Compactification of Extensive Forms and Belief in the Opponents' Future Rationality

We introduce an operation, called compactification, to reduce an extensive form to a compact one where each decision node in the game tree can be assigned to more than one player. Motivated by Thompson (1952)'s interchange of decision nodes, we attempt to capture the notion of a faithful representation of the chronological order of the moves in a dynamic game which plays a vital role in fields like epistemic game theory. The compactification process preserves perfect recall and the unambiguity of the order among information sets. We specify an algorithm, called leaves-to-root process, which compactifies at least as many information sets as any other compactification process. The compact extensive form provides an approach to avoid problems in dynamic game theory due to the vague definition of the chronological order of the moves, for example, belief in the opponents' future rationality (Perea (2014))'s sensitivity to the specific extensive form representation. We show that any strategy which can rationally be chosen under common belief in future rationality in a minimal compact game if and only if it satisfies this property in every extensive form game which is related to it via some compactification process.