Perfect Regular Equilibrium

3 Example: Incapable simple perfect Bayesian equilib-
rium in a general game

This section shows that an extension of the perfect Bayesian equilibrium in general multi-
period games with observed actions might be incapable of satisfying the weak consistency
and the subgame perfect Nash equilibrium condition. We first provide a simple and for-
mal extension of the perfect Bayesian equilibrium in general games. Originally, the perfect
Bayesian equilibrium was defined in finite games. However, it has been extended and ap-
plied to general games on various economic issues, such as the Auction, Bargaining game,
and Signaling game. Here, we try to present a universal definition of the perfect Bayesian
equilibrium that can be commonly applied to such general games. Next, we describe the
setting of the example which is the famous signaling game by Crawford and Sobel (1982).
Then, based on this setting, we show that the simple extension of the perfect Bayesian equi-
librium might be incapable of satisfying the weak consistency and the subgame perfect Nash
equilibrium condition.

3.1 Simple perfect Bayesian equilibrium in a general game

According to Fudenberg and Tirole (1991), a perfect Bayesian equilibrium in a finite game
is defined as an assessment (μ, δ), which is a pair consisting of a system of beliefs μ and a
strategy profile δ, such that (μ, δ) is both 1) reasonable and 2) sequentially rational. Here, an
assessment (μ, δ) is said to be reasonable i) if μ is updated from period to period with respect
to δ and μ itself according to Bayes’ rule whenever possible and ii) if it satisfies the “no-
signaling-what-you-don’t-know” condition that constrains μ off the equilibrium path which

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