Perfect Regular Equilibrium

As a result, no player prefers to change its strategy at any of its information sets. Originally,
Kreps and Wilson (1982) defined the sequential rationality in finite games. We adapt their
definition to general multi-period games with observed actions.

Finally, Definition 6 defines the perfect regular equilibrium.

Definition 6 An assessment (μ, δ) is a perfect regular equilibrium if (μ, δ) is both 1)
regularly consistent and 2) sequentially rational.

5 Properties of the perfect regular equilibrium

The first property of the perfect regular equilibrium is that it always satisfies the weak
consistency, which is a criterion of the rational beliefs. Since the weak consistency was
originally defined in finite games, we start by extending its definition to general multi-period
games with observed actions.

Definition 7 An assessment (μ,δ) is weakly consistent¹⁶ if given each i, 1) μ∣ is the
same as y__i and 2) for each t > 2, μ^t_i satisfies the following functional equation: f f_A1 ∙ ∙ ∙
f_At-i Ib×a(Θ,o¹, ....a^{t l} A ^l⅛θ.a^l. ....a ^y ; da^{t l} ) ∙∙∙ δ¹(θ; da¹ )η(dθ) = f_&f_Ai ∙∙∙ J_a^₁ Ц^о¹.
.... a^i-1)μ((θ_i. a^l. .... a^{t l} ; B)δⁱ~¹(θ. a^l. .... a^t '²; da^t~¹f∙∙δ¹(θ; da^r)η(dθ') for every B ∈ ×j≠_iβ(f∂j)
and A ∈β(θ_i) × (×t', ^l_l ×(, _lA(Tt-)) where I_b×a{∙) and Ia{∙) are indicator functions, i.e.

I_b×a(Θ. a^l..... a^{t l} ) = 1 if (θ. a^l..... a^{t l} ) ∈ B × A and I_b×a(Θ, a^l..... a^{t l} ) = O otherwise.

In plain words, an assessment (μ. δ) is weakly consistent if 1) in the first period, each
player correctly forms its beliefs μ^l 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 all the previous action plans δ^l, ...,δ^{t l} and the probability
measure η given the information about its type and the previous actions (θ_i. a^l. .... a^{t l} ).

¹⁶ We adapt Myerson’s (1991, 4-3) definition to general multi-period games with observed actions.

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