setting12 that the sender has a continuum of types and signals. Accordingly, most games
having similar settings can testify that there exist simple perfect Bayesian equilibria that
break the weak consistency and the subgame perfect Nash equilibrium condition. Since
this setting represents a usual situation, there is a large class of games including similar
settings. Therefore, we conclude that this incapability is a ubiquitous problem with the
solution concept of the simple perfect Bayesian equilibrium. In the next section, we revise
this simple perfect Bayesian equilibrium to develop a solution concept that is capable of
satisfying both the weak consistency and the subgame perfect Nash equilibrium condition.
4 Perfect Regular Equilibrium
The incapability of a simple perfect Bayesian equilibrium is due to the limited application
of Bayes’ rule in general multi-period games with observed actions. Bayes’ rule cannot be
employed if a conditional event, whose probability becomes a denominator in a conditional
probability formula according to Bayes’ rule, takes place with probability zero. In general
games, however, it is possible for every conditional event to take place with probability zero.
In this case, we cannot employ Bayes’ rule at all either on the equilibrium path or off the
equilibrium path. Hence, no system of beliefs is considered to violate Bayes’ rule, which
means that every system of beliefs satisfies the reasonable consistency for a simple perfect
Bayesian equilibrium. As a result, some intuitively inconsistent system of beliefs could be
part of a simple perfect Bayesian equilibrium, and this system of beliefs could lead the simple
12 Jung (2010) showed that, under this setting, an extension of the sequential equilibrium in general games
can cause the same problem, namely, the incapability to satisfy both the weak consistency and the Nash
equilibrium condition.
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