Rationalizable Strategies in Games With Incomplete Preferences

Games with incomplete preferences are normal-form games where the preferences of the players are defined as partial orders over the outcomes of the game. We define rationality in these games as follows. A rational player forms a set-valued belief of possible strategies selected by the opponent(s) and selects a strategy that is not dominated with respect to this belief. Here, we say a strategy is dominated with respect to the set-valued belief if the player has another strategy that would yield a better outcome according to the player's preference relation, no matter which strategy combination the opponent(s) play among those contained in the belief. We define rationalizable strategies as the logical implication of common knowledge of this rationality. We show that the sets of rationalizable strategies are the maximal mutually nondominated sets, i.e., the maximal sets that contain no dominated strategies with respect to each other. We show that no new rationalizable strategies appear when additional preference information is included. We consider multicriteria games as a special case of games with incomplete preferences and introduce a way of representing incomplete preference information in multicriteria games by sets of feasible weights of the criteria.