Externalities, Nonconvexities, and Fixed Points

(F1) ϕ(·, (·, ·)) is continuous on the compact metric space, Z × (X × X);(F2) ϕ(z,(·,x)) is quasiconcave in y on X.Given parameter z ∈ Z, the collective security mapping is given byy → Λ(z, y) :={x ∈ Φ(z) :ϕ(z,(y, x)) ≤ 0} .Moreover, given z, the graph of the collective security mapping is given byGrΛ(z, ·) :={(y,x) ∈ Φ(z) × Φ(z) :x ∈ Λ(z, y)}. (37)Thus, if (y 0 ,x 0 ) ∈ GrΛ(z, ·), then strategy profile x 0 ∈ Φ(z) is secure against defectionprofile y 0 ∈ Φ(z) and we haveϕ(z,(y 0 ,x 0 )) ≤ 0.Henceforth, for each z ∈ Z, wewilldenotebyK(z) the graph of the CS function. Wewill refer to the mapping, z → K(z), astheGCS function. Thus, the GCS functionis given byz → K(z) :=GrΛ(z,·) for all z ∈ Z. (38)The mapping K(·) is ρ Z -w ∗ × w ∗ -upper semicontinuous (i.e., usc) if for every setF, w ∗ × w ∗ -closed in Φ(z) × Φ(z), thesetK − (F ):={z ∈ Z : K(z) ∩ F 6= ∅}is ρ Z -closed in Z, or equivalently, if for every set G, w ∗ × w ∗ -open in Φ(z) × Φ(z),the setK + (G) :={z ∈ Z : K(z) ⊂ G}is ρ Z -open in Z. Also, recall that because Φ(z)×Φ(z) is w ∗ ×w ∗ -compact, K(·) is usc ifand only if K(·) has a ρ Z -w ∗ ×w ∗ -closed graph. Finally, note that (z,(y, x)) ∈ GrK(·)if and only ifϕ(z, (y, x)) ≤ 0.Thus, it follows immediately from the continuity of the function ϕ(·, (·, ·)) on Z ×(X × X) (see property F1 above) that K(·) has a ρ ρZ ×w ∗ ×w∗-closed graph whereρ ρZ ×w ∗ ×w ∗ := ρ Z + d w ∗ ×w ∗ and hence that K(·) is usc.Lemma 7 (The KFC mapping, K(·), isuppersemicontinuous)Suppose assumptions [A-1] hold. Then the mappingis ρ Z -w ∗ × w ∗ -upper semicontinuous.z → K(z)21