Loeb-measurable solutions to ∗finite games

Abstract

Let ∗M be a denumerately comprehensive enlargement of a set-theoretic structure sufficient to model R. If F is an internal ∗finite subset of ∗N such that F = {1,…,γ}, γϵ ∗N−N , we define a class of ∗finite cooperative games having the form Γ F( ∗ν) = 〈F,A(F), ∗ν〉 , where A( F) is the internal algebra of the internal subsets of F, and ∗ν is a set-function with Dom ∗ν=A(F) , Rng ∗ν = ∗R + , and ∗ν(Ø) = 0 . If SI( ∗ν) is the space of S-imputations of a game Γ F( ∗ν) such that ∗ν(F)<η , for some ηϵ ∗N , then we prove that SI( ∗ν) contains two nonempty subsets: QK(Γ F( ∗ν)) and SM 1(Γ F( ∗ν)) , termed the quasi-kernel and S-bargaining set, respectively. Both QK(Γ F( ∗ν)) and SM 1(Γ F( ∗ν)) are external solution concepts for games of the form Γ F ( ∗ν) and are defined in terms of predicates that are approximate in infinitesimal terms. Furthermore, if L( Θ) is the Loeb space generated by the ∗finitely additive measure space 〈 F, A( F), U F 〉, and if a game Γ F( ∗ν) has a nonatomic representation ψ( ∗ ν −0) on L( Θ) with respect to S-bounded transformations, then the standard part of any element in QK(Γ F( ∗ν)) is Loeb-measurable and belongs to the quasi-kernel of ψ( ∗ ν −0) defined in standard terms.