Externalities, Nonconvexities, and Fixed Points

cally approximated by continuous functions, we establish a fixed point result for alarge class of nonconvex-valued Nash correspondences (see De Blasi and Myjak, 1986,for an extensive summary of results on the graphical approximation of convex-valuedUSCO correspondences by continuous functions).To illustrate the usefulness of our results, we use them to prove the existenceof a fulfilled expectations Nash equilibria for a collection of probability-belief parameterizedstrategic form games of network formation over random sender-receivernetworks.We have provided 4 Appendices. In Appendix 1, we present the fundamentalsof USCO mappings and minimal USCO mappings needed for our analysis of Nashcorrespondences. Appendices 2, 3, and 4 contain the proofs of three results containedin the body of the paper.3

with the Hausdorff metric h ρZ ×w ∗ (induced by ρ Z × w ∗ )isalsoacompactmetricspace. We note that P ρZ ×w ∗ f(Z ×X) is a h ρZ ×w ∗-closed subspace of P ρ Z ×w ∗ f(Z ×X)(see the Aliprantis and Border, 2006, for a more detailed development of the compactmetric hyperspaces in general).Because the compact metric spaces Z and X are Peano continua, the compactmetric space, Z × X, equipped with the metric ρ Z × w ∗ is also a Peano continuum(see Willard, 1970, Sections 26 - 28). 6 Because ρ Z and ρ w ∗ are convex metrics, it iseasy to see that ρ Z × w ∗ := ρ Z + ρ w ∗ is a convex metric. Thus, (Z × X, ρ Z × w ∗ ) isa convex Peano continuum.Recall that the Hausdorff metric h ρZ ×w ∗ on P ρ Z ×w ∗ f(Z×X) induced by the metricρ Z × w ∗ on Z × X is given byh ρZ ×w ∗(E0 ,E 1 ):=max{e ρZ ×w ∗(E0 ,E 1 ),e ρZ ×w ∗(E1 ,E 0 )},for any E 0 and E 1 in P ρZ ×w ∗ f(Z × X), where the excess of E i over E i0the excess functionalis given bye ρZ ×w ∗(Ei ,E i0 ):=sup (z i ,x i )∈E i dist ρ Z ×w ∗((zi ,x i ),E i0 ) (5)for i =0or 1 and i 0 ∈ {0, 1}\{i}, and where the distance from (z i ,x i ) to E i0 is givenbydist ρZ ×w ∗((zi ,x i ),E i0 ):=inf (z i 0 ,x i0 )∈E ρ i0 ρ Z ×w ∗((zi ,x i ), (z i0 ,x i0 )),o(6)=inf (z i 0 ,x i0 )∈Enρ i0 Z (z i ,z i0 )+ρ w ∗(x i ,x i0 ) .for i, i 0 =0or 1 and i 6= i 0 (Chapter 3 in Beer, 1993). Making the obvious modificationsto the above, we obtain the definition of the Hausdorff metric, h Z , on P ρZ f(Z)and the Hausdorff metric, h w ∗, on P w ∗ f(X).Because P ρZ ×w ∗ f(Z × X) is compact, any sequence, {E n } n ,inP ρZ ×w ∗ f(Z × X)h ρZ ×w ∗-converges to E0 ∈ P ρZ ×w ∗ f(Z × X), denotedbyE n→ E 0 or h ρZ ×w ∗(En ,E 0 ) → 0,hρZ ×w ∗if and only if {E n } n converges to E 0 in the sense of Kuratowski-Painleve, denoted byLi ρZ ×w ∗En = E 0 = Ls ρZ ×w ∗En . (7)Expression (7) holds if and only if (i) Li ρZ ×w ∗En = E 0 :foreache 0 ∈ E 0 there existsa positive integer, N, and a sequence of elements {e n } n ρ Z ×w ∗ -converging to e 0 suchthat e n ∈ E n for all n ≥ N, and(ii)Ls ρZ ×w ∗En = E 0 : whenever n 1

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

defined on some probability space, (Ω,B(Ω), μ), of uncertain, mutually observableoutcomes (e.g., firm profit levels), where Ω is a separable metric space, B(Ω) is theBorel σ-field of events (i.e., a countably generated σ-field), and μ is a probabilitymeasure (a common prior) defined on B(Ω). In this example, firm i 0 s strategy set,X i , is a convex, w ∗ -compact and metrizable subset of the norm dual, L ∞ , consistingof essentially bounded measurable functions, where L ∞ is the separable norm dualof the separable Banach space L 1 of μ-equivalence classes of functions, f : Ω → R,with finite expectations.We might also suppose that Z is given by X := X 1 ×···×X m the convex, w ∗ -compact and metrizable subset of compensation function profiles. Thus parameter,g e := (g e 1 ,...,ge m) ∈ Z := X := X 1 ×···×X m ,can be thought of as representing firms’ expectations concerning what compensationcontracts might be offered in equilibrium.Thus by Theorem 4, both Z and X i are dendrites.(2) Noncooperative Network Formation Games: Random Sender and ReceiverNetworksHere we give an example of a parameterized collection of network formation gameswhere strategy spaces and the parameter space are convex, w ∗ -compact sets of probabilitymeasures. we will return to this example later in the paper.Consider m individuals seeking noncooperatively to form a network of local connections.Each connection is represented by a 3-tuple, (a, (i, j)) ∈ A × (N × N),where a ∈ A isthetypeofconnection(thearctype)playeri ∈ N sends to playerj ∈ N. Fromthepointofviewofthei th player, connection (a, (i, j)) can be thoughtof as an i th sender connection. Conversely, viewed from the point of view of the j thplayer connection (a, (i, j)) can be thought of as a j th receiver connection. The setof all i-sender connections is given by the Cartesian productA × ({i}×N), (13)while the set of all j-receiver connections is given by the Cartesian productA × (N ×{j}). (14)The set of i-loops, i.e., connections sent and received by player i isgivenbytheCartesian productA × ({i}×{i}). (15)Formally, we need the following notation:N := {1, 2,...,m} the set of nodes, with each node representing a player, withtypical elements denoted by i and j;A := the set of arc types (or actions) potentially available to at all players;A ij := a subset of A containing all i-sender arcs (or actions) potentially available toplayer i for initiating connections with player j;12

K + iK − j:= A×({i}×N) the set of all possible i-sender connections with typical element(a, (i, j)) ∈ A ij × ({i}×{j});:= A × (N ×{j}) the set of all possible j-receiver connections with typicalelement (a, (i, j)) ∈ A ij × ({i}×{j});K := A × (N × N) be the set of all connections with typical element denoted by(a, (i, j)).Assume that the set of arc types, A, is a compact metric space with metric d A ,and that for each ordered player pair, (i, j), the feasible set of i-sender arcs, A ij ,isad A -closed(andhencead A -compact) subset of A. Equipping the set of nodes N withthe discrete metric, d N (so that, d N (i, j) =0if i = j and 1 otherwise), we can thenequip the set of connections, K, with the metricd K := d A + d N + d N .Thus, the d K -distance between connections (a, (i, j)) and (a 0 , (i 0 ,j 0 )) in K is given byd K ((a, (i, j)), (a 0 , (i 0 ,j 0 ))) := d A (a, a 0 )+d N (i, i 0 )+d N (j, j 0 ).Letting P f (K) denote the collection of all nonempty, d K -closed subsets of connections(i.e., nonempty, closed subsets of A×(N ×N)), equip P f (K) with the Hausdorffmetric, h K , generated by the metric d K and note that because (K, d K ) is a compactmetric space, so too is (P f (K),h K ). Moreover, because Ki+ is a d K -closed subset ofK, the collection P f (Ki + ) of all nonempty, d K-closed subsets of i-sender connections(i.e., nonempty, closed subsets of A × ({i}×N)), is an h K -closed subset of P f (K).Similarly, P f (Kj − ) the collection of all nonempty, d K-closed subsets of j-receiver connections(i.e., nonempty, closed subsets of A×(N ×{j})) equipped with the Hausdorffmetric, h K ,isanh K -closed subset of P f (K).Definition 1 (i)(FeasibleNetworks)AnetworkG is a nonempty closed subset ofconnections (i.e., G ∈ P f (K)) such that (i) if (a, (i, j)) ∈ G, then a ∈ A ij ,and(ii)for all node pairs, (i, j), thesectionofG at (i, j) given byG(ij) :={a ∈ A ij :(a, (i, j)) ∈ G} (16)contains at most one arc (i.e., |G(ij)| ≤ 1, where|G(ij)| denotes the cardinality ofthe set G(ij)). Let G ⊂ P f (K) denote the collection of all feasible networks.(ii) (Feasible Sender Networks) An i-sender network, gi + , is a nonempty closedsubset of i-sender connections (i.e., gi + ∈ P f (Ki + )) such that (i) if (a, (i, j)) ∈ g+ i ,then a ∈ A ij and (ii) for all nodes j ∈ N, thesectionofg i at (i, j) given bygi + (ij) :=© a ∈ A ij :(a, (i, j)) ∈ gi+ ª(17)13

