Order Independence and Rationalizability

Two natural strategy elimination procedures have been studied for strategic games. The first one involves the notion of (strict, weak, etc) dominance and the second the notion of rationalizability. In the case of dominance the criterion of order independence allowed us to clarify which notions and under what circumstances are robust. In the case of rationalizability this criterion has not been considered. In this paper we investigate the problem of order independence for rationalizability by focusing on three naturally entailed reduction relations on games. These reduction relations are distinguished by the adopted reference point for the notion of a better response. Additionally, they are parametrized by the adopted system of beliefs. We show that for one reduction relation the outcome of its (possibly transfinite) iterations does not depend on the order of elimination of the strategies. This result does not hold for the other two reduction relations. However, under a natural assumption the iterations of all three reduction relations yield the same outcome. The obtained order independence results apply to the frameworks considered in Bernheim 84 and Pearce 84. For finite games the iterations of all three reduction relations coincide and the order independence holds for three natural systems of beliefs considered in the literature.