Externalities, Nonconvexities, and Fixed Points

with the Hausdorff metric h ρZ ×w ∗ (induced by ρ Z × w ∗ )isalsoacompactmetricspace. We note that P ρZ ×w ∗ f(Z ×X) is a h ρZ ×w ∗-closed subspace of P ρ Z ×w ∗ f(Z ×X)(see the Aliprantis and Border, 2006, for a more detailed development of the compactmetric hyperspaces in general).Because the compact metric spaces Z and X are Peano continua, the compactmetric space, Z × X, equipped with the metric ρ Z × w ∗ is also a Peano continuum(see Willard, 1970, Sections 26 - 28). 6 Because ρ Z and ρ w ∗ are convex metrics, it iseasy to see that ρ Z × w ∗ := ρ Z + ρ w ∗ is a convex metric. Thus, (Z × X, ρ Z × w ∗ ) isa convex Peano continuum.Recall that the Hausdorff metric h ρZ ×w ∗ on P ρ Z ×w ∗ f(Z×X) induced by the metricρ Z × w ∗ on Z × X is given byh ρZ ×w ∗(E0 ,E 1 ):=max{e ρZ ×w ∗(E0 ,E 1 ),e ρZ ×w ∗(E1 ,E 0 )},for any E 0 and E 1 in P ρZ ×w ∗ f(Z × X), where the excess of E i over E i0the excess functionalis given bye ρZ ×w ∗(Ei ,E i0 ):=sup (z i ,x i )∈E i dist ρ Z ×w ∗((zi ,x i ),E i0 ) (5)for i =0or 1 and i 0 ∈ {0, 1}\{i}, and where the distance from (z i ,x i ) to E i0 is givenbydist ρZ ×w ∗((zi ,x i ),E i0 ):=inf (z i 0 ,x i0 )∈E ρ i0 ρ Z ×w ∗((zi ,x i ), (z i0 ,x i0 )),o(6)=inf (z i 0 ,x i0 )∈Enρ i0 Z (z i ,z i0 )+ρ w ∗(x i ,x i0 ) .for i, i 0 =0or 1 and i 6= i 0 (Chapter 3 in Beer, 1993). Making the obvious modificationsto the above, we obtain the definition of the Hausdorff metric, h Z , on P ρZ f(Z)and the Hausdorff metric, h w ∗, on P w ∗ f(X).Because P ρZ ×w ∗ f(Z × X) is compact, any sequence, {E n } n ,inP ρZ ×w ∗ f(Z × X)h ρZ ×w ∗-converges to E0 ∈ P ρZ ×w ∗ f(Z × X), denotedbyE n→ E 0 or h ρZ ×w ∗(En ,E 0 ) → 0,hρZ ×w ∗if and only if {E n } n converges to E 0 in the sense of Kuratowski-Painleve, denoted byLi ρZ ×w ∗En = E 0 = Ls ρZ ×w ∗En . (7)Expression (7) holds if and only if (i) Li ρZ ×w ∗En = E 0 :foreache 0 ∈ E 0 there existsa positive integer, N, and a sequence of elements {e n } n ρ Z ×w ∗ -converging to e 0 suchthat e n ∈ E n for all n ≥ N, and(ii)Ls ρZ ×w ∗En = E 0 : whenever n 1