Should i remember more than you? On the best response to factor-based strategies

In this paper we offer a new approach to modeling strategies of bounded complexity, the so-called factor-based strategies. In our model, the strategy of a player in the multi-stage game does not directly map the set of histories H to the set of her actions. Instead, the player's perception of H is represented by a factor [...] where X reflects the cognitive complexity of the player. Formally, mapping [...] sends each history to an element of a factor space X that represents its equivalence class. The play of the player can then be conditioned just on the elements of the set X. From the perspective of the original multi-stage game we say that a function [...] from H to X is a factor of a strategy [...] if there exists a function [...] from X to the set of actions of the player such that [...]. In this case we say that the strategy [...] is [...]-factor-based. Stationary strategies and strategies played by finite automata and strategies with bounded recall are the most prominent examples of factor-based strategies. In the discounted infinitely repeated game with perfect monitoring, a best reply to a profile of [...]-factor-base strategies need not be a [...]-factor-base strategy. However, if the factor [...] is recursive, namely its value [...] on a finite string of action profiles [...] is a function of [...] and [...], then for every profile of factor-based strategies there is a best reply that is a pure factor-based strategy. We also study factor-based strategies in the more general case of stochastic games..