contains at most one arc (i.e., ¯¯g+ i(ij)¯¯ ≤ 1). Let G+i⊂ P f (Ki + ) denote the collectionof all feasible i-sender networks. 8(iii) (Feasible Receiver Networks) A j-receiver network, gj − , is a nonempty closedsubset of j-receiver connections (i.e., gj − ∈ P f (Ki − )) such that (i) if (a, (i, j)) ∈ g− j ,then a ∈ A ij and (ii) for all nodes i ∈ N, thesectionofgj− at (i, j) given byogj na − (ij) := ∈ A ij :(a, (i, j)) ∈ gj−¯contains at most one arc (i.e., ¯gj − (ij)¯¯¯ ≤ 1). Let G−iof all feasible i-receiver networks.⊂ P f (K − i) denote the collectionNote that each feasible network G ∈ G is equal to the union of all its i-sendernetworks i := 1, 2 ...,m, and conversely. In particular,G := © ∪ i∈N g + i: g + i∈ G + ifor all i ª .Alternatively, each feasible network G ∈ G is equal to the union of all its j-receivernetworks j := 1, 2 ...,m, and conversely. In particular,noG := ∪ j∈N gj − : gj − ∈ G − jfor all j .For G ∈ G, rather than write G = ∪ i gi + , we will sometimes write G := (g+ i ,g+ −i ).9Note that for any sequence of networks, {G n } n in G converging to G 0 ∈ G, wehaveh K (G n ,G 0 ) → 0 if and only if P i h K(gi +n ,gi +0 ) → 0 where G n =(g 1 +n ,g+n 2 ,...,g+n m )for all n and G 0 =(g 1 +0 ,g+0 2 ,...,g+0 m ).For each player i ∈ N, letP i denote the set of all probability measures withsupport contained in the set of i-sender networks, G + i. The set of probability measureson G + iis a convex, w ∗ -compact and metrizable subset of ca i , the set of all bounded,countably additive signed measures on G + i ,whereca i is the separable norm dualof the separable Banach space, C(G + i ,R), of real-valued, h K-continuous functionsdefined on G + i .8 Under our definition of a directed network, in any given network G ∈ G some node pairs, (i, j),may not be connected at all (i.e., (a, (i, j)) /∈ G for all a ∈ A ij). This means that for any givennode i, there may not be a sender connection from i to any other node - and thus node i 0 s senderconnections may not be representable by an m-tuple (i.e., a tuple of fixed length m). Here, ourdefinition identifies a network with the graph of its representing function. In particular, thinking ofa directed network G as a function from ordered node pairs N × N into arcs A, sayγ G : N × N → A,we note that as we move through the set of networks, G, the domain and range of the function γ Grepresenting the network G may vary and in particular, the domain may not be equal to all of N ×N.Thus, for such a set of networks - because the domain varies across networks - it is not possible torepresent the sender connections from any particular node i to all other nodes by a tuple of fixedlength m.9 Alternatively, every feasible network, G ∈ G can be written as G = ∪ jg − j , or adopting ournotational convention for i-sender networks, we can also write G := (g − j ,g− −j ).14

In this example, let the ith player’s strategy set be given by P i with typicalelement μ i (i.e., X i := P i ). We will refer to the probability measure μ i ∈ P i as arandom i-sender network (i.e., each player’s strategy is to choose a random sendernetwork which in turn determines the realized i-sender network g i ∈ G + iaccording tothe probability measure μ i ). We note that μ i is definedontheBorelσ-field, B(G + i ),in G + igenerated by the h K -open sets.Next letP := P 1 ×···×P mbe the compact metric space of product probability measures defined on the productσ-field, B(G 1 ) ×···×B(G m ), with typical elementμ = μ 1 ×···×μ m .We note that μ ∈ P has support contained in G. We will refer to μ ∈ P as a randomnetwork.Taking P as the parameter space, we can think of the parameter μ e := (μ e i , μe −i ) ∈P, as players’ consensus probability beliefs concerning the likelihood with which variousnetworks might emergence in equilibrium (i.e., Z := P). We will assume that eachplayer’s feasible set of random networks is given by a w ∗ -continuous correspondence,Φ i (·) :P −i → P w ∗ fc(P i ),from consensus probability beliefs into the collection of all nonempty, w ∗ -closed, andconvex subsets of random i-sender networks, denoted by P w ∗ fc(P i ).Thus, Z = X = P, satisfying [A-1](2) and (3) - and thus by Theorem 4, Z andX are dendrites.2.7 Descriptive Summary of Main ResultsLet {G z : z ∈ Z} be any parameterized collection of strategic form games satisfyingassumptions [A-1]. Under [A-1] for each parameter value (or externality value) z ∈ Z,the corresponding strategic form game, G z , has a nonempty, compact subset of Nashequilibria, N (z). Our main objective is to show that the Nash correspondence, z →N (z), is approximable. Because the Nash correspondence may have nonconvex - andpossibly disconnected - values, the problem is difficult and requires a new approach(again see De Blasi and Myjak, 1986, for an extensive summary of results on thegraphical approximation of convex-valued correspondences by continuous functions).Here is a descriptive summary of our results, assuming [A-1] throughout. In ourdescription we will use the terms USCO and CUSCO. An USCO is an upper semicontinuous,set-valued mapping with nonempty, compact values, while a CUSCO is anupper semicontinuous, set-valued mapping with nonempty, compact and connectedvalues (e.g., see Hola and Holy, 2009).(1) (Properties of Nash Correspondences) We show that the Nash USCO, N (·),is a composition of two USCOs: an USCO mapping Ky Fan sets into the collectionof sets of Nash equilibria (the KFC mapping) composed with an USCO mapping Z15

into the collection of Ky Fan sets (the GCS mapping). The basic intuition behindour decomposition of the Nash USCO is the following: The GCS mapping essentiallyencodes the details of the game and maps them into the correct Ky Fan set, whilethe KFC mapping carries the game’s Ky Fan set to the correct set of Nash equilibria.Because the game is parameterized, the domain of the game’s Ky Fan set movesabout or varies with the parameter z. Specifically then, we establish that the NashUSCO, N (·), isgivenbyN (z) =N((K(z)) := N ◦ K(z) for all z,where the GCS mapping, K(·) :Z → S, isanUSCOdefined on Z taking values inthe collection of Ky Fan sets, S, andN(·) :S → collection of sets of Nash equilibriais the KFC mapping is an USCO defined on S taking values in the collection of setsof Nash equilibria.Next, we establish that the KFC mapping, N(·), corresponding to any parameterizedcollection, {G z : z ∈ Z}, satisfying [A-1] consists of strands of essential Nashequilibria (essential in the sense of Fort, 1950), with each strand being given by a minimalUSCO, E → n(E), contained in the KFC, N(·), and defined on the hyperspaceof Ky Fan sets, S. 10 Moreover, we show that the KFC mapping, N(·), correspondingto any parameterized collection, {G z : z ∈ Z}, satisfying [A-1] has the three missesproperty or the 3M property. 11Finally, letting [N(·)] and [N (·)] denote the collection of minimal USCOs correspondingto N(·) and N (·), we then show that for the GCS mapping, K(·), correspondingto the parameterized collection, {G z : z ∈ Z},ifn(·) ∈ [N(·)],thenn(K(·)) ∈[N (·)]. Thus we show that(a) the KFC USCO is 3M and each minimal USCO contained in N(·), denoted byn(·) :S → collection of sets of Nash equilibria, is such that for each E ∈ S, thesubset n(E) of Nash equilibria is minimally essential for the set N(E) of Nashequilibria corresponding to Ky Fan set E ∈ S; and10 The USCO, n(·), is minimal in the sense that it has the smallest graph possible. Thus, if en(·) isanother USCO whose graph is contained in the graph of n(·), thenη(·) and en(·) are the same USCO.11 The KFC USCO, N(·) is 3M if for any Ky Fan set E 0 the following implication holds: if for E 1and E 2 in some open ball B hρZ ×w ∗ (δ,E 0 ) are such thatN(E 1 ) ∩ F 1 = ∅andN(E 2 ) ∩ F 2 = ∅for disjoint w ∗ -closed sets F 1 and F 2 , then there exists a third Ky Fan set E 3 containedinthelargeropen ball B hρZ ×w ∗ (3δ,E 0 ) such thatSo, three misses.N(E 3 ) ∩ [F 1 ∪ F 2 ]=∅.16

