Noting that if E ∈ D eE ,thenn eE (E) ∩ B w ∗(ε 0 ,m(E 0 )) = {x E } for some x E ∈ D(E) =D( e E) <strong>and</strong>wehaveforeachE ∈ B w ∗ ×w ∗(δ0 ,E 0 ) ∩ D e E ,m(E) =n eE (E) ∩ B w ∗(ε 0 ,m(E 0 )) = {x E } ∈ D(E). (85)Because m(E 0 ) is a proper subset of n e E (E0 ),wecanchooseε 0 > 0 so that for k EE0such that1k EE0 < δ 0 ,wehaveforallk ≥ k EE0 <strong>and</strong> all E ∈ B w ∗ ×w ∗( 1 k ,E0 ) ∩ D eEn eE (E) ∩ B w ∗(ε 0 ,m(E 0 )), a nonempty proper subset of n eE (E 0 ).Also, because n e E (E0 ) is minimally essential for N(E 0 ) on S, [n e E (E0 )∩B w ∗(ε 0 ,m(E 0 ))]is not essential for N(E 0 ) on S. Thus, there exists some ε 1 > 0 such that for allk HE0 > 0, there exists for each k ≥ k HE0 , H k ∈ B hw ∗ ×w ∗ ( 1 k ,E0 ) ∩ D such thatN(H k ) ∩ B w ∗(ε 1 ,n e E (E0 ) ∩ B w ∗(ε 0 ,m(E 0 ))) = ∅. (86)Similarly, because for all E ∈ S, n(E) is minimally essential for N(E), giventhis ε 1 there is also for each E ∈ B hw ∗ ×w ∗ ( 1 k ,E0 ) ∩ D eE , a positive integer k FE > 0depending on the E ∈ B hw ∗ ×w ∗ (1k EE0 ,E 0 ) ∩ D eE chosen such that for each k ≥ k FEN(F ) ∩ B w ∗(ε 1 ,n e E (E)) 6= ∅, for all F ∈ B h w ∗ ×w ∗ ( 1 k ,E) ∩ D<strong>and</strong> because n eE (E) is a singleton,N(F ) ∩ B w ∗(ε 1 ,n e E (E) ∩ B w ∗(ε0 ,m(E 0 ))) 6= ∅.Given (86) we have for some ε 2 ∈ (0, ε 1 )Ls © N(H k ) ª ∩ B w ∗(ε 2 ,n e E (E0 ) ∩ B w ∗(ε 0 ,m(E 0 ))) = ∅.Finally, let ε 3 > 0 be such that<strong>and</strong> letB w ∗(ε 3 ,Ls © N(H k ) ª ) ∩ B w ∗(ε 2 ,n e E (E0 ) ∩ B w ∗(ε 0 ,m(E 0 ))) = ∅, (87)ε ∗ X:= min © h w ∗({b}, {a}) :b ∈ Ls © N(H k ) ª ; a ∈ B w ∗(ε 2 ,n e E (E0 ) ∩ B w ∗(ε 0 ,m(E 0 ))) ª ,where recall, h w ∗is the Hausdorff metric on P w ∗ f(X). We have0 < ε ∗ X < ε2 + ε 3 .But now we have a contradiction. To see this, choose a sequence of Ky Fan sets,©E ν ,H ν ,F vkª ν,k ⊂ D e E× S × D,52
as follows: First, let γ be such that ε∗ Xγ≤ min{ 1 1, , δ 0 },forsomeγ =1, 2,...,k EE0 k HE0<strong>and</strong> choose k γ so thatk 1γ< ε∗ Xγ. Second, for each ν >k γ ,chooseE ν ∈ B hw ∗ ×w ∗ ( 1 ν ,E0 ) ∩ D eE <strong>and</strong> H ν ∈ B hw ∗ ×w ∗ ( 1 ν ,E0 ) ∩ S<strong>and</strong> for each ν <strong>and</strong> k ≥ k ν := max{ν,k FEν },chooseF vk ∈ B hw ∗ ×w ∗ ( 1 ν ,Eν ) ∩ D.−→ E 0 , H νh w ∗ ×w ∗E 0 as ν →∞, <strong>and</strong> that for each ν, F vk −→h w ∗ ×w ∗By compactness, WLOG we can assume that E νF vk v−→h w ∗ ×w ∗Thus, we haveh w ∗ ×w ∗(F vk ,H ν ) ≤ h w ∗ ×w ∗(F vk ,E ν )+h w ∗ ×w ∗(Ev ,H ν ) → 0 (*)as ν →∞<strong>and</strong> k →∞,k ≥ k v ,whereh w ∗ ×w ∗(Ev ,H ν ) ≤ h w ∗ ×w ∗(Ev ,E 0 )+h w ∗ ×w ∗(E0 ,H ν ) → 0. (**)−→ E 0 ,<strong>and</strong>h w ∗ ×w ∗E ν as k →∞, k ≥ k v .Now observe that for all v, F vk ∈ B hw ∗ ×w ∗ (k 1,Eν ) ∩ D together with the h w ∗ ×w ∗-w ∗ -upper semicontinuity of minimal USCO n(·) on S imply that {z vk } = n(F vk ) →w ∗z v0 ∈ n(E ν ) as k → ∞, k ≥ k v . Moreover, E ν ∈ B hw ∗ ×w ∗ (ν 1 ,E0 ) ∩ DE e for allv together with the h w ∗ ×w ∗-d w∗-upper semicontinuity of minimal USCO n(·) implythat {x ν } = n(E ν ) → x0 ∈ n(E 0 ) as kw ∗ v →∞. Also by compactness, WLOG we canassume that N(H ν ) → N 0 ⊆ N(E 0 ) implying that for ν sufficiently large,hw ∗N(H ν ) ⊂ B w ∗(ε 3 ,Ls{N(H ν )}).But now it follows from (88) <strong>and</strong> the continuity of the excess functions, e w ∗(N(·),N(·))on S × S that there exists a δ 4 > 0 <strong>and</strong> positive integers, k ν 3 <strong>and</strong> ν 3 , such that forν ≥ ν 3 ,<strong>and</strong>fork ≥ k ν 3,h w ∗ ×w ∗(F vk ,H ν ) < δ 4 n implying that e w ∗(N(F vk ),N(H ν )) < ε 4 ,so thatdist w ∗(z vk ,N(H ν )) < ε 4 or z vk ∈ B w ∗(ε 4 ,N(H ν )).Choosing ε 4 so that B w ∗(ε 4 ,N(H ν )) ⊆ B w ∗(ε 3 ,Ls{N(H ν )}), wehave⎫⎪⎬⎪⎭(88)z vk ∈ B w ∗(ε 3 ,Ls{N(H ν )}). (89)But we can also choose ν ≥ ν 3 ,<strong>and</strong>k ≥ k ν 3large enough so thatρ w ∗(z vk ,x 0 ) ≤ ρ w ∗(z vk ,x ν )+ρ w ∗(x ν ,x 0 ) < ε 2 ,where x 0 ∈ nE e (E0 ) ∩ B w ∗(ε 0 ,m(E 0 )). Thuswehaveforν ≥ ν 3 ,<strong>and</strong>k ≥ k ν 3z vk ∈ B w ∗(ε 2 ,n eE (E 0 ) ∩ B w ∗(ε 0 ,m(E 0 ))). (90)Together, (89) <strong>and</strong> (90) contradict (87).53
- Page 5: 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 56 and 57: 12 Appendix 4: The Proof That All K
- Page 58 and 59: Letting E 1 =[E 1 \(X × U 2 )] ∪
- 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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