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Unformatted text preview: David Scoones 2010 Page 1 Economics 112 Lecture Notes Lecture 7. Repeated Games Repeated games are a special class of sequential games in which the same stage game is played repeatedly. These are sometimes called supergames . Because the same basic game is played over and over, these repeated games maintain a simple structure even when we allow them to go on indefinitely. On the other hand, repetition leaves room for much more complex strategies than exist in a single stage. Strategies can be history dependent , and thus incorporate punishments and rewards for cooperation. These games can also be used to think about evolution and learning. Key words: stage game, Folk Theorem, tit-for-tat, grim trigger, time preference, discount rate 1. Repeated Prisoners Dilemma One of the obvious criticisms of our previous treatment of the prisoners dilemma (and social dilemmas more generally) is that they ignore the possibility for cooperation sustained over a long-term relationship. Recall the story: Two suspects, Albert and Brian, have been arrested They are put in separate holding rooms by the arresting detectives Each is offered a deal if he provides evidence against the other Without this testimony, conviction on the main charge is impossible, but some lesser offence can be proven The extensive form for this simultaneous move game is David Scoones 2010 Page 2 In the one shot game, both players have dominant strategies to confess. If we repeat the game twice, the game has four proper subgames, one starting at each of the possible histories of play from the first round. The extensive form is: With even one repetition, the number of strategies grows dramatically (each now has thirty two to choose from). In particular, this introduces the possibility that players use contingent strategies (of the ( ifthen ) variety) to punish non-cooperative behaviour. Some are very ignore the history of play and are very simple, for example always confess. Others use the history but remain fairly simple. For example, the diagram above illustrates the contingent strategy tit-for-tat Round 1: Dont confess Round 2: Confess if the other player confessed in round 1; Dont confess if the other player didnt confess in round 1. 2. Tit-for-Tat Tit-for-tat does condition on play, but only the choices of the other player. More generally, tit -for-tat starts by cooperating and then conditions only on the most recent move by the other player, and (in this context) cooperates following cooperation and defects following defection. (That is, plays Dont following Dont, and Confess following Confess.) So even as histories lengthen with further repetition, tit-for-tat remains simple. Tit-for-tat is quite forgiving, but does punish defection. David Scoones 2010...
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