(b) given any minimal USCO, n(·), of the KFC USCO, N(·), z → n(K(z)), isaminimal USCO of N (·).(2) (Approximation) Next, we establish that any Nash USCO, N (·), possessesan approximating triple, (η,f ∗ , {f n } n ) N ,whereη(·) =n(K(·)) ∈ [N (·)], f ∗ (·) is aquasicontinuous selection of η(·), and{f n } n is a sequence of continuous functionsconverging pointwise to f ∗ . The existence of the minimal USCO η(·) is guaranteedby Drewnowski and Labuda (1990), the existence of a quasicontinuous selection f ∗ isguaranteed by Jayne and Rogers (2002) and Hola and Holy (2009), and the existenceof a pointwise converging sequence of continuous functions is guaranteed by Spruny(2007). We then show that for anyF ∗ ∈ Ls ρZ ×w ∗{Grf n } ⊂ P ρZ ×w ∗ f(Z × X)with induced USCO, F ∗ (·), the following statements are true:(a) For each z ∈ Z, F ∗ (z) is a dendrite (locally connected, connected, and withoutclosed curves) and is the unique irreducible subcontinua containing η(z).(b) For each z ∈ Z, F ∗ (z) ⊆ N (z) for all z ∈ Z.(c) For each z ∈ Z, F ∗ (z) is minimally essential for N (z) in C w ∗ f(X).The proof of (a) rests on new convergence results for minimal USCOs due toAnguelov and Kalenda (2009), a new result due to Hola and Holy (2011) establishingthe equi-quasicontinuity of the pointwise approximating sequence of continuous functions,{f n } n , in the approximating triple, (η,f ∗ , {f n } n ) N , and the dense selectionresult due to Beer (1983). The proof of part (b) rests on the fact that the inducedminimal CUSCO, F ∗ (·), is dendritically valued and the fact that the KFC mapping,N(·), underlying the Nash correspondence, N (·), is3M.(3) (Fixed Points) We show that the fact that the Nash correspondence is approximableimplies that if the parameter space, Z, and the space of strategy profiles,X, are one in the same, then the Nash USCO, N (·), hasfixed points. Thus, we showthat there exists at least one z ∗ ∈ Z such that z ∗ ∈ N (z ∗ ).(4) (An Application to Network Formation Games) In a belief-parameterizedcollection of random sender-receiver network formation games, we use our fixed pointresult to show that there exists a fulfilled expectations Nash equilibrium.3 The Nash CorrespondenceGiven parameter z and given the profile of strategy choices made by other players,player i 0 s choice problem is given byx −i ∈ Q i 0 6=i Φ i0(z), (18)max xi ∈Φ i (z) u i (z,(x i ,x −i )). (19)17

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

4.1 Nikaido-Isoda FunctionsWith each z-game,G z := {Φ i (z),u i (z,(·, ·))} i∈N, (29)we can associate a Nikaido-Isoda function (Nikaido and Isoda, 1955) given by⎫ϕ(z, (y, x)) := u(z,(y, x)) − u(z,(x, x)) ⎬:= P i∈N u i(z, (y i ,x −i )) − P (30)i∈N u ⎭i(z, (x i ,x −i )).LetF := {ϕ(z, (·, ·)) : z ∈ Z} , (31)denote the collection of Nikaido-Isoda functions associated with the collection of z-games, {G z : z ∈ Z}.4.2 The Graph of the Collective Security MappingsCorresponding to each Nikaido-Isoda function, ϕ(z,(·, ·)), thereisacollective securitymapping (i.e., a CS mapping), Λ(z,·) :Φ(z) → P w ∗ f(Φ(z)), givenbyy → Λ(z, y) :={x ∈ Φ(z) :ϕ(z,(y, x)) ≤ 0} . (32)Here, P w ∗ f(Φ(z)) is the collection of all nonempty, w ∗ -closed(andhencew ∗ -compact)subsets of Φ(z). For each choice profile y ∈ Φ(z) of the form y =(y i ,x −i ), Λ(z, y)is the (w ∗ -closed) set of choice profiles, x =(x i ,x −i ),inΦ(z) which are collectivelysecure against the potential noncooperative defection represented by the profiley =(y i ,x −i ) ∈ Φ(z) := Q i Φ i(z) (33)(i.e., ϕ(z, (y,x)) ≤ 0, fory =(y i ,x −i ) and x =(x i ,x −i ) implies that x =(x i ,x −i ) issecure against defection y =(y i ,x −i )). Note that if, given parameter z, x is containedin Λ(z,y) for all possible defection profiles y ∈ Φ(z), thatis,ifx ∈∩ y∈Φ(z) Λ(z,y) (34)then for each player i, x =(x i ,x −i ) is secure against any defection, including thoseof the form y =(y i ,x −i ).Thus,x ∈∩ y∈Φ(z) Λ(z,y) impliesu i (z,(y i ,x −i )) ≤ u i (z,(x i ,x −i )),for all pairs y =(y i ,x −i ) and x =(x i ,x −i ) - and conversely. Thus, the set of Nashequilibria given parameter z can be fully characterized as follows:x ∈ N (z) if and only if x ∈∩ y∈Φ(z) Λ(z, y). (35)Thus, the Nash USCO is given by,⎫z → N (z) =∩ y∈Φ(z) Λ(z, y). ⎬(36)⎭= {x ∈ Φ(z) :x ∈ Γ(z,x)}.Under assumptions [A-1], the function, ϕ(·, (·, ·)) which specifies for each parameterz ∈ Z a particular Nikaido-Isoda function (see (30)), has the following properties:20

