Externalities, Nonconvexities, and Fixed Points

We will denote by M N( e E)(S) the collection of all minimal essential sets of N( e E).Note that for any E ∈ S, ifB is a proper subset of m(E), thenB/∈ E N(E) (S).The following lemma establishes a fundamental fact about minimal USCOs: anyminimal USCO corresponding to any KFC mapping is minimally essentially valued.Lemma 10 (The Connection Between a KFC’s Minimal USCO and Minimal EssentialSets)Suppose assumption [A-1] holds. If n(·) is a minimal USCO of KFC N(·), thenfor any E ∈ S, n(E) ∈ M N(E) (S).Proof. Because n(·) is an USCO, for each E ∈ S, n(E) ∈ E n(E) (S) (i.e., n(E) is anessential subset of itself). To see that n(E) is minimal, suppose that for some E 0 ∈ Sthere is a nonempty, closed, and proper subset m(E 0 ) of n(E 0 ) such that for anyε > 0 there exists a δ > 0 such that for all E contained in B hw ∗ ×w ∗ (δ,E 0 ) ∩ Sn(E) ∩ B w ∗(ε,m(E 0 )) 6= ∅. (43)Because m(E 0 ) is a proper subset of n(E 0 ),wecanchooseε 0 > 0 so that n(E 0 )∩B w ∗(ε 0 ,m(E 0 )) is a proper subset of n(E 0 ), and by (51) we can choose δ 0 > 0 sothatn(E) ∩ B w ∗(ε 0 ,m(E 0 )) 6= ∅ for all E ∈ B hw ∗ ×w ∗ (δ 0 ,E 0 ) ∩ S,define the mapping q(·) as follows:q(E) :=½ n(E) ∩ Bw ∗(ε 0 ,m(E 0 ))n(E)if E ∈ B hw ∗ ×w ∗ (δ 0 ,E 0 ) ∩ Sotherwise.By Lemma 2(ii) in Anguelov and Kalenda (2009), q(·) is an USCO, and moreover,Grq(·) is a proper subset of Grn(·), a contradiction of the fact that n(·) ∈ [N(·)].Thus, for each E ∈ S, n(E) ∈ M n(E) (S), implying that for all E ∈ S, n(E) ∈M N(E) (S).Given E e ∈ S, letNE e (·) denote the restriction of KFC N(·) to the D-equivalenceclass of Ky Fan sets, S eE , determined by E. e Notethatifn(·) is a minimal USCO ofKFC N(·), thenforanyKyFansetE e ∈ S, n eE (·) (the restriction of n(·) to S eE )isaminimal USCO of N eE (·) - a fact established in the following Lemma whose proof isgiven in Appendix 3.Lemma 11 (The Connection Between an KFC’s Minimal USCO and a D-restrictedUSCO’s Minimal Essential Sets)Suppose assumption [A-1] holds. If n(·) is a minimal USCO of KFC N(·), thenfor any e E ∈ S and any E ∈ S eE , n eE (E) ∈ M N eE (E)(S eE ).5.1 The 3M Property of KFCsWebeginwiththedefinition.25