When order matters for iterated strict dominance
by
Martin Dufwenberg
Stockholm University
Coauthors: Mark Stegeman (VPI&SU)

The most basic rule for predicting behavior in non-cooperative games is that players should not adopt strictly dominated strategies. Eliminating such strategies from consideration may permit additional eliminations, and iterated elimination of strictly dominated strategies (IESDS) leads to a fundamental solution concept: the maximal reduction of a game. In some cases, the maximal reduction comprises a single strategy profile. For example, in a standard Cournot duopoly, after eliminating outputs that exceed a monopolist's output, small and large outputs may be eliminated sequentially until, in the limit, only the Cournot-Nash equilibrium remains. Game theorists often seem to assume, explicitly or implicitly, that play should be confined to the maximal reduction of a game. It is surprising therefore, that so little is known about this basic solution concept. This paper studies the conditions under which maximal reductions exist and are unique. Since IESDS entails the shrinking of strategy sets, and shrinking sets inevitably reach a limit, the question of existence amounts to: is the limit nonempty; and if so does it own only undominated strategies? The answers are clearly yes for finite games, but we show by simple examples that infinite games need not have maximal reductions. Uniqueness, on the other hand, concerns the speed and order of reductions: do all paths lead, in the limit, to the same maximal reduction? Again, the answer is known to be yes for finite games, but we show, again through simple examples, that order may matter for infinite games. We next search for classes of games for which IESDS is a well-behaved procedure. Reny (1999) has recently proved the existence of Nash equilibrium for a large class of games: those with finite player sets, compact and convex strategy sets, and payoff functions which are bounded and satisfy a condition of better-reply security. We show that this class contains games where order matters for IESDS. We show, however, that positive results concerning IESDS can be derived for other classes of games. We consider games having arbitrary numbers of players and strategy sets in arbitrary Hausdorff spaces, and we call games compact and continuous if the strategy sets are compact and the payoff functions continuous. This class differs from Reny's because it requires neither convex strategy sets nor finite (or countable) players, but it does require continuous payoffs. We show that, for any compact and continuous game, IESDS satisfies the most obvious desiderata: a maximal reduction always exists and is unique. For games outside this class, these properties routinely fail and the concept of strict dominance is intrinsically problematic. We also identify a larger class of games, for which IESDS preserves the set of Nash equilibria. For this property, it is sufficient that each player have a well-defined best-response correspondence. We show that the existence and uniqueness of maximally reduced games does not extend to this larger class.

Date received: May 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 Conferences Inc. Document # cafc-65.