Externalities, Nonconvexities, and Fixed Points

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Letting E 1 =[E 1 \(X × U 2 )] ∪ £ E 1 ∩ (X × U 2 ) ¤ ,wehaveforall(y, x) ∈ E 1 \(X × U 2 ),ρ w ∗ ×w ∗((y, x),E∗ )= ρ w ∗ ×w ∗((y, x), [E1 \(X × U 2 )] ∪ [E 2 \X × U 1 )])≤ ρ w ∗ ×w ∗((y, x), [E2 \(X × U 1 )] ∪ [E 2 ∩ (X × U 1 )])= ρ w ∗ ×w ∗((y, x),E2 ).Moreover, we have for all(y, x) ∈ E 1 ∩ (X × U 2 ),ρ w ∗ ×w ∗((y, x),E∗ )= ρ w ∗ ×w ∗((y, x), [E1 \(X × U 2 )] ∪ [E 2 \(X × U 1 )])and= ρ w ∗ ×w ∗((y, x), [E2 \(X × U 1 )]),ρ w ∗ ×w ∗((y, x),E2 )Thus, for all (y, x) ∈ E 1 ,= ρ w ∗ ×w ∗((y, x), [E2 \(X × U 1 )] ∪ [E 2 ∩ (X × U 1 )])= ρ w ∗ ×w ∗((y, x), [E2 \(X × U 1 )]).ρ w ∗ ×w ∗((y, x),E∗ ) ≤ ρ w ∗ ×w ∗((y, x),E2 ),implying that e w ∗ ×w ∗(E1 ,E ∗ ) ≤ e w ∗ ×w ∗(E1 ,E 2 ). Together,e w ∗ ×w ∗(E1 ,E ∗ ) ≤ e w ∗ ×w ∗(E1 ,E 2 )ande w ∗ ×w ∗(E∗ ,E 1 ) ≤ e w ∗ ×w ∗(E2 ,E 1 )imply thatThus, we haveh w ∗ ×w ∗(E∗ ,E 1 ) ≤ h w ∗ ×w ∗(E2 ,E 1 ) < 2δ 0 .h w ∗ ×w ∗(E∗ ,E 0 ) ≤ h w ∗ ×w ∗(E∗ ,E 1 )+h w ∗ ×w ∗(E1 ,E 0 )≤ h w ∗ ×w ∗(E2 ,E 1 )+h w ∗ ×w ∗(E1 ,E 0 )< 2δ 0 + δ 0 < 3δ 0 .56
(3) N eE (E ∗ ) ∩ £ U 1 ∪ U 2¤ 6= ∅⇒N eE (E i ) ∩ U i 6= ∅ for some i =1and/or 2:WLOG suppose that x ∈ N eE (E ∗ ) ∩ U 1 . Give the definition of the KFC, N eE (·),we have for each y ∈ D(E ∗ ),(y, x) ∈ ¡ E 1 ∩ ¡ X × U 2¢ c ¢ ∪ ¡ E 2 ∩ ¡ X × U 1¢ c ¢ ,and because x ∈ U 1 , this implies that for each y ∈ D(E ∗ ),and specifically, that for each y ∈ D(E ∗ ),(y, x) ∈ E 1 ∩ ¡ X × U 2¢ c,(y, x) ∈ E 1 ∩ ¡ X × U 1¢ .(*)Thus, given the definition of the D-Restricted KFC, N eE (·), (*) implies thatx ∈ N e E (E1 ) ∩ U 1 ,contradicting the fact that N eE (E 1 ) ∩ U 1 = ∅. Thus we must conclude that N eE (·) hasthe 3M property.References[1] Aliprantis, C. D. and Border, K. C. (2006) Infinite Dimensional Analysis: AHitchhiker’s Guide, 3rd Edition, Springer-Verlag, Berlin-Heidelberg.[2] Amir, R. (1996) “Cournot oligopoly and the theory of supermodular games,”Games and Economic Behavior 15, 132-148.[3] Amir, R. and Lazzati, N. (2011) “Network Effects, Market Structure and IndustryPerformance,” Journal of Economic Theory 146, 2389-2419.[4] Anguelov, R. and Kalenda, O. F. K. (2009) “The Convergence Space of MinimalUSCO Mappings,” Czechoslovak Mathematical Journal 134, 101-128.[5] Beer, G. (1983) “Dense Selections,” Journal of Mathematical Analysis and Applications95, 416-427.[6] Beer, G. (1993) Topologies on Closed and Closed Convex Sets, KluwerAcademicPublishers, Dordrecht.[7] Berge, C. (1997) Topological Spaces, First Dover Edition, Dover Publishers, Mineola(first published by Dunod, Paris, 1962).[8] Bing, R. H. (1949) “A Convex Metric for a Locally Connected Continuum”Bulletin of the American Mathematical Society 55, 812-819.[9] Blumenthal, M. (1953) Distance Geometry, Oxford University Press, Oxford.57
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cally approximated by continuous fu
Page 12 and 13: the set x 0 x 1 ∈ C w ∗ f(X) is
Page 14 and 15: defined on some probability space,
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 24 and 25: Because Φ(z) × Φ(z) is w ∗ ×
Page 26 and 27: where for all n, C n ∈ S E n,then
Page 28 and 29: Definition 4 (The 3M Property - The
Page 30 and 31: Given (48) and the fact that for n
Page 32 and 33: We note that if m(z) ∈ P w ∗ f(
Page 34 and 35: 6.3 AK Convergence of Minimal Nash
Page 36 and 37: and Gr ρZ ×w ∗f ∗ (·) is the
Page 38 and 39: In particular, for all n ≥ N 0 an
Page 40 and 41: the cutting defined by the cut poin
Page 42 and 43: According to our main result, under
Page 44 and 45: By our main approximation result, f
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 60 and 61: [10] Bryant, V. W. (1970) “The Co
Page 62: [42] Ward, L. E., Jr. (1958) “A F
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Externalities, Nonconvexities, and Fixed Points

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