4 Eliminating Dominated Strategies

Suppose Si is a finite set of pure strategies for players i D 1; : : : ; n in normal form game G, so that S D S1 : : : Sn is the set of pure strategy profiles for G, and i .s/ is the payoff to player i when strategy profile s 2 S is chosen by the players. We denote the set of mixed strategies with support in S as mS D S1 : : : Sn, where Si is the set of mixed strategies for player i with support in Si , or equivalently, the set of convex combinations of members of Si . We denote the set of vectors of mixed strategies of all players but i by mS i . We say s 0 i 2 Si is strongly dominated by si 2 Si if, for every i 2 mS i , i.si ; i/ > i.s 0 i ; i/. We say s 0 i is weakly dominated by si if for every i 2 mS i , i.si ; i/ i .s 0 i ; i /, and for at least one choice of i the inequality is strict. A strategy may fail to be strongly dominated by any pure strategy, but may nevertheless be strongly dominated by a mixed strategy (4.8). Suppose si is a pure strategy for player i such that every mixed strategy 0 i ¤ si for player i is weakly (resp. strongly) dominated by si . We call si a weakly (resp. strongly) dominant strategy. for i . If there is a Nash equilibrium in which all players use a dominant strategy, we call this a dominant strategy equilibrium. Once we have eliminated dominated strategies for each player, it often turns out that a pure strategy that was not dominated at the outset is now dominated. Thus, we can undertake a second round of eliminating dominated strategies. Indeed, this can be repeated until pure strategies are no longer eliminated in this manner. In a finite game, this will occur after a finite number of rounds, and will always leave at least one pure strategy remaining for each player. If strongly (resp. weakly) dominated strategies are eliminated, we call this the iterated elimination of strongly (resp. weakly) dominated strategies. We will refer to strategies that are elimi-