Externalities, Nonconvexities, and Fixed Points

a fixed point theorem for the Nash correspondence of a parameterized game satisfyingassumptions [A-1]. Our method of proof is similar in spirit to that of Cellina(1969), who proved the Glicksberg-Kakutani Fixed Point Theorem (1952) via graphicalapproximation by continuous functions. But here, rather than requiring that thecorrespondence be convex-valued in order to construct our approximating sequenceof continuous functions, we instead rely on the machinery developed in the sectionsabove to provide us with such a sequence. Thus, here we prove a proof of a new fixedpoint theorem without convexity - closely related to Ward’s Fixed Point Theorem(1958, 1961) - via graphical approximation by continuous functions. The problem ofgraphically approximating, by continuous functions, a nonconvex valued USCO hasbeen a long standing open question. As Cellina is to Glicksberg-Kakutani, our resultis to Ward-type of fixed point theorems.We will maintain the following assumptions [A-2] in our treatment the fixed pointproblem:(1) assumptions [A-1] remain in force;(2) the strategy profile space, X := X 1 ×···×X m , and the parameter space,Z, areoneandthesamespace.Note that under assumptions [A-2], the space Z (and hence the space X) satisfyour prior assumptions [A-1].We will take as our starting point the convex, dendrite continuum (Z, ρ w ∗) whereρ w ∗ is the convex metric compatible with the compact metrizable weak star topologyin Z. Our main result is the following:Theorem 17 (All Nash USCOs have fixed points)Let {G z : z ∈ Z} be a collection of parameterized strategic form games satisfyingassumptions [A-2] with Nash correspondence, z → N (z). ThenN has fixed points.Proof. Let(η,f ∗ , {f n } n ) N ,be an approximating triple for the Nash USCO N ∈ U w ∗ -w ∗,whereη ∈ [N ] w ∗ -w ∗, f ∗ isa w ∗ -s ∗ B 1 selector of η, and{f n } n ⊂ C w ∗ -s∗(Z, Z) is an USCO bounded sequence suchthat kf n (z) − f ∗ (z)k ∗ → 0 for all z. The existence of such an approximating triplefor N ∈ U w ∗ -w ∗ is guaranteed by assumptions [A-2], Proposition 4.3 in Drewnowskiand Labuda (1990), Theorem 3.3 in Jayne and Rogers (2002), and Theorem 1.2 inSpruny (2007).Let {Grf n k} k be an h w ∗ ×w ∗-convergent subsequence with h w ∗ ×w ∗-limit F ∗ ⊂GrN such that for all k,h hw ∗ ×w ∗ (Grf n k,F ∗ ) < 1 k .By h w ∗ ×w ∗-compactness of P w ∗ ×w ∗ f(Z ×Z), such a subsequence exists. Thus, for anyε > 0, there exists a K ε large enough so that for every z ∈ Z there exists (z 0 ,z 00 ) ∈ F ∗such thatρ w ∗(z,z 0 )+ρ w ∗(f n k(z),z 00 ) < 1 k . (72)41