The Prisoner's Dilemma and Dynamical Systems Associated to Non-Cooperative Games

A new way of looking at repeated games is introduced which incorporates a bounded memory and rationality. In these terms, a resolution of the prisoner's dilemma is given. THE GOAL HERE is to give a natural way of introducing dynamics into game theory, or at least for non-cooperative games. Perhaps the main idea in this treatment of dynamics is the way the past is taken into account. We suppose for both mathematical and model theoretic considerations that the agents only keep some kind of summary or average of the past outcomes (or payoffs) in their memory. Decisions are based on this summary. This kind of modeling reflects the fact that there exist substantive bounds to the storing and organizing of information. We give an axiomatization of bounded memory and rationality, with both institutions and people in mind. On the other hand, the hypothesis used in this treatment leads to a tractable mathematics. Differential equations on function spaces which contain little geometry are replaced by a dynamics on a finite dimensional space. And yet dynamics takes the past into account as a kind of substitute for the theory of delay equations. The perspective in this paper is that of no finite horizon and no discounting of the future. There is always a tomorrow in our plans, and it is as important as today. Also there is a history, a beginning of history, but no end. Decisions are based on the effect of past actions of agents, not on promises or binding agreements. However communication is certainly not precluded. Solutions in our games are asymptotic solutions. To be important for us, they must meet the criteria of stability. This criterion is well-defined by virtue of the dynamical foundations of the models. The first section deals with an example, the repeated prisoner's dilemma, in the language of an arms race. Here a class of strategies, is given where the solution is Pareto optimal, stable, and a Nash equilibrium. Thus at least asymptotically, we have a rather robust resolution of the prisoner's dilemma. We show how good strategies with optimal solutions might bifurcate into strategies with the worst solutions.