Externalities, Nonconvexities, and Fixed Points

and where the KFC, N(·) :S → P w ∗ f(X), is an USCO with values given byN(E) =∩ y∈D(E) {x ∈ R(E) :(y, x) ∈ E} ,foreachE ∈ S. (46)Thus, given the GCS function K(·), wehaveforallz ∈ ZN (z) =N(K(z)) = ∩ y∈Φ(z) {x ∈ Φ(z) :(y, x) ∈ K(z)} ,where K(z) ∈ S z ⊆ S and D(K(z)) = R(K(z)) = Φ(z).We will denote byU 3M := U 3M (S,P w ∗ f(X)) (47)the set of all 3M USCOs defined on the hyperspace of Ky Fan sets, S, withvaluesinP w ∗ f(X).We have shown that all KFCs are 3M and that all minimal USCOs contained ina KFC have minimally essential values. Thus, we know that because K(z) ∈ S forall z, the induced USCO, z → n(K(z)), has minimally essential values on Z. For ourfinal result of this section, we show that if n(·) ∈ [N(·)], thenn(K(·)) ∈ [N (·)] forGCS mapping, K(·), corresponding to the parameterized collection.Theorem 13 (Minimal KFCs and Minimal Nash USCOs)Suppose the collection of z-games, {G z : z ∈ Z}, satisfies assumption [A-1] withcorresponding Nash USCO N (·) =N(K(·)) whereN(·) ∈ U 3M (S,P w ∗ f(X))is the KFC and K(·) is the GCS function. Then,n(K(·)) ∈ [N (·)] for all n(·) ∈ [N(·)].Proof. Suppose not and let m(·) be a minimal USCO of n(K(·)) ∈ U(Z, P w ∗ f(X))(i.e., an USCO defined on Z with values in P w ∗ f(X) such that m(·) ∈ [n(K(·))]) suchthat for some z 0 ∈ Z, m(z 0 ) is a proper subset of n(K(z 0 )). By Lemma 10, becausem(·) ∈ [n(K(·))], m(z 0 ) ∈ M n(K(z 0 ))(P w ∗ f(X)) implying that for any ε 0 > 0 thereexists δ 0 > 0 such that for allz δ0 ∈ B ρZ (δ 0 ,z 0 ⎫) ∩ Z, ⎬(48)n(K(z δ0 )) ∩ B w ∗(ε 0 ,m(z 0 ⎭)) 6= ∅.Becausen(K(z δ0 )) ∩ B w ∗(ε 0 ,m(z 0 ))is a closed subset of both n(K(z δ0 )) and B w ∗(ε 0 ,m(z 0 )), n(K(z δ0 )) ∩ B w ∗(ε 0 ,m(z 0 ))is not an essential for n(K(z δ0 )) in Z. Therefore, there is some ε 1 > 0 such that foreach n there exists z δn ∈ B ρZ ( 1 n ,zδ0 ) ∩ Z such thathin(K(z δn )) ∩ B w ∗ ε 1 ,n(K(z δ0 )) ∩ B w ∗(ε 0 ,m(z 0 )) = ∅. (49)27