 1 y u 2 (a 1, a 2, y)ρ(y a 1 )α 1 (a 1 h t 2 ). 18

19 Back to reputations! Fix ˆα 1 Δ(A 1 ) and suppose μ(ξ(ˆα 1 )) > 0. In a Nash eq, at history h2 t, σ 2(h2 t ) is a best response to α 1 (h t 2 ) := E P[σ 1 (h t 1, ξ) ht 2 ] Δ(A 1), that is, σ 2 (h t 2 ) maximizes a 1 y u 2 (a 1, a 2, y)ρ(y a 1 )α 1 (a 1 h t 2 ). At h t 2, 2 s predicted dsn on the signal y t is p(h t 2 ) := ρ( α 1(h t 2 )) = a 1 ρ( a 1 )α 1 (a 1 h t 2 ). 19

20 Back to reputations! Fix ˆα 1 Δ(A 1 ) and suppose μ(ξ(ˆα 1 )) > 0. In a Nash eq, at history h2 t, σ 2(h2 t ) is a best response to α 1 (h t 2 ) := E P[σ 1 (h t 1, ξ) ht 2 ] Δ(A 1), that is, σ 2 (h t 2 ) maximizes a 1 y u 2 (a 1, a 2, y)ρ(y a 1 )α 1 (a 1 h t 2 ). At h t 2, 2 s predicted dsn on the signal y t is p(h t 2 ) := ρ( α 1(h t 2 )) = a 1 ρ( a 1 )α 1 (a 1 h t 2 ). If player 1 plays ˆα 1, true dsn on y t is ˆp := ρ( ˆα 1 ) = a 1 ρ( a 1 )ˆα 1 (a 1 ). 20

21 Back to reputations! Fix ˆα 1 Δ(A 1 ) and suppose μ(ξ(ˆα 1 )) > 0. In a Nash eq, at history h2 t, σ 2(h2 t ) is a best response to α 1 (h t 2 ) := E P[σ 1 (h t 1, ξ) ht 2 ] Δ(A 1), that is, σ 2 (h t 2 ) maximizes a 1 y u 2 (a 1, a 2, y)ρ(y a 1 )α 1 (a 1 h t 2 ). At h t 2, 2 s predicted dsn on the signal y t is p(h t 2 ) := ρ( α 1(h t 2 )) = a 1 ρ( a 1 )α 1 (a 1 h t 2 ). If player 1 plays ˆα 1, true dsn on y t is ˆp := ρ( ˆα 1 ) = a 1 ρ( a 1 )ˆα 1 (a 1 ). Player 2 s one-step ahead prediction error is d (ˆp p(h t 2 )). 21

22 Bounding prediction errors Player 2 is best responding to an action profile α 1 (h t 2 ) that is d (ˆp p(h t 2 )) -close to ˆα 1 (as measured by the relative entropy of the induced signals). To bound player 1 s payoff, it suffices to bound the number of periods in which d (ˆp p(h t 2 )) is large. 22

23 Bounding prediction errors Player 2 is best responding to an action profile α 1 (h t 2 ) that is d (ˆp p(h t 2 )) -close to ˆα 1 (as measured by the relative entropy of the induced signals). To bound player 1 s payoff, it suffices to bound the number of periods in which d (ˆp p(h t 2 )) is large. For any t, P t 2 is the marginal of P on Y t. Then, P t 2 = μ(ˆξ) P t 2 + (1 μ(ˆξ)) P t 2, and so d( P t 2 Pt 2 ) log μ(ˆξ). 23

24 Applying the chain rule: log μ(ˆξ) d( P t 2 Pt 2 ) = d( P t 1 2 P t 1 2 ) + E Pd(ˆp p(h t 1 2 )) t 1 = E Pd(ˆp p(h2 τ )). τ=0 24

25 Applying the chain rule: log μ(ˆξ) d( P t 2 Pt 2 ) = d( P t 1 2 P t 1 2 ) + E Pd(ˆp p(h t 1 2 )) t 1 = E Pd(ˆp p(h2 τ )). τ=0 Since this holds for all t, τ=0 E Pd(ˆp p(h2 τ )) log μ(ˆξ). 25

