Perfect Regular Equilibrium

Next, Proposition 3 reveals the relationship between the regular consistency for the per-
fect regular equilibrium and the reasonable consistency for the simple perfect Bayesian equi-
librium in finite games. In general games, Bayes’ rule in the reasonable consistency might
give rise to the incapability problem with a simple perfect Bayesian equilibrium. In finite
games, however, Bayes’ rule does not cause this problem, and it functions as well as the
regular conditional probability does. As a result, the reasonable consistency based on Bayes’
rule is equivalent to the regular consistency based on the regular conditional probability in
finite games.

Proposition 3 In finite games, an assessment is regularly consistent if and only if it is
reasonably consistent.

Proof. The result directly follows from the definitions. ■

Finally, Theorem 1 aggregates all the results. Propositions 1 and 2 together ensure that
the perfect regular equilibrium satisfies both conditions, namely, the weak consistency and
the subgame perfect Nash equilibrium condition, which we have suggested as criteria of
rational solution concepts in general games. So, we conclude that the perfect regular equi-
librium successfully extends the perfect Bayesian equilibrium to general games. In addition,
Proposition 3 guarantees that the perfect regular equilibrium is equivalent to the simple
perfect Bayesian equilibrium in finite games. Note that the simple perfect Bayesian equi-
librium is defined as a simple version of the perfect Bayesian equilibrium. Therefore, as a
corollary of these propositions, Theorem 1 brings all the properties of the perfect regular
equilibrium together and provides evidence that it is indeed an extended and simple version
of the perfect Bayesian equilibrium in general multi-period games with observed actions.

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