(F1) ϕ(·, (·, ·)) is continuous on the compact metric space, Z × (X × X);(F2) ϕ(z,(·,x)) is quasiconcave in y on X.Given parameter z ∈ Z, the collective security mapping is given byy → Λ(z, y) :={x ∈ Φ(z) :ϕ(z,(y, x)) ≤ 0} .Moreover, given z, the graph of the collective security mapping is given byGrΛ(z, ·) :={(y,x) ∈ Φ(z) × Φ(z) :x ∈ Λ(z, y)}. (37)Thus, if (y 0 ,x 0 ) ∈ GrΛ(z, ·), then strategy profile x 0 ∈ Φ(z) is secure against defectionprofile y 0 ∈ Φ(z) and we haveϕ(z,(y 0 ,x 0 )) ≤ 0.Henceforth, for each z ∈ Z, wewilldenotebyK(z) the graph of the CS function. Wewill refer to the mapping, z → K(z), astheGCS function. Thus, the GCS functionis given byz → K(z) :=GrΛ(z,·) for all z ∈ Z. (38)The mapping K(·) is ρ Z -w ∗ × w ∗ -upper semicontinuous (i.e., usc) if for every setF, w ∗ × w ∗ -closed in Φ(z) × Φ(z), thesetK − (F ):={z ∈ Z : K(z) ∩ F 6= ∅}is ρ Z -closed in Z, or equivalently, if for every set G, w ∗ × w ∗ -open in Φ(z) × Φ(z),the setK + (G) :={z ∈ Z : K(z) ⊂ G}is ρ Z -open in Z. Also, recall that because Φ(z)×Φ(z) is w ∗ ×w ∗ -compact, K(·) is usc ifand only if K(·) has a ρ Z -w ∗ ×w ∗ -closed graph. Finally, note that (z,(y, x)) ∈ GrK(·)if and only ifϕ(z, (y, x)) ≤ 0.Thus, it follows immediately from the continuity of the function ϕ(·, (·, ·)) on Z ×(X × X) (see property F1 above) that K(·) has a ρ ρZ ×w ∗ ×w∗-closed graph whereρ ρZ ×w ∗ ×w ∗ := ρ Z + d w ∗ ×w ∗ and hence that K(·) is usc.Lemma 7 (The KFC mapping, K(·), isuppersemicontinuous)Suppose assumptions [A-1] hold. Then the mappingis ρ Z -w ∗ × w ∗ -upper semicontinuous.z → K(z)21

Because Φ(z) × Φ(z) is w ∗ × w ∗ -compact, K(·) is usc if and only if the excessfunction, e w ∗ ×w ∗(K(·),K(z0 )), isρ Z -continuous at each z 0 ∈ Z (see remark 1.5 inDe Blasi and Myjak, 1986). The mapping e w ∗ ×w ∗(K(·),K(z0 )), isρ Z -continuous ateach z 0 ∈ Z if for all ε > 0, thereexistsaδ > 0 such thate w ∗ ×w ∗(K(z),K(z0 )) < ε for all z ∈ B δ (z 0 ),where e w ∗ ×w ∗(K(z),K(z0 )) is the excess of K(z) over K(z 0 ) or the sup over distancesfrom points in K(z) to points in K(z 0 ) (see 6 and 5 above).By Proposition 6.3.11 in Beer (1993), because Z is a ρ Z -compact metric space(and hence a Baire space) and because Φ(z) × Φ(z) is w ∗ × w ∗ -compact (and henceseparable), there is a w ∗ -dense G δ subset C of Z where K(·) is ρ Z -w ∗ ×w ∗ -continuous(i.e., K(·) is both usc and lsc - lower semicontinuous on C - also see Fort, 1951-1952).Moreover, the mapping K(·) is ρ Z -w ∗ × w ∗ -continuous on C if and only if K(·) isρ Z -h w ∗ ×w∗-continuous on C (see remark 1.9 in De Blasi and Myjak, 1986). K(·) isρ Z -h w ∗ ×w ∗-continuous (or Hausdorff continuous) at z0 ∈ C if for all ε > 0 there is aδ > 0 such that4.3 Ky Fan Setsh w ∗ ×w ∗(K(z),K(z0 )) < ε for all z ∈ B δ (z 0 ).Given any set E ∈ P w ∗ ×w ∗ f(X × X), define the domain of E to be the set 14D(E) :={y ∈ X :(y, x) ∈ E for some x ∈ X} ∈ P w ∗ f(X).Define the range of E to be the setR(E) :={x ∈ X :(y, x) ∈ E for some y ∈ X} ∈ P w ∗ f(X).The mappings D(·) and R(·) are h w ∗ ×w ∗-h w∗-continuous. To see this, simply notethat if h w ∗ ×w ∗(En ,E 0 ) → 0, thenforevery(y 0 ,x 0 ) ∈ E 0 there exists a sequence{(y n ,x n )} n such that (y n ,x n ) −→w ∗ ×w (y0 ,x 0 ) and (y n ,x n ) ∈ E n .∗Definition 2 (Ky Fan Sets)AsetE ∈ P w ∗ ×w ∗ f(X ×X) is a Ky Fan set if E satisfies the following properties:(Z1) D(E) is nonempty, w ∗ -closed, and convex and D(E) =R(E);(Z2) for all y ∈ D(E), (y, y) ∈ E;(Z3) for all x ∈ R(E), {y ∈ D(E) :(y, x) /∈ E} is convex (possibly empty).We will denote by S the collection of all Ky Fan sets in P w ∗ ×w ∗ f(X × X). Thus,S := {E ∈ P w ∗ ×w ∗ f(X × X) : E satisfies (Z1)-(Z3)} ,and it follows from Lemma 4 in Ky Fan (1961) that if E ∈ S, then∩ y∈D(E) {x ∈ R(E) :(y, x) ∈ E} 6= ∅.14 Here, P w ∗ ×w ∗ f (X × X) denotes the collection of all nonempty, w ∗ × w ∗ -closed subsets of X × X(see Appendix 6).22

Lemma 8 (The Space of Ky Fan Sets)Suppose assumptions [A-1] hold. The following statements are true.(1) S is a h w ∗ ×w ∗-closed subspace of P w ∗ ×w ∗ f(X × X).(2) For all z ∈ Z, K(z) is a Ky Fan set such that for all z ∈ Z,D(K(z)) = Φ(z) and R(K(z)) = Φ(z).For a proof of Lemma 8 see Appendix 2.4.4 Ky Fan CorrespondencesConsider the correspondence,E → N(E) :=∩ y∈D(E) {x ∈ R(E) :(y,x) ∈ E} , (39)defined on S taking values in P w ∗ f(X). We will call the correspondence N(·) from Sinto P w ∗ f(X) the Ky Fan Correspondence (i.e., the KFC).Lemma 9 (The KFC is an USCO)Under assumption [A-1], the KFC, N(·) is an USCO, that is,N(·) ∈ U(S,P w ∗ f(X)). (40)Proof. By Ky Fan (1961) N(E) is nonempty for all E ∈ S and it is easy to seethat N(E) is compact for all E ∈ S. To see that N(·) is upper semicontinuousconsider a sequence {(E n ,x n )} n ⊂ GrN(·) where {E n } n ⊂ S and WLOG assumethat E n E 0 ,andx n → x0 . By(39)wehaveforeachn, (y, x n ) ∈ E n forw ∗→hw ∗ ×w ∗any y ∈ D(E n ).Bytheh w ∗ ×w ∗-h w ∗-continuity of D(·), wehaveforanyy0 ∈ D(E 0 )a sequence {y n } n with y n ∈ D(E n ) for all n and y n →w ∗ y0 . This, together withE n→hw ∗ ×w ∗E 0 and x n →w ∗x0 ,implythat(y 0 ,x 0 ) ∈ E 0 for any y 0 ∈ D(E 0 ). Thus,(E 0 ,x 0 ) ∈ GrN(·). By compactness, the fact that GrN(·) is closed implies that N(·)is upper semicontinuous - with nonempty, w ∗ -compact values.4.5 D-Equivalence Classes of Ky Fan SetsGivenKyFansetE ∈ S, wedefine the D-equivalence class of Ky Fan sets, S E ,asfollows:S E := {E 0 ∈ S : D(E 0 )=D(E)} . (41)Because D(·) h w ∗ ×w ∗-h w ∗-continuous, it is easy to show that S E is a h w ∗ ×w ∗-closedsubset of S. Thus, if {E n } n is a sequence of Ky Fan sets in SE e for some E e ∈ S,then E n E 0 implies that E 0 ∈ S eE . Also, viewing S (·) as a mapping from S→hw ∗ ×w ∗into D-equivalence classes of Ky Fan sets, it is easy to show that S (·) has a h w ∗ ×w ∗-h w ∗ ×w∗-closed graph. In particular, ifh w ∗ ×w ∗(En ,E 0 )+h w ∗ ×w ∗(Cn ,C 0 ) → 0,23

