Externalities, Nonconvexities, and Fixed Points

6.4 Graph ConvergenceBecause the hyperspace of all nonempty ρ ρZ ×w ∗-closed subsets of Z×X, P ρ Z ×w ∗ f(Z×X), is a compact metric space with Hausdorff metric h ρZ ×w∗, we can assume WLOGthat the sequence of graphs, {Grf n } n ⊂ P ρZ ×w ∗ f(Z × X), corresponding to ourpointwise converging, USCO bounded sequence of ρ Z -w ∗ -continuousfunctionsissuchthatGrf n → F ∗hρZ ×w ∗for some F ∗ ∈ P ρZ ×w ∗ f(Z×X). Moreover, because the sequence {f n } n AK-convergesto minimal Nash USCO η(·) =n(K(·)), we have corresponding to {f n } n a sequenceof USCOs, {ψ n } n ⊂ U ρZ -w ∗ with graphs {Grψn } n ⊂ P ρZ ×w ∗ f(Z × X) such thatGrf n ⊆ Grψ n for all n and such that {Grψ n } n decreases to the graph of the exteriorAK-limit USCO ψ ∞ ∈ QM ρZ -w ∗,whereGrψ∞ := ∩Grψ n . Thus, we have thatand becausewithwe can conclude thatGrf nGrψ n→ Grψ ∞ := ∩Grψ n ,hρZ ×w ∗→ F ∗ and Grψ n → Grψ ∞hρZ ×w ∗ hρZ ×w ∗Grf n ⊆ Grψ n for all n,F ∗ ⊆ Grψ ∞ .By Hola and Holy (2009 - e.g., see Corollary 3.3 - also see Appendix 1), our Baire1 selection, f ∗ ∈ ρ Z -s ∗ B 1 of the minimal Nash USCO, η(·) =n(K(·)), isinfactρ Z -w ∗ -quasicontinuous (i.e., f ∗ ∈ QC ρZ -w ∗)withGr ρZ ×w ∗f∗ = Grη.Also, because f ∗ (z) =w ∗ -lim n f n (z) for all z ∈ Z, wheref ∗ ∈ QC w ∗ -w ∗η(z) for all z ∈ Z, wealsohaveGr ρZ ×w ∗f ∗ ⊆ F ∗ implying thatand f ∗ (z) ∈Gr ρZ ×w ∗f ∗ = Grη ⊆ F ∗ .Thus, we haveor equivalently,Gr ρZ ×w ∗f ∗ = Grη ⊆ F ∗ ⊆ Grψ ∞ , (57)Gr ρZ ×w ∗f ∗ (z) =η(z) ⊆ F ∗ (z) ⊆ ψ ∞ (z) for all z ∈ Z, (58)where F ∗ (·) is the USCO induced by F ∗ ∈ P ρZ ×w ∗ f(Z × X), givenbyz → F ∗ (z) :={x ∈ X :(z,x) ∈ F ∗ } ,33