Lectures Road Map

Transcription

1 Lectures 0 - Repeated Games 4. Game Theory Muhamet Yildiz Road Map. Forward Induction Examples. Finitely Repeated Games with observable actions. Entry-Deterrence/Chain-store paradox. Repeated Prisoners Dilemma 3. A general result 4. When there are multiple equilibria 3. Infinitely repeated games with observable actions. Discounting / Present value. Examples 3. The Folk Theorem 4. Repeated Prisoners Dilemma, revisited tit for tat 5. Repeated Cournot oligopoly 4. Infinitely repeated games with unobservable actions

2 Forward Induction Strong belief in rationality: At any history of the game, each agent is assumed to be rational if possible. (That is, if there are two strategies s and s of a player i that are consistent with a history of play, and if s is strictly dominated but s is not, at this history no player j believes that i plays s.) urning Money 0 D S S SS 3, S.,., S -.9,. 0 0S 3,.,. 3,.,..,.,3.,.,3 S.,.,3 S -.9,. 0,3 D, -.9,., -.9,. DS -.9,. 0,3 -.9,. 0,3 O T H E R

3 Repeated Games Entry deterrence Enter Acc. (,) X Fight (0,) (-,-)

4 Entry deterrence, repeated twice, many times Enter Acc. Enter Acc. (,) (,3) Acc. Fight Enter X X Fight X Fight (,3) (0,0) Enter Acc. (0,0) (-,) (0,4) X Fight (-,) (-,-) What would happen if repeated n times? Prisoners Dilemma, repeated twice, many times Two dates T = {0,}; At each date the prisoners dilemma is played: C D C 5,5 6,0 D 0,6, At the beginning of players observe the strategies at 0. Payoffs= sum of stage payoffs.

5 C Twice-repeated PD C D C D C D D C D C D C D C D C D C D C D C D C D C D C D What would happen if T = {0,,,,n}? A general result G = stage game = a finite game T = {0,,,n} At each t in T, G is played, and players remember which actions taken before t; Payoffs = Sum of payoffs in the stage game. Call this game G(T). Theorem: If G has a unique subgame-perfect equilibrium s*, G(T) has a unique subgameperfect equilibrium, in which s* is played at each stage.

6 With multiple equilibria T = {0,} L T, M 5,0 R 0,0 s* = At t = 0, each i play Mi; At t =, play (,R) if (M,M) at t = 0, play (T,L) otherwise. M 0,5 4,4 0,0 L M R 0,0 0,0 3,3 T, 6,, M,6 7,7,,, 4,4 Infinitely repeated Games with observable actions T = {0,,,,t, } G = stage game = a finite game At each t in T, G is played, and players remember which actions taken before t; Payoffs = Discounted sum of payoffs in the stage game. Call this game G(T).

7 Definitions The Present Value of a given payoff stream π = (π 0,π,,π t, ) is PV(π;δ) = Σ t= δ t π t = π 0 + δπ + + δ t π t + The Average Value of a given payoff stream π is ( δ)pv(π;δ) = ( δ)σ t= δ t π t The Present Value of a given payoff stream π at t is PV t (π;δ) = Σ s=t δ s-t π s = π t + δπ t+ + + δ s π t+s + Infinite-period entry deterrence X Enter (0,) (-,-) Acc. Fight (,) Strategy of Entrant: Enter iff Accomodated before. Strategy of Incumbent: Accommodate iff accomodated before.

8 Incumbent: V(Acc.) = V A = /( δ); V(Fight) = V F = /( δ); Case : Accommodated before. Fight => - + δv A Acc. => + δv A. Case : Not Accommodated Entrant: Accommodated Enter => +V AE X => 0 +V AE Not Acc. Enter =>-+V FE X => 0 +V FE Fight => - + δv F Acc. => + δv A Fight - + δv F + δv A V F V A = /( δ) /δ δ /3. Infinitely-repeated PD C D C 5,5 6,0 D 0,6, A Grimm Strategy: Defect iff someone defected before. V D = /( δ); V C = 5/( δ) = 5V D ; Defected before (easy) Not defected D => C => C

