Perfect Regular Equilibrium

Theorem 1 Every perfect regular equilibrium satisfies both the weak consistency and the
subgame perfect Nash equilibrium condition. Furthermore, in finite games, an assessment
(μ, δ) is a perfect regular equilibrium if and only if it is a simple perfect Bayesian equilibrium.

There is another solution concept for general games. Jung (2010) developed complete
sequential equilibria in general finite-period games with observed actions by improving se-
quential equilibria. In general games, the sequential equilibrium might give rise to the inca-
pability problem as in the case of the perfect Bayesian equilibrium. The complete sequential
equilibrium solves this incapability problem by replacing beliefs with complete beliefs. The
complete beliefs are probability measures defined, not on each information set, but on the
whole class of information sets in each period. Note that all strategy profiles lead to the
whole class of information sets in each period with probability one and thus they can well-
define probability distributions over the whole class of information sets. As a result, any
arbitrary strategy profile can properly induce consistent complete beliefs, and therefore the
complete sequential equilibrium can improve the sequential equilibrium in general games.

This complete sequential equilibrium, however, is not closely related to the perfect regular
equilibrium in general games in that it might not be a perfect regular equilibrium and vice
versa. This is because the consistency for the complete sequential equilibrium and the
regular consistency for the perfect regular equilibrium place different restrictions on the
beliefs off the equilibrium path. Consequently, a complete sequential equilibrium might not
be a perfect regular equilibrium in general games and a perfect regular equilibrium might
not be a complete sequential equilibrium either.

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