On Forward Induction

33 Pages Posted: 7 Feb 2007

See all articles by Srihari Govindan

Srihari Govindan

University of Rochester

Robert Wilson

Stanford Graduate School of Business

Date Written: January 2007


We examine Hillas and Kohlberg's conjecture that invariance to the addition of payoff-redundant strategies implies that a backward induction outcome survives deletion of strategies that are inferior replies to all equilibria with the same outcome. That is, invariance and backward induction imply forward induction. Although it suffices in simple games to interpret backward induction as a subgame-perfect or sequential equilibrium, to obtain general theorems we use a quasi-perfect equilibrium, viz. a sequential equilibrium in strategies that are admissible continuations from each information set. Using this version of backward induction, we prove the Hillas-Kohlberg conjecture for two-player extensive-form games with perfect recall. We also prove an analogous theorem for general games by interpreting backward induction as a proper equilibrium, since a proper equilibrium is equivalent to a quasi-perfect equilibrium of each extensive form with the same normal form, provided beliefs are justifed by perturbations invariant to inessential transformations of the extensive form. For a two-player game we prove that if a set of equilibria includes a proper equilibrium of every game with the same reduced normal form then it satisfies forward induction, i.e. it includes a proper equilibrium of the game after deleting strategies that are inferior replies to all equilibria in the set. We invoke slightly stronger versions of invariance and properness to handle nonlinearities in an N-player game.

Keywords: economic theory, game theory

JEL Classification: C72

Suggested Citation

Govindan, Srihari and Wilson, Robert B., On Forward Induction (January 2007). Available at SSRN: https://ssrn.com/abstract=961852 or http://dx.doi.org/10.2139/ssrn.961852

Srihari Govindan

University of Rochester ( email )

Department of Economics
Rochester, NY NY 14627
United States
5852757214 (Phone)

Robert B. Wilson (Contact Author)

Stanford Graduate School of Business ( email )

655 Knight Way
Stanford, CA 94305-5015
United States
650-723-8620 (Phone)
650-725-7979 (Fax)

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