9 Tit for Tat Start with C; thereafter, play what the other player played in the previous round. Is (Tit-for-tat,Tit-for-tat) a SPE? Modified: There are two modes:. Cooperation, when play C, and. Punishment, when play D. Start in Cooperation; if any player plays D in Cooperation mode, then switch to Punishment mode for one period and switch back to the Cooperation period next. Folk Theorem Definition: A payoff vector v = (v,v,,v n ) is feasible iff v is a convex combination of some pure-strategy payoff-vectors, i.e., v = p u(a ) + p u(a ) + + p k u(a k ), where p + p + + p k =, and u(a j ) is the payoff vector at strategy profile a j of the stage game. Theorem: Let x = (x,x,,x n ) be s feasible payoff vector, and e = (e,e,,e n ) be a payoff vector at some equilibrium of the stage game such that x i > e i for each i. Then, there exist δ < and a strategy profile s such that s yields x as the expected average-payoff vector and is a SPE whenever δ > δ.

10 Folk Theorem in PD C C 5,5 D 0,6 A SPE with PV (.,.)? D 6,0, With PV (.,5)? With PV (6,0)? With PV (5.9,0.)? Infinitely-repeated Cournot oligopoly N firms, MC = 0; P = max{-q,0}; Strategy: Each is to produce q = /(n); if any firm defects produce q = /(+n) forever. V C = V D = V(D C) = Equilibrium

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12 IRCD (n=) Strategy: Each firm is to produce q*; if any one deviates, each produce /(n+) thereafter. V C = q*(-q*)/(-δ); V D = /(9(-δ)); V D C = max q(-q*-q) +δv D Equilibrium iff q * = ( q *) / 4 δ + 9 ( q *) ( δ )( q *) / 4 + δ / 9 9 5δ q* 3(9 δ ) ( δ ) 0.4 x = δ, y = (3-5/3 δ)/(9-δ ) 0.3 y x

13 Carrot and Stick Produce ¼ at the beginning; at ant t > 0, produce ¼ if both produced ¼ or both produced x at t-; otherwise, produce x. Two Phase: Cartel & Punishment V C = /8(-δ). V x = x(-x) + δv C. V D C = max q(-/4-q) + δv X = (3/8) + δv X V D x = max q(-x-q) + δv X = (-x) /4 + δv X V C V D C V C (3/8) + δ V C + δ x(-x) (-δ ) V C - (3/8) δx(-x) (+δ)/8 - (3/8) δx(-x) V X V D C (-δ)v x (-x) /4 (-δ)(x(-x) + δ/8(-δ)) (-x) /4 (-δ)x(-x) + δ/8 (-x) /4 x x + /8 9/64δ 0 (9/4-δ)x (3-δ)x +δ/8(-δ) 0 Incomplete information

14 Incomplete information We have incomplete (or asymmetric) information if one player knows something (relevant) that some other player does not know. High p An Example Firm Hire W Work Shirk (, ) (0, ) Nature Low -p Do not hire Hire W Work (, ) Do not hire Shirk (-, )

15 The same example Hire Nature High p W Work Shirk (, ) (0, ) hire Firm Low -p W Work (, ) Do not Shirk (-, ) Another Example uy (p, -p) High 0.5 Seller p p Don t uy (p,-p ) Nature Low.5 What would you ask if you were to choose p from [0,4]? p p Don t uy Don t uy Don t (p, -p) (0,0) (p, -p )

16 Seller Same Another Example p p What would you ask if you were to choose p from [0,4]? Nature High 0.5 Low.5 Nature High 0.5 Low.5 uy Don t uy Don t uy uy Don t Don t (p, -p) (p,-p ) (p, -p) (0,0) (p, -p ) ayes Rule Prob(A and ) Prob(A ) = Prob() Prob(A and ) = Prob(A )Prob() = Prob( A)Prob(A) Prob(A ) = Prob( A)Prob(A) Prob()

17 Example Work µ -p -p p Success Prob(Work Success) = µp/[µp + ( µ)(-p)] Prob(Work Failure) = (-µ)p/[µ( p) + ( µ)p] Shirk µ p Failure 0.9 P(W S) P(w S),P(W F) P(W F) µ