perfect Bayesian equilibrium to the incapability as shown in the example.
To solve this incapability problem with the simple perfect Bayesian equilibrium, we revise
it by replacing Bayes’ rule with a regular version of the conditional probability and refer to
the revised solution concept as a perfect regular equilibrium. The regular version of the con-
ditional probability is another way of formulating a conditional probability and is especially
designed to well-define the probability given probability zero events. It therefore defines a
conditional probability implicitly through a functional equation without referring to a frac-
tion between probabilities of events. Accordingly, it does not show the limited application
problem as Bayes’ rule does, and it can well-define the conditional probabilities given ‘almost
every’ probability zero event. Therefore, the perfect regular equilibrium equipped with this
regular conditional probability13 can solve the incapability problem with the simple perfect
Bayesian equilibrium.
The definition of the perfect regular equilibrium is the same as that of the simple perfect
Bayesian equilibrium except for its approach to the conditional probabilities. Thus, the
perfect regular equilibrium is defined as an assessment (μ,δ) such that 1) μ is updated
from period to period with respect to δ and μ itself according to the regular conditional
probability, and 2) taking μ as given, no player prefers to change its strategy δ⅛ at any of
its information sets. The first condition for the perfect regular equilibrium is referred to as
regular consistency and the second condition is referred to as the sequential rationality. In
this section, we formally define these conditions and the perfect regular equilibrium.
13 For more information regarding the regular version of the conditional probability, please refer to Ash
(1972, 6.6).
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