26 From prediction bounds to payoff bounds Definition An action α 2 Δ(A 2 ) is an ε-entropy confirming best response to α 1 Δ(A 1 ) if there exists α 1 Δ(A 1) such that 1 α 2 is a best response to α 1 ; and 2 d(ρ( α 1 ) ρ( α 1 )) ε. The set of ε-entropy confirming BR s to α 1 is denoted B d ε (α 1 ). 26

27 From prediction bounds to payoff bounds Definition An action α 2 Δ(A 2 ) is an ε-entropy confirming best response to α 1 Δ(A 1 ) if there exists α 1 Δ(A 1) such that 1 α 2 is a best response to α 1 ; and 2 d(ρ( α 1 ) ρ( α 1 )) ε. The set of ε-entropy confirming BR s to α 1 is denoted Bε d (α 1 ). In a Nash eq, at any on-the-eq-path history h t 2, player 2 s action is a d(ˆp p(h t 2 ))-entropy confirming BR to ˆα 1. 27

28 From prediction bounds to payoff bounds Definition An action α 2 Δ(A 2 ) is an ε-entropy confirming best response to α 1 Δ(A 1 ) if there exists α 1 Δ(A 1) such that 1 α 2 is a best response to α 1 ; and 2 d(ρ( α 1 ) ρ( α 1 )) ε. The set of ε-entropy confirming BR s to α 1 is denoted Bε d (α 1 ). Define, for all payoff types ξ Ξ 1, v ξ α 1 (ε) := min α 2 B d ε (α 1) u 1 (α 1, α 2, ξ), and denote by w ξ α 1 the largest convex function below v ξ α 1. 28

29 The product-choice game I c s H 2, 3 0, 2 L 3, 0 1, 1 Suppose ˆα 1 = 1 H. c is unique BR to α 1 if α 1 (H) > 1 2. s is also a BR to α 1 if α 1 (H) = 1 2. d(1 H 1 2 H L) = log 1/2 = log { v ξ H (ε) = 2, if ε < log 2, 0, if ε log 2. 29

30 A picture is worth a thousand words player 1 payoffs v ξ H 2 w ξ H 0 log 2 ε 30

31 The product-choice game II c s H 2, 3 0, 2 L 3, 0 1, 1 Suppose ˆα 1 = 2 3 H L. c is unique BR to α 1 if α 1 (H) > 1 2. s is also a BR to α 1 if α 1 (H) = 1 2. d(ˆα H L) = 2 3 = 5 3 2/3 log 1/ /3 3 log 1/2 log 2 log 3 =: ˉε

32 Two thousand? player 1 payoffs v ξˆα 1 v ξ H w ξˆα 1 w ξ H ˉε log 2 ε 32

33 The reputation bound Proposition Suppose the action type ˆξ = ξ(ˆα 1 ) has positive prior probability, μ(ˆξ) > 0, for some potentially mixed action ˆα 1 Δ(A 1 ). Then, player 1 type ξ s payoff in any Nash equilibrium of the game Γ(μ, δ) is greater than or equal to w ξˆα 1 (ˆε), where ˆε := (1 δ) log μ(ˆξ). The only aspect of the set of types and the prior that plays a role in the proposition is the probability assigned to ˆξ. The set of types may be very large, and other quite crazy types may receive significant probability under the prior μ. 33

34 The proof Since in any Nash equilibrium (σ 1, σ 2 ), each payoff type ξ has the option of playing ˆα 1 in every period, we have U 1 (σ, ξ) = (1 δ) t=0 δt E P [u 1 (σ 1 (ht 1 ), σ 2 (ht 2 ), ξ) ξ] (1 δ) t=0 δt E Pu 1 (ˆα 1, σ 2 (ht 2 ), ξ) 34

