Far-sighted equilibria in 2 × 2, non-cooperative, repeated games

Abstract

Consider a two-person simultaneous-move game in strategic form. Suppose this game is played over and over at discrete points in time. Suppose, furthermore, that communication is not possible, but nevertheless we observe some regularity in the sequence of outcomes. The aim of this paper is to provide an explanation for the question why such regularity might persist for many (i.e., infinite) periods.

Each player, when contemplating a deviation, considers a sequential-move game, roughly speaking of the following form: “if I change my strategy this period, then in the next my opponent will take his strategy ‘b’ and afterwards I can switch to my strategy ‘a’, but then I am worse off since at that outcome my opponent has no incentive to change anymore, whatever I do”. Theoretically, however, there is no end to such reaction chains. In case that deviating by some player gives him less utility in the long run than before deviation, we say that the original regular sequence of outcomes is far-sighted stable for that player. It is a far-sighted equilibrium if it is far-sighted stable for both players.