where for all n, C n ∈ S E n,thenC 0 ∈ S E 0. This too is an immediate consequence ofthe h w ∗ ×w ∗-h w∗-continuity of D(·).Consider again the Ky Fan correspondence,E → N(E) :=∩ y∈D(E) {x ∈ R(E) :(y,x) ∈ E} .Suppose now that we restrict N(·) to the space of Ky Fan sets D-equivalent to theKy Fan set, Φ(z) × Φ(z). 15 We will denote this D-equivalence class byS z := {E ∈ S : D(E) =Φ(z)} ⊂ S.Note that for the GCS function, K(·), wehaveforallz ∈ Z,K(z) ∈ S z ⊂ S.5 Minimal Essential Sets, Minimal KFCs, and the 3MPropertyWe begin with the definition of sets of Nash equilibria that are essential and minimallyessential in the sense of Fort (1950) and Jiang (1962). 16Definition 3 (Essential Sets and Minimal Essential Sets)Let N(·) beaKFCandlet e E ∈ S be a given Ky Fan set.(1) A nonempty, closed subset e( e E) of N( e E), whereN( e E):=∩ y∈D( e E)nx ∈ R( e E):(y, x) ∈ e Eis said to be an essential set of N( e E) in S if for all ε > 0 there exists a h w ∗ ×w ∗-openball of radius δ > 0 centered at e E such that for all E 0 ∈ B hw ∗ ×w ∗ ( e E) ∩ S,o,N(E 0 ) ∩ B w ∗(ε,e( e E)) 6= ∅. (42)We will denote by E N( E) e (S) the collection of all essential sets of N( E) e in S.(2) A nonempty closed subset m( E) e of N( E) e is said to be a minimal essential setin S if (i) m( E) e ∈ E N( E) e (S) and if (ii) m( E) e is a minimal element of E N( E) e (S) orderedby set inclusion (i.e., if e( E) e ∈ E N( E) e (S) and e( E) e ⊆ m( E) e then e( E)=m( e E)). e15 Note also that for the “box” Ky Fan setΦ(z) × Φ(z)we haveD(Φ(z) × Φ(z)) = Φ(z),N(Φ(z) × Φ(z)) = Φ(z),and{y ∈ D(E) :(y, x) /∈ Φ(z) × Φ(z)} = ∅.16 See also, Carbonell-Nicolau (2010), Yu, Yang, and Xiang (2005), and Zhou, Xiang, and Yang(2005).24

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

Definition 4 (The 3M Property - The Three Misses Property)We say that the KFC,N(·) :S → P w ∗ f(X),has the 3M property if given any e E ∈ S, anyE 0 ∈ S eE , any pair of disjoint closedsets F 1 and F 2 in D( e E) ⊆ X, and any open ball B hw ∗ ×w ∗ (δ 0 ,E 0 ) ∩ S eE , δ 0 > 0, theD-restricted KFC,E → N eE (E),is such that if B hw ∗ ×w ∗ (δ 0 ,E 0 ) ∩ SE e contains two D-equivalent Ky Fan sets, E1 andE 2 , such thatN eE (E 1 ) ∩ F 1 = ∅ and N eE (E 2 ) ∩ F 2 = ∅,then the larger open ball B hw ∗ ×w ∗ (3δ 0 ,E 0 ) ∩ SE e contains a third D-equivalent Ky Fanset, E 3 , such thatN eE (E 3 ) ∩ [F 1 ∪ F 2 ]=∅.Conversely, if KFC N(·) fails to have the 3M property, then for some pair of KyFan sets,eE ∈ S and E 0 ∈ S eE ,there exists a pair of disjoint closed sets F 1 and F 2 in D( e E) ⊆ X, and an open ball,B hw ∗ ×w ∗ (δ 0 ,E 0 ) ∩ S eE , δ 0 > 0, containingtwoD-equivalent Ky Fan sets, E 1 and E 2such thatN eE (E 1 ) ∩ F 1 = ∅ and N eE (E 2 ) ∩ F 2 = ∅,but such that for all E 3 ∈ B hw ∗ ×w ∗ (3δ 0 ,E 0 ) ∩ S e EN e E (E3 ) ∩ [F 1 ∪ F 2 ] 6= ∅.Theorem 12 (All KFCs Have the 3M Property)Any KFC,N(·) :S → P w ∗ f(X),corresponding to a parameterized collection of games,{G z : z ∈ Z} ,satisfying assumption [A-1] has the 3M property.The proof of Theorem 12 will be given in Appendix 4.In summary, for the collection of z-games, {G z : z ∈ Z}, satisfying [A-1], the Nashcorrespondence is given byN (z) =N(K(z)) for all z ∈ Z, (44)where the Ky Fan valued GCS mapping, K(·) :Z → S, isw ∗ -w ∗ × w ∗ -upper semicontinuouson Z with values given byK(z) :=GrΛ(z,·) ∈ S for each z ∈ Z, (45)26

and where the KFC, N(·) :S → P w ∗ f(X), is an USCO with values given byN(E) =∩ y∈D(E) {x ∈ R(E) :(y, x) ∈ E} ,foreachE ∈ S. (46)Thus, given the GCS function K(·), wehaveforallz ∈ ZN (z) =N(K(z)) = ∩ y∈Φ(z) {x ∈ Φ(z) :(y, x) ∈ K(z)} ,where K(z) ∈ S z ⊆ S and D(K(z)) = R(K(z)) = Φ(z).We will denote byU 3M := U 3M (S,P w ∗ f(X)) (47)the set of all 3M USCOs defined on the hyperspace of Ky Fan sets, S, withvaluesinP w ∗ f(X).We have shown that all KFCs are 3M and that all minimal USCOs contained ina KFC have minimally essential values. Thus, we know that because K(z) ∈ S forall z, the induced USCO, z → n(K(z)), has minimally essential values on Z. For ourfinal result of this section, we show that if n(·) ∈ [N(·)], thenn(K(·)) ∈ [N (·)] forGCS mapping, K(·), corresponding to the parameterized collection.Theorem 13 (Minimal KFCs and Minimal Nash USCOs)Suppose the collection of z-games, {G z : z ∈ Z}, satisfies assumption [A-1] withcorresponding Nash USCO N (·) =N(K(·)) whereN(·) ∈ U 3M (S,P w ∗ f(X))is the KFC and K(·) is the GCS function. Then,n(K(·)) ∈ [N (·)] for all n(·) ∈ [N(·)].Proof. Suppose not and let m(·) be a minimal USCO of n(K(·)) ∈ U(Z, P w ∗ f(X))(i.e., an USCO defined on Z with values in P w ∗ f(X) such that m(·) ∈ [n(K(·))]) suchthat for some z 0 ∈ Z, m(z 0 ) is a proper subset of n(K(z 0 )). By Lemma 10, becausem(·) ∈ [n(K(·))], m(z 0 ) ∈ M n(K(z 0 ))(P w ∗ f(X)) implying that for any ε 0 > 0 thereexists δ 0 > 0 such that for allz δ0 ∈ B ρZ (δ 0 ,z 0 ⎫) ∩ Z, ⎬(48)n(K(z δ0 )) ∩ B w ∗(ε 0 ,m(z 0 ⎭)) 6= ∅.Becausen(K(z δ0 )) ∩ B w ∗(ε 0 ,m(z 0 ))is a closed subset of both n(K(z δ0 )) and B w ∗(ε 0 ,m(z 0 )), n(K(z δ0 )) ∩ B w ∗(ε 0 ,m(z 0 ))is not an essential for n(K(z δ0 )) in Z. Therefore, there is some ε 1 > 0 such that foreach n there exists z δn ∈ B ρZ ( 1 n ,zδ0 ) ∩ Z such thathin(K(z δn )) ∩ B w ∗ ε 1 ,n(K(z δ0 )) ∩ B w ∗(ε 0 ,m(z 0 )) = ∅. (49)27