35 The proof Since in any Nash equilibrium (σ 1, σ 2 ), each payoff type ξ has the option of playing ˆα 1 in every period, we have U 1 (σ, ξ) = (1 δ) t=0 δt E P [u 1 (σ 1 (ht 1 ), σ 2 (ht 2 ), ξ) ξ] (1 δ) t=0 δt E Pu 1 (ˆα 1, σ 2 (ht 2 ), ξ) (1 δ) t=0 δt E Pv ξˆα 1 (d(ˆp p(h2 t ))) 35

36 The proof Since in any Nash equilibrium (σ 1, σ 2 ), each payoff type ξ has the option of playing ˆα 1 in every period, we have U 1 (σ, ξ) = (1 δ) t=0 δt E P [u 1 (σ 1 (ht 1 ), σ 2 (ht 2 ), ξ) ξ] (1 δ) t=0 δt E Pu 1 (ˆα 1, σ 2 (ht 2 ), ξ) (1 δ) t=0 δt E Pv ξˆα 1 (d(ˆp p(h2 t ))) (1 δ) t=0 δt E Pw ξˆα 1 (d(ˆp p(h2 t ))) 36

37 The proof Since in any Nash equilibrium (σ 1, σ 2 ), each payoff type ξ has the option of playing ˆα 1 in every period, we have U 1 (σ, ξ) = (1 δ) t=0 δt E P [u 1 (σ 1 (ht 1 ), σ 2 (ht 2 ), ξ) ξ] (1 δ) t=0 δt E Pu 1 (ˆα 1, σ 2 (ht 2 ), ξ) (1 δ) t=0 δt E Pv ξˆα 1 (d(ˆp p(h2 t ))) (1 δ) t=0 δt E Pw ξˆα 1 (d(ˆp p(h2 t ))) ( w ξˆα 1 (1 δ) ) t=0 δt E Pd(ˆp p(h2 t )) 37

38 The proof Since in any Nash equilibrium (σ 1, σ 2 ), each payoff type ξ has the option of playing ˆα 1 in every period, we have U 1 (σ, ξ) = (1 δ) t=0 δt E P [u 1 (σ 1 (ht 1 ), σ 2 (ht 2 ), ξ) ξ] (1 δ) t=0 δt E Pu 1 (ˆα 1, σ 2 (ht 2 ), ξ) (1 δ) t=0 δt E Pv ξˆα 1 (d(ˆp p(h2 t ))) (1 δ) t=0 δt E Pw ξˆα 1 (d(ˆp p(h2 t ))) ( w ξˆα 1 (1 δ) ) t=0 δt E Pd(ˆp p(h2 t )) w ξˆα 1 ( (1 δ) log μ(ˆξ) ) = w ξˆα 1 (ˆε). 38

39 Patient player 1 Corollary Suppose the action type ˆξ = ξ(ˆα 1 ) has positive prior probability, μ(ˆξ) > 0, for some potentially mixed action ˆα 1 Δ(A 1 ). Then, for all ξ Ξ 1 and η > 0, there exists a ˉδ < 1 such that, for all δ (ˉδ, 1), player 1 type ξ s payoff in any Nash equilibrium of the game Γ(μ, δ) is greater than or equal to v ξˆα 1 (0) η. 39

40 When does B d 0 (α 1) = BR(α 1 )? Suppose ρ( a 1 ) ρ( a 1 ) for all a 1 a 1. Then pure action Stackelberg payoff is a reputation lower bound provided the simple Stackelberg type has positive prob. Suppose ρ( α 1 ) ρ( α 1 ) for all α 1 α 1. Then mixed action Stackelberg payoff is a reputation lower bound provided the prior includes in its support a dense subet of Δ(A 1 ). 40

41 How general is the result? The same argument (with slightly worse notation) works if the monitoring distribution depends on both players actions (though statistical identifiability is a more demanding requirement, particularly for extensive form stage games). The same argument (with slightly worse notation) also works if the game has private monitoring. Indeed, notice that player 1 observing the signal played no role in the proof. 41