Perfect Regular Equilibrium

δ^t~¹μ^tΓ¹

Next, we formally define the sequential rationality in general games. For notational
simplicity, given a player i and a period t, let a functional Et : Θ_t × A¹ × ∙ ∙ ∙ × A^{t 1} ×
Φ × ∏ —> R (= R U {-∞, ∞}) be a conditional expected utility functional¹⁵ where Φ × ∏
is the set of all assessments. That is, for each (θ_t, a¹,..., a^{t 1} ) ∈ Θ_t × A¹ × ∙ ∙ ∙ × A^{t 1},
Et(θ_t, a¹, ...,a^t~¹; μ, δ) denotes an expected utility value with respect to the system of beliefs μ
and the strategy profile δ conditional on reaching the information set Θ-_t × {(θ_t, a¹,..., a^{t 1})}.
Definition 5 A strategy profile δ is sequentially rational with respect to a system of belief's
μ if, given each i and t, and given each (θ_t, a¹,..., a^{t 1}) ∈ Θ_t × A¹ × ∙ ∙ ∙ × A^{t 1}, we have
Efiθt,a¹,...,a^t-¹∙ μ,δ~) ≥ E^t_i(θ_i, a¹,..., a^t~¹ ; μ, (δ'_i,δ-tfi) for every δ'_i ∈ ∏_i∙

Here, a set Θ-_t × {(θ_t,a¹,...,a^{t 1} )} denotes an information set of player i. Thus, the
sequential rationality requires that, in responding to the other players’ strategies δ~_t, each
player i make its best response δ_t with respect to the system of beliefs μ, which would induce
the greatest expected utility value given any of its information sets Θ-_t × {(θ_t, a¹,..., a^{t 1})}.

¹⁵ Formally, the conditional expected utility functional is implicitly defined just like the expected utility
functional. So, given i and t, the conditional expected utility functional E^t_i is a unique function satisfying
the following conditions for any arbitrary strategy profile δ. First, if Efiθi, a¹,..., Ci^t~¹; μ, δ) is finite, then, for
any ε > 0, there exist both a period t' ∈ {t,..., T} and a sequence of actions (α^{t +1},..., a^τ) ∈ A^{t +1} × ∙ ∙ ∙ × A^τ
such that for any t'' ≥ t',

I ^Eii^{, a ,}...^, a ^{; μ, δ} ʃθ Ja^ ∙ ∙ ∙ JAi^{rr u}fi^θi^{, θ}—i^{, a ,}...^{, a , a ,}...^{, a} ,h ^,...^,
a^τ )δ^t (θ i,θ-i,cι ¹,...,a^{t -1}; da^t ) ∙∙∙ δ^t(θ _i,θ _i,Cι, ¹,...,a ^t~¹; dafiμ^t_i(θ _i,Cι ¹,...,C ^t~¹; d.θ _i) ∣< ε.

Second, if E_i^t(θ i,a ¹,..,Cl ^t~¹; μ,δ) is infinite, then, for any M ∈ N, there exist both a period t' ∈ {t,...,T}
and a sequence of actions (a^{t +1},..., a^τ) ∈ A^{t +1} × ∙ ∙ ∙ × A^τ such that for any t'' ≥ t',

J_h J_Ai ∙ ∙ ∙ J_Ai" ^ui(θi,θ-i, Ci ¹,..., Ci^t~¹,a^t, ...,a^t'', a^t''⁺¹, ...,C^t)δ^t (θi, θ~i,a¹,

..., a^{t -1}; da^t ) ∙ ∙ ∙ δ^t(θ_i, θ~_i, d¹,..., d^t~¹; da^t')μ^t_i(θ_i, a¹,..., d^t~¹; dθ _i )

>M when E*(θ i,a ¹,...,a^t-1; μ, δ) = ∞ and < -M when E^t_i(θ i,a ¹,..,a^t~¹; μ,δ) = -∞.

Again, this definition of the conditional expected utility functional makes sense according to Ash (1972, 2.6).

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