On the Dynamics of Finite Memory Distributed Systems

We consider a kinetic equation approach to dynamical systems with finite memory, based upon a probabilistic approach given in Dorofeenko and Shorish (2004). This approach uses a master equation methodology to analytically model the dynamics of distributed systems with many heterogeneous agents, each agent possessing a fixed (zero-memory) strategy for pairwise interaction. The methodology allows one to consider spatially distributed populations of agents in a continuous space–continuous time model. We extend the techniques of this approach to models where agents may possess a finite memory of their encounters with other agents. This allows the analytical modeling of, for example, the 'tit-for-tat' strategy in a repeated Prinsoner’s Dilemma game, where agents remember their most recent encounter. We demonstrate that models of finite agent memory, where the outcome is determined by pairwise interactions between agents, lead to the emergence of collective (probabilistic) structures in both time and space. We also show that this extension requires a suitable redefinition of the 'characteristic time' of the system, in order to arrive at an analogue of the master equation in the fixed strategy environment.