第八讲 重复博弈

Suppose that the players anticipate that (R1,R2) will be the second-stage outcome if the first-stage outcome is (M1,M2), but that (L1,L2) will be the second-stage outcome if any of the eight other first-stage outcomes occurs. There are three subgame-perfect equilibria: ((L1,L2),(L1,L2)), ((R1,R2),(L1,L2)), ((M1,M2),(R1,R2)).
∞ ∞
三(1)、Definition: T-period repeated games 、
Let G be a strategic game. Denote the set of players by N and the set of actions and payoff function of each player i by Ai and ui respectively. The T -period repeated game of G for the discount factor δ is the extensive game with perfect information and simultaneous moves in which: the set of players is N ; the set of terminal histories is the set of sequences (a1 , a 2 ,⋯ , aT ) of action profiles in G; the player function assigns the set of all players to every history (a1 ,⋯ , a t ) (for every value of t ); the set of actions available to any player i after any history is Ai ; each player i evaluates each terminal history (a1 , a 2 ,⋯ , aT ) according to its discounted average (1 − δ )∑ t =1 δ t −1ui (a t ).
第八讲 Repeated Game
Some examples
重复囚徒困境：纳什均衡与合作解； 市场买卖：一次性买卖和重复买卖； 奥尔森的独裁、民主和发展观：“流窜的匪徒”和 “常驻的匪徒”（Olson：“Dictatorship, Democracy, and
Development”, American Political Science Review，Vol.87， No.3 September 1993.
T
Discounted average: supppose that a player's preferences over streams (w1 , w2 ,
⋯ ) of payoffs are represented by the discounted sum ∑ t =1 δ t −1wt of these payoffs, where 0 < δ < 1. There is a value of c such that the player is indifferent between the stream (w1 , w2 ,⋯ ) and the constant stream (c, c,⋯ ). c /(1 − δ ) = ∑ t =1 δ t −1wt = V ⇒ c = (1 − δ )V .
二、Preferences
Discounting: each player i has a payoff function ui for the strategic game and a discount factor δ i between 0 and 1 such that she evaluates the sequence
(a1 , a 2 ,⋯ , aT ) of outcomes of the strategic game by the sum: ui (a1 ) + δ i ui (a 2 ) + δ i2ui (a 3 ) + ⋯ + δ iT −1ui (aT ) = ∑ t =1 δ it −1u i (a t ).
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一、The main idea
The main idea of the theory is that a player may be deterred from exploiting her short-term advantage by the “threat” of “punishment” that reduces her long-term payoff. So we analyze whether threats and promises about future behavior can influence current behavior in repeated relationships.
Grim trigger strategy
Choose C as long as the other player chooses C; If in any period the other player chooses D, then choose D in every subsequent period.
Conclusion
If a player is sufficiently patient, the strategy that chooses C after every history is a best response to the grim trigger strategy. Thus when the players are sufficiently patient, the strategy pair in which both players use the grim trigger strategy is a Nash equilibrium of the repeated prisoner’s dilemma: neither player can do better by adopting another strategy in the repeated game. The outcome of this equilibrium is (C,C) in every period.
Questions
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How patient do the players have to be? What other outcomes are generated by Nash equilibrium? Is the strategy pair in which each player uses the grim trigger strategy a subgame perfect euqilibrium? Are there Nash equilibria or subgame perfect equilibria in which the players’ strategies punish deviations less severely? How do the arguments apply to games other than the prisoner’s dilemma?
If a player chooses C in every period, then the outcome is (C,C) and her payoff is 2 in every period. If she switches to D in some period, then she obtains a payoff of 3 in that period (a short-term gain) and a payoff of 1 in every subsequent period (a long-term loss).
T
三(2)、Definition: 、
Infinitely repeated game
The infinitely repeated game of G for the discount factor δ differs only in that the set of terminal histories is the set of infinite sequences (a1 , a 2 ,⋯ ) is the payoff of each player i to the terminal history (a1 , a 2 ,⋯ ) is the discounted average (1 − δ )∑ t =1 δ t −1ui (a t ).
Discussion
L2 M2 R2 L2 M2 R2 3，3 3，3 6*，6*
L1 1*，1* 5，0 0，0 M1 0，5 4，4 0，0 R1 0，0 0，0 3*，3*
L1 4*，4* 8，3 M1 3，8 7，7 R1 3，3 3，3
Because the payoff of (R1,R2) is higher than (L1,L2), there may be a renegotiation problem. If (R1,R2) is to be the second-stage outcome after every first-stage outcome, then the incentive to play (M1,M2) in the first-stage is destroy. There are two subgame-perfect equilibria: ((L1,L2),(R1,R2)), ((R1,R2),(R1,R2)).