Lecture 10

Lecture 10 - Repeated Games In these notes, well discuss...

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Repeated Games In these notes, we’ll discuss the repeated games, the games where a particular smaller game is repeated; the small game is called the stage game. The stage game is repeated regardless of what has been played in the previous games. For our analysis, it is important whether the game is repeated finitely or infinitely many times, and whether the players observe what each player has played in each previous game. 1.1 Finitely repeated games with observable actions We will first consider the games where a stage game is repeated finitely many times, and at the beginning of each repetition each player recalls what each player has played in each previous play. Consider the following entry deterrence game, where an entrant (1) decides whether to enter a market or not, and the incumbent (2) decides whether to fight or accommodate the entrant if he enters. 1 Enter 2 X Acc. Fight (0,2) (-1,-1) (1,1) Consider the game where this entry deterrence game is repeated twice, and all the previous actions are observed. Assume that a player simply cares about the sum of his payoffs at the stage games. This game is depicted in the following figure. 1 1 Enter 2 X Acc. Fight 1 Enter 2 X Acc. Fight (1,3) (0,0) (2,2) 1 Enter 2 X Acc. Fight (-1,1) (-2,-2) (0,0) 2 Enter 1 X Acc. Fight (-1,1) (1,3) (0,4) Note that after the each outcome of the first play, the entry deterrence game is played again —where the payoff from the first play is added to each outcome. Since a player’s preferences over the lotteries do not change when we add a number to his utility function, each of the three games played on the second “day” is the same as the stage game (namely, the entry deterrence game above). The stage game has a unique subgame perfect equilibrium, where the incumbent accommodates the entrant, and anticipating this, the entrant enters the market. 1 Enter 2
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X Acc. Fight (0,2) (-1,-1) (1,1) In that case, each of the three games played on the second day has only this equilibrium as its subgame perfect equilibrium. This is depicted in the following. 1 Enter 2 X Acc. Fight 1 Enter 2 X Acc. Fight (1,3) (0,0) (2,2) 1 Enter 2 X Acc. Fight (-1,1) (-2,-2) (0,0) 2 Enter 1 X Acc. Fight (-1,1) (1,3) (0,4) Using backward induction, we therefore reduce the game to the following. 2
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This note was uploaded on 04/05/2012 for the course ECON 406 taught by Professor Sjostrom during the Spring '12 term at Rutgers.

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Lecture 10 - Repeated Games In these notes, well discuss...

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