Amanda Friedenberg

Iterated dominance revisited

A fundamental solution concept in game theory is the iterative deletion of strongly dominated strategies. The concept has a long history in game theory, going back at least to Luce and Raiffa (1957). Bernheim (1984) and Pearce (1984) asserted that (up to issues of correlation) the iteratively undominated (IU) strategies correspond to the strategies consistent with common knowledge of rationality. Since then, many papers have formally investigated the epistemic conditions of IU. (See, e.g., Brandenburger and Dekel (1987), Tan and Werlang (1988), Battigalli and Siniscalchi (2002), among many others.) In this paper we revisit the epistemic conditions for IU. We point out that, in somewhat subtle ways, the literature is incomplete. We go on to provide novel epistemic conditions for IU.

Lexicographic Beliefs and Assumption

Foundations for iterated admissibility (i.e., the iterated removal of weakly dominated strategies) need to confront a fundamental challenge. On the one hand, admissibility requires that a player consider every strategy of their opponents possible. On the other hand, reasoning that the opponents are rational requires ruling out certain strategies. Brandenburger, Friedenberg, Keislers (BFK, Econometrica, 2008) foundations for iterated admissibility address this challenge with two ingredients: lexicographic beliefs and the concept of “assumption.” However, BFK restrict attention to lexicographic beliefs whose supports are essentially disjoint. This restriction does not have a compelling behavioral rationale, or a clear intuitive interpretation. At the same time, it plays a crucial role in BFKs foundations for iterated admissibility—specifically, in their analysis of assumption. We provide an alternate characterization of assumption, which applies to all lexicographic beliefs. We also characterize two variants of assumption, based on two extensions of ‘weak dominance’ to infinite state spaces. These notions of assumption coincide with BFKs notion when the state space is finite and lexicographic beliefs have disjoint support; but they are different in more general settings. Leveraging these characterization results, we show that disjoint supports do not play a role in the foundations for iterated admissibility.