A note on reactive equilibria in the discounted prisoner's dilemma and associated games

Abstract

Necessary and sufficient conditions are derived under which the Tit-for-Tat strategy pair is a subgame perfect equilibrium in the standard discounted prisoner's dilemma. The sensi- tivity of the equilibrium to changes in the discount parameters is shown to be a common feature of all such nontrivial "reactive" equilibria. This appears to rule them out as reasonable solutions to the discounted prisoner's dilemma. In remarks, we indicate how these results can be extended to a fairly general class of games. The purpose of this note is to investigate the subgame perfection (see Selten (7)) properties of certain pure strategy equilibria in infinitely repeated versions of the classic prisoner's dilemma and related classes of games. Concentrating on the prisoner's dilemma for the moment, the equilibria we consider consist of strategy pairs in which at least one of the players bases current period action only on the prior actions of his opponent, ignoring the history of his own actions in the supergame. A strategy of this type is called reactive for obvious reasons. The corresponding equilibria will also be called reactive. In the example below, we demonstrate the existence of a perfect reactive equilibrium in which the collusive outcome obtains along the equilibrium path. This equilibrium is seen to be quite special, however; being extremely sensitive to variations in the discount rates of the players. We show that this is a feature com- mon to all such nontrivial reactive equilibria. In particular, the class of perfect reactive equilibria which are robust to changes in the discount rates consists of the single trivial equilibrium in which both players choose their noncooperative action in every