Game Theory Repeated Games at


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Game Theory Repeated Games at

  1. 1. Game Theory Repeated Games at<br /><br />
  2. 2. Repeated games are an important and simple category of dynamic games. <br />As the name suggests, repeated games are dynamic games generated by the repetition of some static game a certain number of times, either finite or infinite.<br />Game Theory Repeated Games<br /><br />
  3. 3. On the other hand, repeated games are a simple class of dynamic games because the actions and payoffs to players stay the same over time.<br />At the same time, this feature makes repeated games interesting.<br />Game Theory Repeated Games<br /><br />
  4. 4. Any differences between the equilibrium outcomes of the static game must be coming from the fact that there are multiple periods in the interaction.<br />Game Theory Repeated Games<br /><br />
  5. 5. Starting with static game G, called the “stage game” one can construct a new game which is the repetition of G game with T rounds, where T can be either finite or infinite. <br />In each round, all players simultaneously choose actions from their sets of possible actions inherited from game G.<br />The General Setup of the Game<br /><br />
  6. 6. A player observes the outcome of the stage game before the next round of play. <br />Players can thus observe all of the past outcomes of the stage game when they choose their action in any particular round. <br />All of the past outcomes are referred to as ‘history of the game’.<br />The General Setup of the Game<br /><br />
  7. 7. Players can choose different actions in the stage game depending on the history of play up to that point in the repeated game. <br />A strategy for a player is therefore a plan which specifies a particular action of the stage game for each possible history of play. <br />The General Setup of the Game<br /><br />
  8. 8. Taking account of all players’ strategies determines a sequence of outcomes associated with a payoff for each player.<br />Suppose it is period k ≤ T. The strategies chosen by all players lead to a sequence of payoffs for a given player. <br />The General Setup of the Game<br /><br />
  9. 9. The General Setup of the Game<br />The player’s payoff in the repeated game in period<br /> k = Uk + EUk+1 + E2Uk+2 +…+ ET-k UT<br /><br />
  10. 10. In other words, a player’s payoff is discounted sum of the stream of payoffs from the stage game, with payoffs that are received in the future reduced by a factor of E<1. <br />There are two equivalent ways to think of E, as a representation of impatience.<br />The General Setup of the Game<br /><br />
  11. 11. Typically, if one is willing to pay some money to get some benefit in the future, one would be willing to pay more for that benefit today.<br />Conversely, a monetary payoff today can be invested at the risk free interest rate until next year.<br />The General Setup of the Game<br /><br />
  12. 12. The General Setup of the Game<br />Therefore, the same amount of money received next year is worth less than today than the same amount of money received today.<br /><br />
  13. 13. Consider the example of Prisoners’ Dilemma once more, this time with a repetition of the game. <br />In this game C is a dominant strategy for both players and this game has a unique Nash Equilibrium (C,C) worse for either player than (S,S).<br />Prisoners’ Dilemma Replayed<br /><br />
  14. 14. Let’s consider a new game: the repetition of this game two times. <br />A strategy for each player specifies and action in the first period and an action in the second period for each of the four possible outcomes in the first period.<br />Prisoners’ Dilemma Replayed<br /><br />
  15. 15. A sample strategy for player 1 is <br />Period 1: Play S<br />Period 2: Plays S if (S,S) otherwise C<br />If both players adopt this strategy, in period one the outcome will be (S, S) leading to a payoff of (-1, -1) for period one.<br />Prisoners’ Dilemma Replayed<br /><br />
  16. 16. Prisoners’ Dilemma Replayed<br />In period 2, the strategy dictates that each player will play S again, leading to a payoff of (-1, -1). <br />The period one payoff to each player is just the payoff in period one, plus N times the anticipated payoff in period 2. <br /> -1 + (-1)N.<br /><br />
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