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

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 φ: H→ X, where X reflects the “cognitive complexity” of the player, and the strategy chooses its mixed action at history h as a function of φ(h). In this case we say that φ is a factor of a strategy and that the strategy is φ-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 φ is recursive if its value at history h^′ that follows history h is a function of φ(h) and the incremental information h^′\ h. For example, in a repeated game with perfect monitoring, a factor φ is recursive if its value φ(a₁, … , a_t) on a finite string of action profiles (a₁, … , a_t) is a function of φ(a₁, … , 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 φ is recursive, then for every profile of φ-factored strategies there is a pure φ-factored strategy that is a best reply, and if the stochastic game has finitely many states and actions and the factor φ has a finite range then there is a pure φ-factored strategy that is a best reply in all the discounted games with a sufficiently large discount factor.