In this definition, the functional equation can determine a conditional probability of μti
only on the equilibrium path. This is because if a set A ∈ ×( lA(At l ) is off the equilibrium
path, then both sides in the functional equation become zero, so a conditional probability of
μt given A can be arbitrary. Consequently, this definition indicates that the weak consistency
imposes restrictions only on the beliefs on the equilibrium path, and thus it imposes no
restriction on the beliefs off the equilibrium path. Note that the regular consistency places
restrictions on the beliefs on the support of the product measure A lμt l, which includes
all the beliefs on the equilibrium path. As a result, in general multi-period games with
observed actions, if an assessment satisfies the regular consistency, then it also satisfies the
weak consistency. This statement is formulated in Proposition 1.
Proposition 1 If an assessment is regularly consistent, then it is weakly consistent.
Proof. The result directly follows from the definitions. ■
Kreps and Ramey (1987) introduced another criterion of the rational beliefs, convex
structural consistency. According to them, the convex structural consistency is defined as a
consistency criterion under which the beliefs of the players should reflect the informational
structure of a game through a convex combination of players’ strategies. Thus, under this
consistency criterion, if players would be unexpectedly located, they should then form their
beliefs such that a convex combination of strategies can induce the beliefs17 . This criterion
17 In fact, this convex structural consistency is a weak version of the structural consistency of Kreps and
Wilson (1982). Kreps and Wilson defined the structural consistency as a consistency criterion under which
the beliefs of the players should reflect the informational structure of a game through a single strategy profile.
Thus, this structural consistency requires players to use only one strategy profile to form one belief. Because
of the strong requirement, however, most of the solution concepts including the sequential equilibrium and
the perfect equilibrium do not satisfy this criterion even in finite games.
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