On the epistemic foundation for backward induction
by
Geir B. Asheim
Department of Economics, University of Oslo

In recent years, two influential contributions on backward induction in finite generic perfect information games have appeared, namely Aumann (Games Econ. Beh., 1995) and Ben-Porath (Rev. Econ. Stud., 1997). These contributions reach opposite conclusions: While Aumann establishes that common knowledge of rationality implies that the backward induction outcome is reached, Ben-Porath shows that this outcome is not the only one that is consistent with common belief of rationality. The models of Aumann and Ben-Porath are different. One such difference is that Aumann makes use of true `knowledge', while Ben-Porath's analysis is based on `belief' with probability one.

The present paper shows how the conclusions of Aumann and Ben-Porath can be captured by imposing requirements on the players within the same general framework. Furthermore, the interpretations of the present analysis correspond closely to the intuitions that Aumann and Ben-Porath convey in their discussions.

By imposing that each player takes all opponent strategies into account (`caution'), this paper ensures that each player takes the possibility of reaching any subgame of the extensive form into account. This means that a rational choice in the whole game implies a rational choice in all subgames that are not precluded from being reached by the player's own strategy.

The main distinguishing feature of the present analysis is to consider the event that a player believes in opponent rationality rather than the event that the player himself is rational. Strategies surviving the Dekel-Fudenberg procedure, where one round of weak elimination is followed by iterated strong elimination, can be characterized as maximal strategies when there is common certain belief that each player satisfies `caution' and believes in the whole game that the opponent chooses rationally (`belief of opponent rationality'). For generic perfect information games, outcomes that correspond to strategies surviving the Dekel-Fudenberg procedure are exactly those that are promoted by Ben-Porath's analysis.

A perfect information game offers choice situations, not only in the whole game, but also in proper subgames. Hence, one can argue that `belief of opponent rationality' should be replaced by `belief in each subgame of opponent rationality': Each player believes in each subgame that his opponent chooses rationally in the subgame. The main results of this paper show how, for generic perfect information games, common certain belief of `caution' and `belief in each subgame of opponent rationality' is possible and uniquely determines the backward induction outcome. Hence, the present analysis provides an alternative route to Aumann's conclusion, namely that common knowledge (or certain belief) of an appropriate form of (belief of) rationality implies the backward induction outcome.

This epistemic foundation for backward induction requires common certain belief of `caution' and `belief in each subgame of opponent rationality', where the term `certain belief' is being used in the sense that an event is certainly believed if the complement is Savage-null. As shown by a counterexample, the characterization does not obtain if instead common belief (in a sense that generalizes belief with probability one) is considered.

Date received: April 24, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # caez-39.