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
Efiθt,a1,...,at-1∙ μ,δ~) ≥ Eti(θi, a1,..., at~1 ; μ, (δ'i,δ-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 (αt +1,..., aτ) ∈ At +1 × ∙ ∙ ∙ × Aτ
such that for any t'' ≥ t',
I Eii, a ,..., a ; μ, δ ʃθ Ja^ ∙ ∙ ∙ JAirr ufiθi, θ—i, a ,..., a , a ,..., a ,h ,...,
aτ )δt (θ i,θ-i,cι 1,...,at -1; dat ) ∙∙∙ δt(θ i,θ i,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,θ-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; dθ 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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