Given (48) and the fact that for n sufficiently largewe havez δn ∈ B ρZ ( 1 n ,zδ0 ) ∩ Z ⊂ B ρZ (δ 0 ,z 0 ) ∩ Z,n(K(z δn )) ∩ B w ∗(ε 0 ,m(z 0 )) 6= ∅.Therefore, for all n sufficiently large, we have somex n ∈ n(K(z δn )) ∩ B w ∗(ε 0 ,m(z 0 )).Because x n ∈ n(K(z δn )), foralln sufficiently large we have by (49),hix n /∈ B w ∗ ε 1 ,n(K(z δ0 )) ∩ B w ∗(ε 0 ,m(z 0 )) . (50)WLOG, suppose that x n →w ∗ x0 and note that z δn →ρZz δ0 . Thus, because n(K(·)) isan USCO and B w ∗(ε 0 ,m(z 0 )) is closed, we havex 0 ∈ n(K(z δ0 )) ∩ B w ∗(ε 0 ,m(z 0 )).But now we have a contradiction, because by (50) it must be the case that for someε 2 ∈ (0, ε 1 ),hix 0 /∈ B w ∗ ε 2 ,n(K(z δ0 )) ∩ B w ∗(ε 0 ,m(z 0 )) .Therefore, we must conclude that for no z ∈ Z can it be true that m(z) apropersubset of n(K(z)) - and therefore that n(K(·)) ∈ [N (·)].We close this Section by noting that our 3M Property is similar in spirit to Condition(c) in Yu, Yang, and Xiang (2005) (i.e., YYX). Given arbitrary compact metricspaces, (Z, d Z ) and (X, d X ), YYX (2005) show that if an USCO, Γ ∈ U(Z, P dX f(X)),satisfies Condition (c), then for each z ∈ Z, Γ(z) contains at least one minimal essentialset, m(z), andevery minimal essential set is connected. But we have seen thatnot every minimal essential set is connected. Thus, YYX’s conclusions concerningthe connectedness of minimal essential sets appears not to be true.5.2 Minimal Essential Sets Relative to a HyperspaceHere we introduce the notion of a minimal essential set relative to a particular hyperspace.We also introduce the notion of an USCO in the connected class.We will consider minimal essentiality relative to the hyperspace of subcontinua,C w ∗ f(X), as well as minimal essentiality relative to P w ∗ f(X). We begin with thedefinition.Definition 5 (Minimal Essentiality Relative to C w ∗ f(X))Let Γ ∈ U ρZ -w ∗ be any USCO. We say that a set e(z) ∈ C w ∗ f(X) is essential forthe set, Γ(z), ife(z) ⊆ Γ(z) and if e(z) is such that for all ε > 0 there exists a δ > 0,such thatΓ(z 0 ) ∩ B w ∗(ε,e(z)) 6= ∅ for all z 0 ∈ B δ (z).28

We say that m(z) ∈ C w ∗ f(X) is minimally essential in C w ∗ f(X) if for any otherm 0 (z) ∈ C w ∗ f(X) essential for Γ(z), m 0 (z) ⊆ m(z) implies that m 0 (z) =m(z).We will denote by E Γ(z) (C w ∗ f(X)) the collection of sets in C w ∗ f(X) essentialfor Γ(z), and we will denote by M Γ(z) (C w ∗ f(X)) the collection of sets in C w ∗ f(X)minimally essential for Γ(z).Minimal essentiality relative to P w ∗ f(X) is defined similarly. In the followingLemma we establish that for any minimal USCO, ϕ(·) ∈ [Γ] ρZ -w ∗, Γ ∈ U ρ Z -w ∗,andfor any z, ϕ(z) is minimally essential in P w ∗ f(X) for the set, Γ(z). Thus, we willshow thatϕ(z) ∈ M Γ(z) (P w ∗ f(X)) for all z ∈ Z.The following result relating minimal USCOs and minimal essential sets, for minimalUSCOs defined on the parameter space is similar to our prior result, relatingminimal USCOs and minimal essential sets, for minimal USCOs definedonthehyperspaceof Ky Fan sets.Lemma 14 (ϕ(z) is minimally essential in P w ∗ f(X) for Γ(z))Suppose assumptions [A-1] hold and let Γ ∈ U ρZ -w ∗ be an USCO. Then for anyminimal USCO, ϕ ∈ [Γ] ρZ -w ∗,andforanyz ∈ Z, the subset given by ϕ(z) ∈ P w ∗ f(X)is minimally essential for Γ(z) in P w ∗ f(X).Proof. Because ϕ(·) is an USCO, for each z ∈ Z, ϕ(z) is essential for Γ(z). Infact,ϕ(z) is essential for ϕ(z) - i.e., for all ε > 0 there exists a δ > 0, such thatϕ(z 0 ) ∩ B w ∗(ε, ϕ(z)) 6= ∅ for all z 0 ∈ B δ (z).To see that ϕ(z) is minimal, suppose that for some z 0 ∈ Z there is a nonempty,closed, and proper subset m(z 0 ) of ϕ(z 0 ) such that for any ε > 0 there exists a δ > 0such that for all z contained in B δ (z 0 )ϕ(z) ∩ B w ∗(ε,m(z 0 )) 6= ∅. (51)Because m(z 0 ) is a proper subset of ϕ(z 0 ),wecanchooseε 0 > 0 so that ϕ(z 0 )∩B w ∗(ε 0 ,m(z 0 )) is a proper subset of ϕ(z 0 ), and by (51) we can choose δ 0 > 0 so thatdefine the mapping q(·) as follows:ϕ(z) ∩ B w ∗(ε 0 ,m(z 0 )) 6= ∅ for all z ∈ B δ 0(z 0 ),q(z) :=½ ϕ(z) ∩ Bw ∗(ε 0 ,m(z 0 )) if z ∈ B δ 0(z 0 )ϕ(z)otherwise.By Lemma 2(ii) in Anguelov and Kalenda (2009), q(·) is an USCO, and moreover,Grq(·) is a proper subset of Grϕ(·), a contradiction of the fact that ϕ(·) ∈ [Γ] ρZ -w ∗.Thus, for each z ∈ Z, ϕ(z) is minimally essential in P w ∗ f(X) for ϕ(z), and becauseϕ(z) ⊆ Γ(z), ϕ(z) is minimally essential in P w ∗ f(X) for Γ(z).29

We note that if m(z) ∈ P w ∗ f(X) is a proper subset of ϕ(z),thenitisnot minimallyessential for Γ(z) in P w ∗ f(X). In particular, there is some ε m(z) > 0, such that foreach δ > 0 we can find a z δ ∈ B δ (z) such thatΓ(z δ ) ∩ B w ∗(ε m(z) ,m(z)) = ∅.But for this ε m(z) because ϕ(z) is minimally essential for Γ(z) in P f (X), wecanfinda δ m(z) > 0 such that for all z 0 ∈ B δm(z) (z)Γ(z 0 ) ∩ B w ∗(ε m(z) , ϕ(z)) 6= ∅.Therefore, at m(z) there exists some z δ m(z)∈ B δm(z) (z) such thatwhile at ϕ(z) we haveΓ(z δ m(z)) ∩ B w ∗(ε m(z) ,m(z)) = ∅,Γ(z δ m(z)) ∩ B w ∗(ε m(z) , ϕ(z)) 6= ∅.Thus, the miss problem (i.e., the fact that Γ(z δ m(z))∩B w ∗(ε m(z) ,m(z)) = ∅) iscausedby the points in ϕ(z) missing from m(z) - and the miss problem can be arranged nomatter which points in ϕ(z) are missing from m(z). We will return to the missproblem below.6 Approximation6.1 All Nash USCOs are ApproximableOur objective in this section is to show that under assumptions [A-1] all Nash USCOsare approximable. This is a direct consequence of the fact that all KFCs are 3M. Inparticular, we will show that if [A-1] holds and if N ∈ U ρZ -w ∗ is a Nash USCO, thenbecause the underlying KFC, N, has the 3M property (i.e., N ∈ U 3M (S,P w ∗ f(X)))there exists a sequence of continuous functions, f n : Z → X, such that for any ε > 0there exists an N ε such that for all n ≥ N ε ,Grf n ⊆ B ε (GrN ), (52)This is equivalent to saying that for n sufficiently large, the continuous function f nis such that for any (z 0 ,x 0 ) ∈ Grf n there is in the graph of the USCO, GrΓ, atleastone point, (z 00 ,x 00 ), such thatρ ρZ ×w ∗((z0 ,x 0 ), (z 00 ,x 00 )) := ρ Z (z 0 ,z 00 )+ρ w ∗(x 0 ,x 00 ) < ε. (53)where B ρZ ×w ∗ (ε,GrΓ) is the ρ Z × w ∗ -open ε-enlargement of GrΓ given byB ρZ ×w ∗ (ε,GrΓ) := © (z 0 ,x 0 ) ∈ Z × X : dist ρZ ×w ∗((z0 ,x 0 ),GrΓ) < ε ª ,wheredist ρZ ×w ∗((z0 ,x 0 ),GrΓ) :=inf (z 00 ,x 00 )∈GrΓ[ρ Z (z 0 ,z 00 )+ρ w ∗(x 0 ,x 00 )].⎫⎬⎭ (54)30

