Perfect Regular Equilibrium

1 Introduction

We propose a revised version of the perfect Bayesian equilibrium in general multi-period
games with observed actions. Fudenberg and Tirole (1991) formulated the perfect Bayesian
equilibrium in the setting of finite games that allow only a finite number of types and strate-
gies. In finite games, this perfect Bayesian equilibrium satisfies criteria of rational solution
concepts such as weak consistency and the subgame perfect Nash equilibrium condition.
However, it might not satisfy these criteria in general games that allow a continuum of types
and strategies. To solve this problem with the perfect Bayesian equilibrium, we revise its
definition by replacing Bayes’ rule with a regular conditional probability. We refer to this
revised version of the perfect Bayesian equilibrium as the perfect regular equilibrium. We
show that it satisfies these criteria of rational solution concepts in general multi-period games
with observed actions. In addition, this perfect regular equilibrium is equivalent to a simple
version of the perfect Bayesian equilibrium in finite games. Therefore, we conclude that the
perfect regular equilibrium extends the perfect Bayesian equilibrium into general games as
a simple version of it.

In game theory, most of the solution concepts were developed as refinements of the Nash
equilibrium introduced by Nash (1951). The Nash equilibrium, as the most popular solution
concept, embodies the behavior of rational players. So, it consists of a set of strategies for
each player such that each strategy is the best response to the other strategies. This Nash
equilibrium became known as a compelling condition for rational strategies, and thus it
became a necessary condition for rational solution concepts. However, the Nash equilibrium

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