Perfect Regular Equilibrium

condition is fundamentally different from the consistency condition. Hence, we conclude that
its contribution to the reasonability condition is not sufficient compared with the complexity
caused by this condition⁷ . As a result, we simplify the definition of the reasonability in
general games by excluding this no-signaling-what-you-don’t-know condition. We call this
simple extension of the reasonability reasonable consistency.

Definition 2 An assessment (μ,δ) is reasonably consistent if given each i, 1) μ¹ is the
same as y__i and 2) for each (θ_i, a¹, ...,a^t~¹) ∈ Θ_i × A¹ × ∙ ∙ ∙ × A^{t 1} and each B ∈ ×_j∕jβ(Θ_j),
μt(θ_i, a¹,..., a^t~¹∙, B) indicates the same probability as

f_θ~. f_{at-iy^B(θ~i)δ^t~¹(θ, a¹, ...,a^t~²; da^t-¹)μ^tr¹(θ_i, a¹, ...,a^t~²; dθ~f)

J<> f{_a⅛-ι∕ⁱ-¹(θ, a¹, ...,a ² da^t-^r)μ^tr∖θi, a¹, ...,a ² dθ _b

whenever t ≥ 2 and ʃθ βι_{at 1} fδ' ^] (0, oⁱ, ...,a^t '²: da^{l 1})μ^l_i ¹ (θ_i, a¹, ...,a^t '²: dθ _i) > 0 where
I_b(∙) is an indicator function, i.e. I_b(θ~_i) = 1 if θ _i ∈ B and I_b(θ~f) = 0 if θ _i ∈ B.

In other words, an assessment (μ, δ) is reasonably consistent if 1) in the first period, each
player correctly forms its beliefs μ¹ based on the type and state probability measure η, and
2) from the second period, each player employs Bayes’ rule to update its beliefs μ^t_i with
respect to the previous action plans δ^{t 1} and the previous beliefs μ^{t 1} whenever possible.
Here, “whenever possible” means whenever an information set Θ-_i ×{(θ_i, a¹, ...,a^{t 1})} is
reached with positive probability with respect to δ^{t 1} and μ^t~¹, that is, ʃθ β_ва^ ∖ fδ ^{] 1}('). a¹,
..., A' '²'. da^l' ¹)μi ^] (θ_i, a¹, ..., a¹'²; dθ _i) > 0. Note that, in finite games, this definition of the
reasonable consistency represents the same condition as the definition of the reasonability in

⁷ According to Fudenberg and Tirole (1991), this condition requires that “no player i’s deviation be
treated as containing information about things that player i does not know.” Here, “player i’s deviation” is
its behavior off the equilibrium path, so this condition places restrictions on the beliefs off the equilibrium
path. However, the incapability problem with a perfect Bayesian equilibrium occurs more significantly on
the equilibrium path than off the equilibrium path. Therefore, this condition cannot solve the problem with
the solution concept of the perfect Bayesian equilibrium. This is another reason why we conclude that the
contribution of this condition is not sufficient.

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