Extensive Form Rationalizability
The heart has its reasons of which reason
The extensive form of a game is informationally richer than the normal form since players gather information that allows them to update their subjective priors as the game progresses. For this reason, the study of rationalizability in extensive form games is more complex than the corresponding study in normal form games. There are two ways to use the added information to eliminate strategies that would not be chosen by a rational agent: backward induction and forward induction. The latter is relatively exotic (although more defensible) and will be addressed in chapter 9. Backward induction, by far the most popular technique, employs the iterated elimination of weakly dominated strategies, arriving at the subgame perfect Nash equilibria—the equilibria that remain Nash equilibria in all subgames. We shall call an extensive form game generic if it has a unique subgame perfect Nash equilibrium.
In this chapter we develop the tools of modal logic and present Robert Aumann's famous proof (Aumann 1995) that CKR implies backward induction. This theorem has been widely criticized, as well as widely misinterpreted. I will try to sort out the issues, which are among the most important in contemporary game theory. I conclude that Aumann is perfectly correct, and the real culprit is CKR itself.
Backward induction in extensive form games with perfect information (i.e., where each information set is a single node) operates as follows. Choose any terminal node τ ∊ Τ and find the parent node of this terminal node, say node v. Suppose player i chooses at ν and suppose i's highest payoff at ν is attained at terminal node τ' ∊ T. Erase all the branches from ν so ν becomes a terminal node and attach the payoffs from τ' to the new terminal