Repeated games

Important concepts

A repeated game depends on a stage game that is repeated a finite, infinite, or potentially infinite number of times but with a positive probability of ending in finite time. The expected payoffs of the players are the sum of the payoffs from each stage game in the sequence, appropriately weighted. The weights are determined by the price of time to the individual (rate of time preference if no market exists or the rate of interest if a market allows the individual to readjust money values over time) and the probability of the game continuing for another period. The resulting generalized discount factor will be discussed in class.

Describing strategy profiles of the game

The number of strategy profiles to consider can be extremely large if there are many repetitions or if there is no prescribed limit to the number of stages. As a result there has been an emphasis on compact ways of describing a strategy for each player in the game. One set of strategies that may be described relatively briefly contains trigger strategies. Trigger strategies start by focusing on realizing an outcome in the sequence of stage games that result in payoffs higher than a conservative benchmark (the payoffs that the players could guarantee themselves by choosing an appropriate action in the stage game). If a player reneges from the strategy sequence that would achieve this gain a penalty period follows in which the conservative benchmark is realized. This penalty period may be followed after a specified lag with a return to cooperative behavior, which in turn would end if the trigger were again set off. Some examples are the grim trigger, tit for tat, two tits for a tat, and the like.

Nash equilibria of repeated games

The repeated play of profiles of actions that lead to Nash equilibria in the stage game are always a Nash equilibrium of the repeated game.

Nash equilibria of repeated games with a finite number of interactions

Watson concentrates on a two stage game. He shows that for some stage games that a punishment strategy can result in a better outcome that the repeated Nash equilibrium of the stage game. The actions in the second stage game interaction have to be a Nash equilibrium for the stage game but the actions in the first stage may not be. The actions in the first stage must give sufficiently high payoffs to increase the discounted sum of payoffs to each player. The penalty is to threaten to “play” an alternative Nash equilibrium in the stage game of the second stage if the player that would have an incentive to deviate in the first interaction does so. A necessary condition is that the stage game has more than one Nash equilibrium and that the stage game has an action profile outcome with higher joint payoffs than any of the Nash equilibria. Note that the necessary conditions will not frequently occur in two stage games. For details see Watson pp 211-215.

Nash equilibria of repeated games that have no prescribed limit to the number of interactions

Consider the case where the repeated Nash equilibria in the stage game are not Pareto superior (the prisoners’ dilemma is an example of a stage game with this property). If the generalized discount factor is sufficiently close to one, trigger strategies can sustain Nash equilibria for the repeated game that generate Pareto optimal payoffs for the players and an embarrassingly large number of equilibria that are better than the repetition of the actions leading to a Nash equilibrium in each stage game. This result is known as the Folk theorem (See Watson p. 222).

Examples

See problems and text of Chs. 22 and 23 and any introduced in class.