Trimmed Strategies: Achieving Sequentially Rational Equilibria with only Partially Specified Strategies

Abstract

Both subgame-perfect equilibrium and weak-perfect Bayesian equilibrium impose rationality at information sets that are irrelevant for a sequentially rational outcome. In this paper, for each of these equilibrium concepts, I characterize the maximal set of such information sets. These information sets can be trimmed (or removed) from strategy sets without affecting the equilibrium outcome. A trimmed equilibrium is an equilibrium for the game where the irrelevant information sets have been ignored. It is shown that the trimmed version of an equilibrium concept, either subgame-perfect or weak-perfect Bayesian equilibrium, is just sufficiently restrictive to ensure that equilibrium outcomes are consistent with the original concept, i.e., ignoring any additional information sets could change the equilibrium outcome. An example demonstrates that trimming irrelevant information sets can lead to existence of an outcome consistent with subgame-perfect rationality in cases where a subgame perfect equilibrium does not exist.