Repeated games are a simple category of dynamic games where a static game is played multiple times, either finitely or infinitely. In repeated games, actions and payoffs stay the same over time. This allows players' strategies to depend on the history of past actions and outcomes. A player's payoff in a repeated game is the discounted sum of their stream of payoffs from each round, where future payoffs are reduced by a discount factor reflecting time preference. For example, in a repeated Prisoner's Dilemma game, players may be able to sustain cooperative outcomes through conditional strategies.

1. Game Theory Repeated Games at HelpWithAssignment.com www.HelpWithAssignment.com

2. Repeated games are an important and simple category of dynamic games. 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. Game Theory Repeated Games www.HelpWithAssignment.com

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. At the same time, this feature makes repeated games interesting. Game Theory Repeated Games www.HelpWithAssignment.com

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. Game Theory Repeated Games www.HelpWithAssignment.com

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. In each round, all players simultaneously choose actions from their sets of possible actions inherited from game G. The General Setup of the Game www.HelpWithAssignment.com

6. A player observes the outcome of the stage game before the next round of play. Players can thus observe all of the past outcomes of the stage game when they choose their action in any particular round. All of the past outcomes are referred to as ‘history of the game’. The General Setup of the Game www.HelpWithAssignment.com

7. Players can choose different actions in the stage game depending on the history of play up to that point in the repeated game. A strategy for a player is therefore a plan which specifies a particular action of the stage game for each possible history of play. The General Setup of the Game www.HelpWithAssignment.com

8. Taking account of all players’ strategies determines a sequence of outcomes associated with a payoff for each player. Suppose it is period k ≤ T. The strategies chosen by all players lead to a sequence of payoffs for a given player. The General Setup of the Game www.HelpWithAssignment.com

9. The General Setup of the Game The player’s payoff in the repeated game in period k = Uk + EUk+1 + E2Uk+2 +…+ ET-k UT www.HelpWithAssignment.com

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. There are two equivalent ways to think of E, as a representation of impatience. The General Setup of the Game www.HelpWithAssignment.com

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. Conversely, a monetary payoff today can be invested at the risk free interest rate until next year. The General Setup of the Game www.HelpWithAssignment.com

12. The General Setup of the Game Therefore, the same amount of money received next year is worth less than today than the same amount of money received today. www.HelpWithAssignment.com

13. Consider the example of Prisoners’ Dilemma once more, this time with a repetition of the game. 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). Prisoners’ Dilemma Replayed www.HelpWithAssignment.com

14. Let’s consider a new game: the repetition of this game two times. 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. Prisoners’ Dilemma Replayed www.HelpWithAssignment.com

15. A sample strategy for player 1 is Period 1: Play S Period 2: Plays S if (S,S) otherwise C 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. Prisoners’ Dilemma Replayed www.HelpWithAssignment.com

16. Prisoners’ Dilemma Replayed In period 2, the strategy dictates that each player will play S again, leading to a payoff of (-1, -1). The period one payoff to each player is just the payoff in period one, plus N times the anticipated payoff in period 2. -1 + (-1)N. www.HelpWithAssignment.com

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