Externalities, Nonconvexities, and Fixed Points

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the set x 0 x 1 ∈ C w ∗ f(X) is called a segment of X between points x 0 and x 1 in X,and here, because X is strongly convex, x 0 x 1 is unique - it is the only segmentbetween x 0 and x 1 . Because X is a ρ w ∗-compact metric space, convexity can befully characterized in terms of segments. In particular, X is convex if and only if itcontains all its segments.Continuing to focus on the convex Peano continuum (X, ρ w ∗), becauseX is metricallyconvexthew∗ -closed ball, B ρw ∗ (ε,x) about x ∈ X is convex and hence connectedand is given byB ρw ∗ (ε,x)={x 0 ∈ X : ρ w ∗(x, x 0 ) ≤ ε} , (8)for all ε > 0 and x ∈ X. Also,becauseX is a convex metric spaces (in fact, a stronglyconvex metric space), the closed ball, B ρw ∗ (ε,x), aboutx isgivenbytheclosureofopen ball B ρw ∗ (ε,x) about x ∈ X. In particular,B ρw ∗ (ε,x):=cl {x 0 ∈ Z : ρ w ∗(x, x 0 ) < ε} := {x 0 ∈ X : ρ w ∗(x, x 0 ) < ε}andB ρw ∗ (ε,x):={x 0 ∈ X : ρ w ∗(x, x 0 ) < ε} ,(see Theorem 2.8 in Nadler, 1977). Here, “cl” denotes the ρ w ∗-closure of the w ∗ -openball B ρw ∗ (ε,x).For any nonempty subset E of X,wedefine the ρ w ∗-open ε-enlargement, B ρw ∗ (ε,E),of E to be the set of points in X within ρ w ∗-distance less than ε of E. Formally,x ∈ X is within ρ w ∗-distance less than ε of E ifdist ρw ∗ (x, E) :=inf x 0 ∈E ρ w ∗(x, x 0 ) < ε(i.e., if the shortest distance from x to E is less than ε). Thus, the w ∗ -open ε-enlargement, B ρw ∗ (ε,E), isgivenbyB ρw ∗ (ε,E):= © x ∈ X : dist ρw ∗ (x, E) < ε ª := ∪ x∈E B ρw ∗ (ε,x), (10)while the w ∗ -closed ε-enlargement, B ρw ∗ (ε,E), isdefined to beB ρw ∗ (ε,E):= © x ∈ X : dist ρw ∗ (x, E) ≤ ε ª := ∪ x∈E B ρw ∗ (ε,x). (11)In fact, because X is a convex metric space (and therefore has convex open balls),we have by Theorem 3.3 in Nadler (1977) thatB ρw ∗ (ε,E):=∪ x∈E B ρw ∗ (ε,x)=∪ x∈E B ρw ∗ (ε,x).Before moving on, we note that continuing to abuse the notation, we will writeh Z and h w ∗ rather than h c Z and hc w ∗, for convex Hausdorff metrics in C ρ Z f(Z) andC w ∗ f(X). Also, rather than write ρ w ∗, we will very often denote by w ∗ the metricρ w ∗.⎫⎬⎭(9)10
2.5 IrreducibilityOur final preliminary result, due to Goodykoontz (1977), characterizes the hereditaryunicoherence of Peano continua in terms of the continuity properties of a particularmapping. First a definition: a subcontinuum M E ∈ C w ∗ f(X) is irreducible aboutE ∈ P w ∗ f(X) provided E ⊆ M E and no proper subcontinuum of M E contains E.Combining Charatonik (1964) and Goodykoontz (1977), a Peano continuum X ishereditarily unicoherent if and only if for each E ∈ P w ∗ f(X) there is a unique subcontinuumM E ∈ C w ∗ f(X) irreducible about E and given byDue to uniqueness, the expression,M E = ∩ {M ∈ C w ∗ f(X) :E ⊆ M} .κ(E) :=∩ {M ∈ C w ∗ f(X) :E ⊆ M} , (12)defines a function, κ(·) :P w ∗ f(X) → C w ∗ f(X), and the continuity properties of thisfunction characterize hereditary unicoherence (i.e., the absence of closed curves insubcontinua).Theorem 6 (κ(·) is continuous if and only if X is a dendrite)Suppose assumptions [A-1] hold. Then κ(·) is continuous on P w ∗ f(X) with respectto the Hausdorff metric h w ∗ if and only if X is a dendrite.Proof. This result is an immediate consequence of Theorem 1 in Goodykoontz(1977) and the fact that because (X, ρ w ∗) is compact, the Vietoris topology and theHausdorff metric topology coincide on P w ∗ f(X).2.6 Examples of Dendritic Strategy Spaces from Contracting andNetwork Formation GamesIn order to provide for a rich set of potential applications, we have formulated ourparameterized strategic form game, G, assuming that each player’s strategy set, X i ,is a weak star compact subset (i.e., a w ∗ -compact subset) in the norm dual of aBanach space E i . 7 Thus our model covers examples where the ith player’s strategyset, X i , is a closed bounded convex subset of R n i(i.e., the set X i , i ∈ N, is a convex,w ∗ -compact subset in the separable norm dual of the Banach space R n i). Otherexamples which take advantage of the generality allowed by our assumptions are thefollowing:(1) Competitive Executive Compensation ContractingConsider m firms competing for the talents of a particular executive by competitivelyoffering executive compensation contracts. In such an application, we mightsuppose that firm i 0 s strategy set, X i , is given by a set of contracts represented by theset of all μ-equivalence classes of real-valued state-contingent compensation functions,g : Ω → [L i ,H i ],L i
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