General Rationalizability and Its Robustness for Strategic Form Games with Incomplete Information

We extend the $\Delta$-rationalizability (see Battigalli and Siniscalchi 2003) to infinite strategic form games with incomplete information. The most important feature of the $\Delta$-rationalizability is that there is no specified epistemic type space \`{a} la Harsanyi. However, we can impose a collection of exogenous restrictions on first order beliefs over payoff types and strategies represented by a collection of correspondences $\Delta$. When $\Delta$ represents only restrictions on beliefs over payoff types, we show that the $\Delta$-rationalizable sets are nonempty under general topological conditions. Robustness with respect to almost common belief for rationality of $\Delta$-rationalizability is established under general conditions by two alternative approaches. We can approximate common belief by finite order of mutual beliefs; we can approximate common belief by common $p$-belief. One important feature of our analysis in the robustness is that in the second approach, different level of belief is allowed for every order of mutual belief among players