Externalities, Nonconvexities, and Fixed Points

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3.1 Best Response MappingsLetting player i 0 s optimal payoff function be given byu ∗ i (z,x −i) :=max xi ∈Φ i (z) u i (z, (x i ,x −i )), (20)player i 0 s best response correspondence is given byΓ i (z,x −i ):={x i ∈ Φ i (z) :u i (z,(x i ,x −i )) ≥ u ∗ i (z, x −i)} . (21)It can be shown, using The Berge Maximum Theorem (1962), that u ∗ i (·, ·) is continuouson Z × X −i (with respect to the relative w ∗ product topology) and that for eachz the joint best response correspondence,x → Γ(z,x) := Q i Γ i(z,x −i ), (22)is a w ∗ -w ∗ -upper semicontinuous mapping with nonempty, w ∗ -compact values. 12 Asin the literature (e.g., Hola-Holy, 2009), we call such a mapping an USCO. Wewilldenote byU(X, P w ∗ f(X)) (23)the collection of all USCOs. Here, P w ∗ f(X) denotes the collection of all nonempty,w ∗ -closed, and convex subsets of X := X 1 ×···×X m . 13 Thus, for each z ∈ Z, thebest response correspondence Γ(z,·) is an USCO, i.e.,3.2 Nash USCOsΓ(z,·) ∈ U(X, P w ∗ f(X)) for all z ∈ Z.A Nash equilibrium for the z-game, G z := {Φ i (z),u i (z,(·, ·))} i∈N , is a profile ofstrategy choices,x ∗ ∈ Φ(z) := Q i∈N Φ i(z),such that for each player i,u i (z,(x ∗ i ,x∗ −i )) = max x i ∈Φ i (z) u i (z, (x i ,x ∗ −i )),or equivalently, a Nash equilibrium is a profile of strategy choices, x ∗ ∈ Φ(z), suchthatx ∗ ∈ Γ(z, x ∗ ).12 A correspondence, Λ(·), fromZ into X is w ∗ -w ∗ -upper semicontinuous at z if for every w ∗ -opensubset V of X such thatΛ(z) ⊆ V ,there exists a w ∗ -neighborhood U z of z such thatΛ(z 0 ) ⊆ V for all z 0 ∈ U z.Λ(·) is w ∗ -w ∗ -upper semicontinuous (w ∗ -w ∗ -usc) if it is w ∗ -w ∗ -usc at all z ∈ Z. Λ(·) is an USCO ifit is (i) w ∗ -w ∗ -usc and if (ii) for all z ∈ Z, Λ(z) is a nonempty, w ∗ -compact subset of X.13 See Appendix 6.18
The set of all Nash equilibria, N (z), forz-game G z is therefore given byN (z) :={x ∗ ∈ Φ(z) :x ∗ ∈ Γ(z,x ∗ )}. (24)It is well known, and easily shown under assumptions [A-1], that the set-valuedmapping, z → N (z), isw ∗ -w ∗ -upper semicontinuous with nonempty, w ∗ -compactvalues. Thus, the Nash correspondence (or the Nash mapping), N (·), is also anUSCO, but one from the parameter space Z with values in P w ∗ f(X), i.e.,N (·) ∈ U ρZ -w ∗ := U(Z, P w ∗ f(X)). (25)3.3 Approximable Parameterized GamesOur focus will be on Nash USCOs and our main objective will be to show that theNash USCO, N , belonging to any parameterized collection of games, G, satisfyingassumptions [A-1] is approximable. By this we mean that the Nash USCO is suchthat for any ε > 0, there exists a continuous function, f ε : Z → X, such that for any(z 0 ,x 0 ) ∈ Grf ε there is in the graph, GrN , of the Nash USCO at least one point,(z 00 ,x 00 ), such thatThis is equivalent to writing,ρ ρZ ×w ∗((z0 ,x 0 ), (z 00 ,x 00 )) := ρ Z (z 0 ,z 00 )+ρ w ∗(x 0 ,x 00 ) < ε. (26)Grf ε ⊆ B ρZ ×w∗(ε,GrN ), (27)where B ρZ ×w∗(ε,GrN ) is the open ε-enlargement of GrN given byB ρZ ×w ∗(ε,GrN ):=© (z 0 ,x 0 ) ∈ Z × X : dist ρZ ×w ∗((z0 ,x 0 ),GrN ) < ε ª ,wheredist ρZ ×w ∗((z0 ,x 0 ),GrN ):=inf (z 00 ,x 00 )∈GrN [ρ Z (z 0 ,z 00 )+ρ w ∗(x 0 ,x 00 )].⎫⎬⎭ (28)We will refer to any parameterized collection of games, G, satisfying [A-1], and havingan approximable Nash USCO N , as an approximable parametrized collection.4 Ky Fan Sets, GCS Mappings, and KFC MappingsIn this section we will show that the Nash USCO is a composition of two USCOs: theGCS USCO (from parameters Z into Ky Fan sets) and the KFC USCO (from Ky Fansets into Nash equilibria). Second, we will show that the KFC USCO is composed ofa bundle of minimal, usc, KFC strands, each with essential Nash equilibrium values,Third, we will show that a minimal KFC USCO (i.e., a KFC strand) composed withthe GCS mapping is a minimal Nash USCO. Finally, we will show that any KFCUSCO corresponding to a parameterized collection of games satisfying [A-1] has the3M property. Establishing these facts involves three key ingredients: (i) Nikaido-Isoda functions and collective security mappings, (ii) Ky Fan sets, and (iii) minimalsets of essential of Nash equilibria.19
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 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 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

Externalities, Nonconvexities, and Fixed Points

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