Externalities, Nonconvexities, and Fixed Points

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Because Φ(z) × Φ(z) is w ∗ × w ∗ -compact, K(·) is usc if and only if the excessfunction, e w ∗ ×w ∗(K(·),K(z0 )), isρ Z -continuous at each z 0 ∈ Z (see remark 1.5 inDe Blasi and Myjak, 1986). The mapping e w ∗ ×w ∗(K(·),K(z0 )), isρ Z -continuous ateach z 0 ∈ Z if for all ε > 0, thereexistsaδ > 0 such thate w ∗ ×w ∗(K(z),K(z0 )) < ε for all z ∈ B δ (z 0 ),where e w ∗ ×w ∗(K(z),K(z0 )) is the excess of K(z) over K(z 0 ) or the sup over distancesfrom points in K(z) to points in K(z 0 ) (see 6 and 5 above).By Proposition 6.3.11 in Beer (1993), because Z is a ρ Z -compact metric space(and hence a Baire space) and because Φ(z) × Φ(z) is w ∗ × w ∗ -compact (and henceseparable), there is a w ∗ -dense G δ subset C of Z where K(·) is ρ Z -w ∗ ×w ∗ -continuous(i.e., K(·) is both usc and lsc - lower semicontinuous on C - also see Fort, 1951-1952).Moreover, the mapping K(·) is ρ Z -w ∗ × w ∗ -continuous on C if and only if K(·) isρ Z -h w ∗ ×w∗-continuous on C (see remark 1.9 in De Blasi and Myjak, 1986). K(·) isρ Z -h w ∗ ×w ∗-continuous (or Hausdorff continuous) at z0 ∈ C if for all ε > 0 there is aδ > 0 such that4.3 Ky Fan Setsh w ∗ ×w ∗(K(z),K(z0 )) < ε for all z ∈ B δ (z 0 ).Given any set E ∈ P w ∗ ×w ∗ f(X × X), define the domain of E to be the set 14D(E) :={y ∈ X :(y, x) ∈ E for some x ∈ X} ∈ P w ∗ f(X).Define the range of E to be the setR(E) :={x ∈ X :(y, x) ∈ E for some y ∈ X} ∈ P w ∗ f(X).The mappings D(·) and R(·) are h w ∗ ×w ∗-h w∗-continuous. To see this, simply notethat if h w ∗ ×w ∗(En ,E 0 ) → 0, thenforevery(y 0 ,x 0 ) ∈ E 0 there exists a sequence{(y n ,x n )} n such that (y n ,x n ) −→w ∗ ×w (y0 ,x 0 ) and (y n ,x n ) ∈ E n .∗Definition 2 (Ky Fan Sets)AsetE ∈ P w ∗ ×w ∗ f(X ×X) is a Ky Fan set if E satisfies the following properties:(Z1) D(E) is nonempty, w ∗ -closed, and convex and D(E) =R(E);(Z2) for all y ∈ D(E), (y, y) ∈ E;(Z3) for all x ∈ R(E), {y ∈ D(E) :(y, x) /∈ E} is convex (possibly empty).We will denote by S the collection of all Ky Fan sets in P w ∗ ×w ∗ f(X × X). Thus,S := {E ∈ P w ∗ ×w ∗ f(X × X) : E satisfies (Z1)-(Z3)} ,and it follows from Lemma 4 in Ky Fan (1961) that if E ∈ S, then∩ y∈D(E) {x ∈ R(E) :(y, x) ∈ E} 6= ∅.14 Here, P w ∗ ×w ∗ f (X × X) denotes the collection of all nonempty, w ∗ × w ∗ -closed subsets of X × X(see Appendix 6).22
