Collective Action Continued

Note that in a repeated game there is no link between

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Unformatted text preview: he players simultaneously choose actions knowing all the actions chosen by everybody at dates 1 through t ␣ 1 (history). In other words, a simple simultaneous ­move game is repeated T times. Note that in a repeated game there is no link between the periods, i.e. the game that is played in each period is not affected by actions taken in previous period (differently from dynamic games). Current strategies, however, can depend on the past. The prisoner's dilemma again: provision of a public good Let's consider the prisoner's dilemma and assume now that the players play this same game repeatedly (and learn past moves along the way).  ­ Each player's payoff is equal to the present discounted value of his per ­period payoff over the time horizon: preferences are separable.  ­ The discount factor is 2 (0; 1): Solution with T finite  ­ To solve, work backwards from the end.  ­ At date T, the strategies must specify a NE for any history. Remember: payoffs at T are not affected by history: strategies must specify a NE for the simple one ­period game ) both players free ride at T.  ­ At date T ␣ 1, strategies must form a two ­period NE for any history. How ­ ever, the last two periods' payoffs are independent of history and T 's outcome will not depend on what happens in period T ␣ 1 ) both players free ride: General Result: if the one period NE is unique, then the T ­periods game equilibrium is simply a repetition of this equilibrium T times. Solution with T infnite The strategic considerations in an infnitely ­repeated game are different from those in a one ­shot game because the introduction of time permits the players to reward and punish their opponents for their behaviour in the past. In the prisoner's dilemma there are now many possible equilibria. Both players free riding at each period is still an equilibrium, but there exist other equilibria....
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