equilibrium based on the regular conditional probability can always satisfy the two criteria
of the rational solution concepts in general games. Moreover, in finite games, the perfect
regular equilibrium is equivalent to a simple version of the perfect Bayesian equilibrium.
Therefore, this perfect regular equilibrium extends the perfect Bayesian equilibrium into
general games as a simple version of it.
The rest of the paper is organized as follows. Section 2 formulates a general multi-period
game with observed actions. Section 3 provides a simple extension of the perfect Bayesian
equilibrium in general games and then illustrates its incapability to satisfy the two criteria of
the rational solution concepts, namely, the weak consistency and the subgame perfect Nash
equilibrium condition. Section 4 formally defines the perfect regular equilibrium. Finally,
Section 5 shows that every perfect regular equilibrium satisfies these two criteria of the
rational solution concepts and concludes that a perfect regular equilibrium is an extended
and simple version of the perfect Bayesian equilibrium in general multi-period games with
observed actions.
2 General multi-period game with observed actions
We adopt the “multi-period games with observed actions” from Fudenberg and Tirole (1991)
and adapt it to general games that allow infinite actions and types, but only finite players.
Hence, like the game from Fudenberg and Tirole (1991), a general multi-period game with
observed actions is represented by five items: players, a type and state space, a probability
measure on the type and state space, strategies, and utility functions. Based on these items,