of the rational beliefs imposes restrictions on all the beliefs off the equilibrium path as well
as on the equilibrium path. Note that the perfect regular equilibrium, just like the perfect
Bayesian equilibrium in finite games, places restrictions only on the beliefs on the support
of the product measure δt lμt l. Since the support of the product measure δt lμt 1 might
not cover all the beliefs off the equilibrium path, the perfect regular equilibrium might not
put restrictions on all the beliefs off the equilibrium path. Accordingly, the perfect regular
equilibrium might not satisfy the convex structural consistency18 .
The second property of the perfect regular equilibrium is that it always satisfies the
subgame perfect Nash equilibrium condition, which is a criterion of the rational strategies.
This property is due to the sequential rationality, which is one of the two conditions for the
perfect regular equilibrium. If an information set initiates a subgame, then the conditional
probability on the information set given the information set itself is uniquely determined
as one. Then, the sequential rationality condition, given the information set, becomes the
same as the Nash equilibrium condition, which means that the perfect regular equilibrium
induces a Nash equilibrium in the subgame. As a result, the perfect regular equilibrium
satisfies the subgame perfect Nash equilibrium condition. Proposition 2 formally presents
this second property of the perfect regular equilibrium in general multi-period games with
observed actions.
Proposition 2 Every perfect regular equilibrium is a subgame perfect Nash equilibrium.
Proof. The result directly follows from the definitions. ■
18 Jung (2010) introduced a new solution concept, complete sequential equilibrium, in general finite-period
games with observed actions and presented conditions under which the complete sequential equilibrium
satisfies the convex structural consistency.
24
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