The Many Faces of Rationalizability

The rationalizability concept was introduced in \cite{Ber84} and \cite{Pea84} to assess what can be inferred by rational players in a non-cooperative game in the presence of common knowledge. However, this notion can be defined in a number of ways that differ in seemingly unimportant minor details. We shed light on these differences, explain their impact, and clarify for which games these definitions coincide. Then we apply the same analysis to explain the differences and similarities between various ways the iterated elimination of strictly dominated strategies was defined in the literature. This allows us to clarify the results of \cite{DS02} and \cite{CLL05} and improve upon them. We also consider the extension of these results to strict dominance by a mixed strategy. Our approach is based on a general study of the operators on complete lattices. We allow transfinite iterations of the considered operators and clarify the need for them. The advantage of such a general approach is that a number of results, including order independence for some of the notions of rationalizability and strict dominance, come for free.