Recursive Markov Process for Iterated Games with Markov Strategies

Abstract

The dynamics in games involving multiple players, who adaptively learn from their past experience, is not yet well understood. We analyzed a class of stochastic games with Markov strategies in which players choose their actions probabilistically. This class is formulated as a $k^{\text{th}}$ order Markov process, in which the probability of choice is a function of $k$ past states. With a reasonably large $k$ or with the limit $k \to \infty$, numerical analysis of this random process is unfeasible. This study developed a technique which gives the marginal probability of the stationary distribution of the infinite-order Markov process, which can be constructed recursively. We applied this technique to analyze an iterated prisoner's dilemma game with two players who learn using infinite memory.