Evolutionary Stability in the Infinitely Repeated Prisoners' Dilemma Played by Two-State Moore Machines

Abstract

A large literature which attempts to explain how cooperative outcomes can be supported in the repeated Prisoners' Dilemma (RPD) has emerged in recent years. An interesting branch of this literature has analyzed the infinite RPD as a game in which two metaplayers choose a strategy which is implemented by a finite automaton, or Moore machine. This line of research has been especially interesting because it allows us to capture the notion of bounded rationality in the players. Also, the use of these machines as devices to implement strategies permits us to quantify the notion of strategic complexity. We can then apply the idea that complexity is costly to the analysis, and the results have proven very interesting. So far, the work in this area has applied the concepts of Nash equilibrium and evolutionary stability in determining what reasonable outcomes are in these games. However, it has frequently been that pointed out these equilibrium concepts may be either too restrictive or not restrictive enough to be useful in analyzing the infinitely repeated Prisoners' Dilemma. That is, if we use the Nash equilibrium as the appropriate equilibrium concept, the set of equilibrium outcomes is infinitely large. On the other hand, if we require the equilibrium to be evolutionarily stable, the set of such equilibrium outcomes is frequently empty. It has been suggested that the requirement for evolutionary stability may be too stringent a criterion. The purpose of this essay is to examine what happens in these games in an evolutionary framework under various conditions. The results of the simulations I performed and report here indicate certain combinations of strategies may exist which, although not evolutionarily stable in the sense of Maynard Smith [14], prove to be invasion-proof against permissible mutant strategies. This is akin to saying a set of possible strategies may exist which cannot be invaded, but neither the individual strategies nor the mixed strategy represented by the population is evolutionarily stable. We can imagine the mix of strategies changing as various mutants attempt to invade the population but the set of strategies remaining the same as the one with which we started. In fact, it may well be the case that this set of strategies will not last forever, but it may last for a very long time. I formalize this idea later in the essay. In order to test the robustness of