Externalities, Nonconvexities, and Fixed Points

(b) given any minimal USCO, n(·), of the KFC USCO, N(·), z → n(K(z)), isaminimal USCO of N (·).(2) (Approximation) Next, we establish that any Nash USCO, N (·), possessesan approximating triple, (η,f ∗ , {f n } n ) N ,whereη(·) =n(K(·)) ∈ [N (·)], f ∗ (·) is aquasicontinuous selection of η(·), and{f n } n is a sequence of continuous functionsconverging pointwise to f ∗ . The existence of the minimal USCO η(·) is guaranteedby Drewnowski and Labuda (1990), the existence of a quasicontinuous selection f ∗ isguaranteed by Jayne and Rogers (2002) and Hola and Holy (2009), and the existenceof a pointwise converging sequence of continuous functions is guaranteed by Spruny(2007). We then show that for anyF ∗ ∈ Ls ρZ ×w ∗{Grf n } ⊂ P ρZ ×w ∗ f(Z × X)with induced USCO, F ∗ (·), the following statements are true:(a) For each z ∈ Z, F ∗ (z) is a dendrite (locally connected, connected, and withoutclosed curves) and is the unique irreducible subcontinua containing η(z).(b) For each z ∈ Z, F ∗ (z) ⊆ N (z) for all z ∈ Z.(c) For each z ∈ Z, F ∗ (z) is minimally essential for N (z) in C w ∗ f(X).The proof of (a) rests on new convergence results for minimal USCOs due toAnguelov and Kalenda (2009), a new result due to Hola and Holy (2011) establishingthe equi-quasicontinuity of the pointwise approximating sequence of continuous functions,{f n } n , in the approximating triple, (η,f ∗ , {f n } n ) N , and the dense selectionresult due to Beer (1983). The proof of part (b) rests on the fact that the inducedminimal CUSCO, F ∗ (·), is dendritically valued and the fact that the KFC mapping,N(·), underlying the Nash correspondence, N (·), is3M.(3) (Fixed Points) We show that the fact that the Nash correspondence is approximableimplies that if the parameter space, Z, and the space of strategy profiles,X, are one in the same, then the Nash USCO, N (·), hasfixed points. Thus, we showthat there exists at least one z ∗ ∈ Z such that z ∗ ∈ N (z ∗ ).(4) (An Application to Network Formation Games) In a belief-parameterizedcollection of random sender-receiver network formation games, we use our fixed pointresult to show that there exists a fulfilled expectations Nash equilibrium.3 The Nash CorrespondenceGiven parameter z and given the profile of strategy choices made by other players,player i 0 s choice problem is given byx −i ∈ Q i 0 6=i Φ i0(z), (18)max xi ∈Φ i (z) u i (z,(x i ,x −i )). (19)17