Perfect Regular Equilibrium

We first provide a formal definition of the regular consistency in general multi-period
games with observed actions.

Definition 4 An assessment (μ,δ) is regularly consistent if given each i, 1) μ1 is the
same as η_-t and 2) for each t > 2 and each (θ_t. a¹..... a^t-2) ∈ Θ_t × A¹ ×∙∙∙× A², μ^t_i satisfies
the following functional equation: J_j∂^{t l}(θ.v^l. ....a^{t 2}: da^{t 1})μ∕∙ ¹(θ_t.a¹. ....a^{t 2}: dθ _i ') =
ʃe /а^%(^.°¹. ....°^t~¹: B)δ^t~∖θ.a¹. ...,a^t~²; da^t-1')μ^t- (θ_t.a¹. ....a^{t 2}: dθ-f) for every B ∈
×_j≠_iβ(Θ_j) and A ∈ ×L1fi(At-¹).

That is, an assessment (μ,δ) is regularly 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 the regular conditional probability to update its
beliefs μ^t_i with respect to the previous action plans δ^t-1 and the previous beliefs μ^t-1 given
the information about its type and the previous actions (θ_i,a¹, .... a^l' ²). Here, Definition
4 implicitly defines μ^t_t as a regular conditional probability measure through the functional
equation governed by δ^t-1 and μ^t-1. In this way, Definition 4 can avoid the limited applica-
tion problem since the functional equation is well-defined for any arbitrary set B ×{(θ_t.a¹.
....a^t-2)} ×A where B ∈ ×j≠_tβ(Θj) and A ∈ ×f=₁β(A^t-1). As a result, beliefs μ^t can be
properly updated with respect to δ^t-1 and μ^t-1. Note that the functional equation can de-
termine a conditional probability of μ^t_i only within the support¹⁴ of the product measure of
δ^t-1 and μt^-1. This is because if a set A ∈ ×{=₁β(A^t-1) is outside the support of the product
measure δ^t-1μ^t-1, then both sides in the functional equation become zero, and so the con-
ditional probability of μ^t_i given A can be arbitrary. Consequently, a conditional probability
of μ^t_i is only meaningful given a set of actions within the support of the product measure

¹⁴ In a metric space, a support of a measure is defined as the smallest closed set within which the measure
lives.

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