6.2 Approximating TriplesWebeginwiththedefinition of a Baire 1 function.Definition 6 (Baire 1 Functions)Afunctionf : Z → X is ρ Z -s ∗ Baire 1, denoted byf ∈ ρ Z -s ∗ B 1 := ρ Z -s ∗ B 1 (Z, X),if for every s ∗ -open set G ⊂ X, f −1 (G) is an F σ in Z (a countable union of ρ Z -closedsets in Z).Given that X is w ∗ -compact and metrizable and hence norm separable (see Wilansky,1978, p. 145), f ∈ ρ Z -s ∗ B 1 (Z, X) if and only if f is the pointwise limit of asequence of ρ Z -s ∗ -continuous functions (e.g., see Lee, Tang, and Zhao, 2000).Let N (·) ∈ U ρZ -w ∗ be a Nash USCO and let η(·) =n(K(·)) ∈ [N (·)] ρ Z -w ∗ be aminimal Nash USCO for minimal KFC, n(·), and GCS mapping, K(·). ByTheorem3.3 in Jayne and Rogers (2002), we know that for any USCO, and in particular, forany minimal Nash USCO η(·), there exists a Baire 1 function f ∗ ∈ ρ Z -s ∗ B 1 such thatf ∗ (z) ∈ η(z) for all z ∈ Z (i.e., there exists a Baire 1 selector f ∗ of η). Moreover, weknow that this Baire 1 selector comes equipped with a pointwise converging sequenceof continuous functions, and we also know by Theorem 1.2 in Spruny (2007) (see alsoKalenda, 2007), that this sequence of pointwise approximating, continuous functions,{f n } n ⊂ C ρZ -s ∗(Z, X), can be chosen so it is USCO bounded - that is, so that Grf n ⊂Grβ for all n for some USCO, β ∈ U ρZ -w ∗. The minimal Nash USCO, η ∈ [N ] ρ Z -w ∗,together with its Baire 1 selection, f ∗ ∈ ρ Z -s ∗ B 1 , and the Baire 1’s USCO-bounded,point-wise approximating sequence of continuous functions, {f n } n ⊂ C ρZ -s∗(Z, X),form what we will call an approximating triple,(η,f ∗ , {f n } n ) N ,for Nash USCO N ∈ U ρ∗Z-w ∗. Thus, given approximating triple, (η,f∗ , {f n } n ) N ,forN ∈ U ρZ -w ∗ we have η ∈ [N ] ρ Z -w ∗, f ∗ ∈ ρ Z -s ∗ B 1 , and USCO bounded {f n } n ⊂∗(Z, X) such that for all z ∈ ZC ρZ -sf ∗ (z) ∈ η(z) =n(K(z)) and kf n (z) − f ∗ (z)k ∗ → 0.Because k·k ∗ -convergence (or s ∗ -convergence) implies w ∗ -convergence (i.e., convergencewith respect to the convex metric ρ w ∗), the sequence, {f n } n ⊂ C ρZ -s∗(Z, X),is also a sequence of ρ Z -w ∗ -continuous functions (see Theorem 2.6.7 in Megginson,1998). Thus, given approximating triple, (η,f ∗ , {f n } n ) N , for the Nash USCON ∈ U ρZ -w ∗ we have for all z ∈ Zf ∗ (z) ∈ η(z) =n(K(z)) and ρ w ∗( f n (z),f ∗ (z)) → 0.Note that the existence of an approximating triple for any Nash USCO, N ∈ U ρZ -w ∗,isguaranteed by assumptions [A-1], Proposition 4.3 in Drewnowski and Labuda (1990),Theorem 3.3 in Jayne and Rogers (2002), and Theorem 1.2 in Spruny (2007).31

6.3 AK Convergence of Minimal Nash USCOsWebeginwithadefinition of AK-convergence of minimal USCOs. Under our assumptions[A-1] this definition is equivalent to the more general notion of convergence forminimal USCOs introduced by Anguelov and Kalenda (2009) - see Theorem 11 inAnguelov and Kalenda (2009).Definition 7 (AK Convergence)Let {η n (·)} n ⊂ M ρZ -w ∗(Z, P w ∗ f(X)) be any sequence of minimal USCOs. We saythat {η n (·)} n AK-converges to minimal USCO η ∗ ∈ M ρZ -w ∗,denotedbyηn →AKη ∗ ,ifand only if there exists a sequence of USCOs {ψ n (·)} n ⊂ U ρZ -w ∗ such that(1) Grη n ⊆ Grψ n for all n,(2) Grψ n0 ⊆ Grψ n for all n 0 ≥ n,(3) η ∗ (·) is the unique minimal USCO whose graph is contained in the graph ofthe mapping ψ ∞ (·) given byz → ψ ∞ (z) :=∩Grψ n (z) :={x ∈ X :(z, x) ∈∩Grψ n } . (55)Therefore, if η n → η ∗ for {η n } n ⊂ M ρZ -w ∗ and η∗ ∈ M ρZ -w∗, then there exists aAKsequence {ψ n } n ⊂ M ρZ -w ∗ such that (1) Grηn ⊆ Grψ n for all n, (2)Grψ n0 ⊆ Grψ nfor all n 0 ≥ n, and(3){η ∗ } =[ψ ∞ ] ρZ -w ∗. We will refer to the mapping ψ∞ (·) asthe exterior AK-limit USCO. Note that the exterior AK-limit USCO is quasiminimal(i.e., ψ ∞ (·) ∈ QM ρZ -w ∗).Because the space of all continuous functions, C ρZ -w ∗ := C ρ Z -w∗(Z, X) is containedin the space of all minimal USCOs, M ρZ -w ∗ := M(Z, P w ∗ f(X)), wehaveC ρZ -w ∗ ⊂ M ρ Z -w ∗ ⊂ QM ρ Z -w ∗ ⊂ U ρ Z -w∗. (56)In fact, Anguelov and Kalenda (2009) show that M ρZ -w ∗ is the AK-closure of C ρ Z -w ∗(see Theorem 34 in Anguelov and Kalenda, 2009).The following result is due to Anguelov and Kalenda (2009) (see Corollary 16(ii)and Theorem 34 in Anguelov and Kalenda, 2009) relates pointwise convergence ofcontinuous functions and AK-convergence of minimal USCOs.Theorem 15 (Pointwise Convergence in C ρZ -w ∗ and AK-Convergence in M ρ Z -w ∗)Suppose [A-1] holds. Let {f n } n ⊂ C ρZ -w ∗ be an USCO bounded sequence of ρ Z-w ∗ -continuous functions and let η ∈ M ρZ -w ∗. If for all z ∈ Z, w∗ -lim n f n (z) existsand is contained in η(z), then{f n } n AK-converges to η.Here,wehavefortheapproximatingtriple,(η,f ∗ , {f n } n ) N ,f n (z) →w ∗ f ∗ (z) ∈ η(z) for all z.Thus, we havef n →AKη ∈ [N ] ρZ -w ∗.32

