Perfect Regular Equilibrium



δt~1μtΓ1

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 : Θt × A1 × ∙ ∙ ∙ × At 1 ×
Φ ×
∏ —> R (= R U {-∞, ∞}) be a conditional expected utility functional15 where Φ × ∏
is the set of all assessments. That is, for each (θt, a1,..., at 1 ) Θt × A1 × ∙ ∙ ∙ × At 1,
Et(θt, a1, ...,at~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 × {(θt, a1,..., at 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
(θt, a1,..., at 1) Θt × A1 × ∙ ∙ ∙ × At 1, we have
Efiθ
t,a1,...,at-1μ,δ~)Eti(θi, a1,..., at~1 ; μ, (δ'i-tfi) for every δ'i i

Here, a set Θ-t × {(θt,a1,...,at 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 δt with respect to the system of beliefs μ, which would induce
the greatest expected utility value given any of its information sets
Θ-t × {(θt, a1,..., at 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 Eti is a unique function satisfying
the following conditions for any arbitrary strategy profile
δ. First, if Efiθi, a1,..., Cit~1; μ, δ) is finite, then, for
any
ε > 0, there exist both a period t' {t,..., T} and a sequence of actions t +1,..., aτ) At +1 × ∙ ∙ ∙ × Aτ
such that for any t'' t',

I Eii, a ,..., a ; μ, δ ʃθ Ja^ ∙ ∙ ∙ JAirr ufiθi, θi, a ,..., a , a ,..., a ,h ,...,
aτ )δt (θ i-i,cι 1,...,at -1; dat ) ∙∙∙ δt(θ ii,Cι, 1,...,a t~1; dafiμti(θ i,Cι 1,...,C t~1; d.θ i) < ε.

Second, if Eit(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 (at +1,..., aτ) At +1 × ∙ ∙ ∙ × Aτ such that for any t''t',

Jh JAi ∙ ∙ ∙ JAi" ui(θi-i, Ci 1,..., Cit~1,at, ...,at'', at''+1, ...,Ct)δt (θi, θ~i,a1,

..., at -1; dat ) ∙ ∙ ∙ δt(θi, θ~i, d1,..., dt~1; dat')μti(θi, a1,..., dt~1; i )

>M when E*(θ i,a 1,...,at-1; μ, δ) =and -M when Eti(θ i,a 1,..,at~1; μ,δ) = -∞.

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

21



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