Econometrica, Submission #6956, revised ON FORWARD INDUCTION

A player’s pure strategy is called relevant for an outcome of a game in extensive form with perfect recall if there exists a weakly sequential equilibrium with that outcome for which the strategy is an optimal reply at every information set it does not exclude. The outcome satisfies forward induction if it results from a weakly sequential equilibrium in which players’ beliefs assign positive probability only to relevant strategies at each information set reached by a profile of relevant strategies. We prove that if there are two players and payoffs are generic then an outcome satisfies forward induction if every game with the same reduced normal form after eliminating redundant pure strategies has a sequential equilibrium with an equivalent outcome. Thus in this case forward induction is implied by decision-theoretic criteria. This paper has two purposes. One is to provide a general definition of forward induction for games in extensive form with perfect recall. As a refinement of weakly sequential equilibrium, forward induction restricts the support of a player’s belief at an information set to others’ strategies that are optimal replies to some weakly sequential equilibrium with the same outcome, if there are any that reach that information set. The second purpose is to resolve a conjecture by Hillas and Kohlberg [27, §13.6], of which the gist is that ‘invariant backward induction outcomes satisfy forward induction.’ An outcome is invariant if every extensive form representing the same strategic situation (i.e. they have the same reduced normal form) has a sequential equilibrium with an equivalent outcome. For a game with two players and generic payoffs we prove that an invariant outcome satisfies forward induction. The definitions and theorem are entirely decision-theoretic. None of the technical devices invoked in game theory, such as perturbations of players’ strategies or payoffs, are needed. Date: December 2004. Submitted January 2007. Revised January 29, 2008.