Should I remember more than you? Best responses to factored strategies

Abstract

In this paper we offer a new, unifying approach to modeling strategies of bounded complexity. In our model, the strategy of a player in a game does not directly map the set H of histories to the set of her actions. Instead, the player’s perception of H is represented by a map $\varphi :H \rightarrow X,$ where X reflects the “cognitive complexity” of the player, and the strategy chooses its mixed action at history h as a function of $\varphi (h)$. In this case we say that $\varphi$ is a factor of a strategy and that the strategy is $\varphi$-factored. Stationary strategies, strategies played by finite automata, and strategies with bounded recall are the most prominent examples of factored strategies in multistage games. A factor $\varphi$ is recursive if its value at history $h'$ that follows history h is a function of $\varphi (h)$ and the incremental information $h'\setminus h$. For example, in a repeated game with perfect monitoring, a factor $\varphi$ is recursive if its value $\varphi (a_1,\ldots ,a_t)$ on a finite string of action profiles $(a_1,\ldots ,a_t)$ is a function of $\varphi (a_1,\ldots ,a_{t-1})$ and $a_t$.We prove that in a discounted infinitely repeated game and (more generally) in a stochastic game with finitely many actions and perfect monitoring, if the factor $\varphi$ is recursive, then for every profile of $\varphi$-factored strategies there is a pure $\varphi$-factored strategy that is a best reply, and if the stochastic game has finitely many states and actions and the factor $\varphi$ has a finite range then there is a pure $\varphi$-factored strategy that is a best reply in all the discounted games with a sufficiently large discount factor.