Lemma 8 (The Space of Ky Fan Sets)Suppose assumptions [A-1] hold. The following statements are true.(1) S is a h w ∗ ×w ∗-closed subspace of P w ∗ ×w ∗ f(X × X).(2) For all z ∈ Z, K(z) is a Ky Fan set such that for all z ∈ Z,D(K(z)) = Φ(z) and R(K(z)) = Φ(z).For a proof of Lemma 8 see Appendix 2.4.4 Ky Fan CorrespondencesConsider the correspondence,E → N(E) :=∩ y∈D(E) {x ∈ R(E) :(y,x) ∈ E} , (39)defined on S taking values in P w ∗ f(X). We will call the correspondence N(·) from Sinto P w ∗ f(X) the Ky Fan Correspondence (i.e., the KFC).Lemma 9 (The KFC is an USCO)Under assumption [A-1], the KFC, N(·) is an USCO, that is,N(·) ∈ U(S,P w ∗ f(X)). (40)Proof. By Ky Fan (1961) N(E) is nonempty for all E ∈ S and it is easy to seethat N(E) is compact for all E ∈ S. To see that N(·) is upper semicontinuousconsider a sequence {(E n ,x n )} n ⊂ GrN(·) where {E n } n ⊂ S and WLOG assumethat E n E 0 ,andx n → x0 . By(39)wehaveforeachn, (y, x n ) ∈ E n forw ∗→hw ∗ ×w ∗any y ∈ D(E n ).Bytheh w ∗ ×w ∗-h w ∗-continuity of D(·), wehaveforanyy0 ∈ D(E 0 )a sequence {y n } n with y n ∈ D(E n ) for all n and y n →w ∗ y0 . This, together withE n→hw ∗ ×w ∗E 0 and x n →w ∗x0 ,implythat(y 0 ,x 0 ) ∈ E 0 for any y 0 ∈ D(E 0 ). Thus,(E 0 ,x 0 ) ∈ GrN(·). By compactness, the fact that GrN(·) is closed implies that N(·)is upper semicontinuous - with nonempty, w ∗ -compact values.4.5 D-Equivalence Classes of Ky Fan SetsGivenKyFansetE ∈ S, wedefine the D-equivalence class of Ky Fan sets, S E ,asfollows:S E := {E 0 ∈ S : D(E 0 )=D(E)} . (41)Because D(·) h w ∗ ×w ∗-h w ∗-continuous, it is easy to show that S E is a h w ∗ ×w ∗-closedsubset of S. Thus, if {E n } n is a sequence of Ky Fan sets in SE e for some E e ∈ S,then E n E 0 implies that E 0 ∈ S eE . Also, viewing S (·) as a mapping from S→hw ∗ ×w ∗into D-equivalence classes of Ky Fan sets, it is easy to show that S (·) has a h w ∗ ×w ∗-h w ∗ ×w∗-closed graph. In particular, ifh w ∗ ×w ∗(En ,E 0 )+h w ∗ ×w ∗(Cn ,C 0 ) → 0,23
Page 5: cally approximated by continuous fu
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Page 16 and 17: contains at most one arc (i.e., ¯
Page 18 and 19: into the collection of Ky Fan sets
Page 20 and 21: 3.1 Best Response MappingsLetting p
Page 22 and 23: 4.1 Nikaido-Isoda FunctionsWith eac
Page 26 and 27: where for all n, C n ∈ S E n,then
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Page 32 and 33: We note that if m(z) ∈ P w ∗ f(
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Page 38 and 39: In particular, for all n ≥ N 0 an
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Page 46 and 47: 9 Appendix 1: USCO FundamentalsIn t
Page 48 and 49: 9.3 Equi-QuasicontinuityIn order to
Page 50 and 51: 9.5 Dense SelectionsFor each F ∈
Page 52 and 53: 10 Appendix 2: The Proof of Lemma 8
Page 54 and 55: Noting that if E ∈ D eE ,thenn eE
Page 56 and 57: 12 Appendix 4: The Proof That All K
Page 58 and 59: Letting E 1 =[E 1 \(X × U 2 )] ∪
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Externalities, Nonconvexities, and Fixed Points

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