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 to the set of her actions. Instead, the player's perception of is represented by a factor : -> where reflects the "cognitive complexity" of the player. Formally, mapping sends each history to an element of a factor space that represents its equivalence class. The play of the player can then be conditioned just on the elements of the set From the perspective of the original multi-stage game we say that a function from o is a factor of a strategy if there exists a function from to the set of actions of the player such that = In this case we say that the strategy is -factor-asedStationary 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 (1 , . . . , ) on a finite string of action profiles ( , . . . , ) is a function of (1 , . . . , - ) 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.