Perfect Regular Equilibrium

players would not reach if they would play according to δ. In addition, δ is sequentially
rational with respect to μ if, taking μ as given, no player prefers to change its strategy δ⅛ at
any of its information sets.

Of these two conditions for a perfect Bayesian equilibrium, the first condition, reason-
ability, might lead it to being incapable of satisfying the weak consistency and the subgame
perfect Nash equilibrium condition in general multi-period games with observed actions. To
be precise, the incapability of a perfect Bayesian equilibrium is caused by the weakness of
Bayes’ rule. Bayes’ rule is a way of formulating a conditional probability or a conditional
probability density function and defines them as a fraction between two probabilities or a
fraction between two probability density functions. So, Bayes’ rule can be employed only
when the probability of a given event, which becomes a denominator in the fraction, is pos-
itive or when the probability density functions are well-defined. This limited application of
Bayes’ rule consequently gives rise to the incapability of a perfect Bayesian equilibrium in
general games.

To clearly see this incapability of a perfect Bayesian equilibrium, we formally extend
the definition of the reasonability into general multi-period games with observed actions.
Notice that, in this extension, we omit the “no-signaling-what-you-don’t-know” condition
for simplicity’s sake. This condition was designed to improve the reasonability condition
so that this reasonability condition might become as plausible as the consistency condition
introduced by Kreps and Wilson (1982). As shown by Osborne and Rubinstein (1994,
234.3), however, the reasonability condition including the no-signaling-what-you-don’t-know

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