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Game Theory – Repeated Games

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. 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. Any differences between the equilibrium outcomes of the static game must be coming from the fact that there are multiple periods in the interaction.

The General Setup of the Game:

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. 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’. 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. 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 player’s payoff in the repeated game in period

k = Uk + EUk+1 + E2Uk+2 +…+ ET-k UT

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. 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. Therefore, the same amount of money received next year is worth less than today than the same amount of money received today.

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).

Player 2
C S
Player 1 C -5, -5 0, -12
S -12, 0 -1, -1

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.

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. 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.

This article is in continuation with our previous articles on Economics and Game Theory which include Prisoners’ Dilemma, Battle of the sexes, Cutting a Cake, Solow’s Growth Model

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