Perfect Regular Equilibrium

Fudenberg and Tirole (1991, Definition 3.1) except for the no-signaling-what-yon-don’t-know
condition.

There is another version of Bayes’ rule, a continuous version of Bayes’ rule, but we cannot
use this version to extend the definition of the reasonability into general multi-period games
with observed actions. A continuous version of Bayes’ rule defines a conditional probability
density function as a fraction between two probability density functions. Accordingly, this
version requires well-defined probability density functions. In general games with a contin-
uum of actions, however, only mixed strategies that assign zero probability to every single
action can be represented as probability density functions. As a result, this version of Bayes’
rule is not well-defined for any strategies that assign positive probability to a single action.
In particular, this version is not well-defined for any of the pure strategies under which
players would play a single action at each information set. Therefore, we cannot extend the
definition of the reasonability into general games by using this continuous version of Bayes’
rule⁸ .

Based on the reasonable consistency condition, a perfect Bayesian equilibrium is extended
in general multi-period games with observed actions. We call this simple extension of the
perfect Bayesian equilibrium a simple perfect Bayesian equilibrium.

Definition 3 An assessment (μ,δ) is a simple perfect Bayesian equilibrium if (μ,δ)

⁸ There is a way to combine these two versions of Bayes’ rule. This way does not solve the limited
application problem with Bayes’ rule in general games, either. This is because it requires well-defined
probability or probability density functions. In general games, however, players’ strategies might induce
neither probability nor a probability density function. For example, the sender’s strategy introduced in the
next subsection induces neither probability nor a probability density function. As a result, this combined
version of Bayes’ rule still has limited application, and therefore it could result in the incapability of a perfect
Bayesian equilibrium in general games.

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