6.4 Graph ConvergenceBecause the hyperspace of all nonempty ρ ρZ ×w ∗-closed subsets of Z×X, P ρ Z ×w ∗ f(Z×X), is a compact metric space with Hausdorff metric h ρZ ×w∗, we can assume WLOGthat the sequence of graphs, {Grf n } n ⊂ P ρZ ×w ∗ f(Z × X), corresponding to ourpointwise converging, USCO bounded sequence of ρ Z -w ∗ -continuousfunctionsissuchthatGrf n → F ∗hρZ ×w ∗for some F ∗ ∈ P ρZ ×w ∗ f(Z×X). Moreover, because the sequence {f n } n AK-convergesto minimal Nash USCO η(·) =n(K(·)), we have corresponding to {f n } n a sequenceof USCOs, {ψ n } n ⊂ U ρZ -w ∗ with graphs {Grψn } n ⊂ P ρZ ×w ∗ f(Z × X) such thatGrf n ⊆ Grψ n for all n and such that {Grψ n } n decreases to the graph of the exteriorAK-limit USCO ψ ∞ ∈ QM ρZ -w ∗,whereGrψ∞ := ∩Grψ n . Thus, we have thatand becausewithwe can conclude thatGrf nGrψ n→ Grψ ∞ := ∩Grψ n ,hρZ ×w ∗→ F ∗ and Grψ n → Grψ ∞hρZ ×w ∗ hρZ ×w ∗Grf n ⊆ Grψ n for all n,F ∗ ⊆ Grψ ∞ .By Hola and Holy (2009 - e.g., see Corollary 3.3 - also see Appendix 1), our Baire1 selection, f ∗ ∈ ρ Z -s ∗ B 1 of the minimal Nash USCO, η(·) =n(K(·)), isinfactρ Z -w ∗ -quasicontinuous (i.e., f ∗ ∈ QC ρZ -w ∗)withGr ρZ ×w ∗f∗ = Grη.Also, because f ∗ (z) =w ∗ -lim n f n (z) for all z ∈ Z, wheref ∗ ∈ QC w ∗ -w ∗η(z) for all z ∈ Z, wealsohaveGr ρZ ×w ∗f ∗ ⊆ F ∗ implying thatand f ∗ (z) ∈Gr ρZ ×w ∗f ∗ = Grη ⊆ F ∗ .Thus, we haveor equivalently,Gr ρZ ×w ∗f ∗ = Grη ⊆ F ∗ ⊆ Grψ ∞ , (57)Gr ρZ ×w ∗f ∗ (z) =η(z) ⊆ F ∗ (z) ⊆ ψ ∞ (z) for all z ∈ Z, (58)where F ∗ (·) is the USCO induced by F ∗ ∈ P ρZ ×w ∗ f(Z × X), givenbyz → F ∗ (z) :={x ∈ X :(z,x) ∈ F ∗ } ,33

and Gr ρZ ×w ∗f ∗ (·) is the USCO induced by Gr ρZ ×w ∗f ∗ ∈ P ρZ ×w ∗ f(Z × X), givenbyz → Gr ρZ ×w ∗f ∗ (z) := © x ∈ X :(z,x) ∈ Gr ρZ ×w ∗f ∗ª .Because the exterior AK-limit USCO, ψ ∞ ,is quasiminimal, [ψ ∞ ] ρZ -w ∗ = {η},and because Z is a Baire space and X is a metric space, we have by Lemma 7in Anguelov and Kalenda (2009) that ψ ∞ (·) is single-valued on a dense, open G δsubset D of Z. Also, because Z is locally convex, and hence locally connected, andbecause Grf n F ∗ for the sequence of ρ Z -w ∗ -continuous functions, {f n } n ,by→hρZ ×w ∗Theorem 3 in Hola (1987), we have for the induced USCO, z → F ∗ (z), thatF ∗ (·) isconnected-valued. Thus, F ∗ (·) is a CUSCO and thusF ∗ (z) ∈ C w ∗ f(X) for all z ∈ Z.Summarizing then, we have for Nash USCO, N ∈ U ρZ −w∗, an approximatingtriple,(η,f ∗ , {f n } n ) N ,such that for all z ∈ ZGr ρZ -w ∗f ∗ (z) =η(z) ⊆ F ∗ (z) ⊆ ψ ∞ (z), (59)where η(z) and F ∗ (z) are continua for each z and [ψ ∞ ] ρZ -w∗ = {η}. Butforz in theρ Z -dense, open G δ set D ⊂ Z we haveGr ρZ -w ∗f ∗ (z) =η(z) =F ∗ (z) =ψ ∞ (z). (60)6.5 MainResultonApproximationOur main results on the structure and properties of the induced CUSCO, F ∗ (·), arethe following:Theorem 16 (All USCOs with the 3M Property are approximable)Suppose assumptions [A-1] hold and let N ∈ U ρZ -w ∗ beaNashUSCOwithapproximatingtriple,(η,f ∗ , {f n } n ) N ,where η ∈ [N ] ρZ -w ∗, f ∗ ∈ QC ρZ -w ∗,and{f n } n ⊂ C ρZ -w∗(Z, X) are such that forall z ∈ Zf n (z) → f ∗ (z) ∈ η(z).w ∗If F ∗ ∈ h ρZ ×w ∗-Ls{Grf n } is any h ρZ ×w∗-limit point of the corresponding graph sequence,{Grf n } n ⊂ P ρZ ×w ∗ f(Z × X),with induced USCO, F ∗ (·), then the following statements are true:(1) For all z ∈ Z, F ∗ (z) ∈ C w ∗ f(X) is a strongly convex dendrite (locally connectedand hereditarily unicoherent with unique segments).34

(2) For all z ∈ Z, F ∗ (z) is the unique, irreducible continuum about η(z) andF ∗ (z) ⊆ N (z).(3) For all z ∈ Z, F ∗ (z) is minimally essential for N (z) in C w ∗ f(X) -i.e.,F ∗ (z) ∈ M N (z) (C w ∗ f(X)).Proof. (1) Because Z is a locally connected, compact metric space, the connectednessof F ∗ (z) follows Theorem 3 in Hola (1987). Thus, we have F ∗ (z) ∈ C w ∗ f(X) for allz ∈ Z. Because the hyperspace of subcontinua, C w ∗ f(X), is a convex metric space(see Theorems 2 and 3 and Corollary 4 above), the fact that F ∗ (z) is a dendrite (i.e.,a locally connected, hereditarily unicoherent subcontinua) for all z is an immediateconsequence of Theorem 5 above. Moreover, by Whyburn (1942) (also see Charatonikand Charatonik, Theorem 1.3, 1998), all subcontinua of F ∗ (z) are dendrites. Thefact that all subcontinua of F ∗ (z) are also dendrites implies, by Theorem 2.5 of Duda(1962), that F ∗ (z) is strongly convex for all z.(2) We have by expression (59) that η(z) ⊆ F ∗ (z). We will show thatF ∗ (z) =κ(η(z)) := ∩ {M ∈ C w ∗ f(X) :η(z) ⊆ M} for all z ∈ Z,where κ(·) is the Goodykoontz (1977) mapping, continuous with respect to the Hausdorffmetric h w ∗ on P w ∗ f(X) because X is a dendrite (see expression 12), and wherefor each z ∈ Z, κ(η(z)) ∈ C w ∗ f(X) is the unique subcontinuum irreducible aboutη(z). Because F ∗ (z) ∈ C w ∗ f(X) and η(z) ⊆ F ∗ (z) for all z ∈ Z, the equalityF ∗ (z) =κ(η(z)) canfailonlyifforsomez 0 , κ(η(z 0 )) is a proper subset of F ∗ (z 0 ).Letx 0 ∈ F ∗ (z 0 )\κ(η(z 0 )). Because (z 0 ,x 0 ) ∈ F ∗ there exists a sequence {(z n ,f kn (z n ))} nsuch that (z n ,f kn (z n )) → (z 0 ,x 0 ). By compactness of X we can assume WLOG that{f ∗ (z n )} n converges to some l 0 . Therefore, (z 0 ,l 0 ) ∈ Gr ρZ ×w ∗f∗ = Grη. Because(z 0 ,x 0 ) /∈ Gr ρZ ×w ∗f ∗ = Grη,there exists open balls, B δ 0(z 0 ) and B ε 0(x 0 ),ofsufficiently small radii, δ 0 > 0 andε 0 > 0, such that£Bδ 0(z 0 ) × B ε 0(x 0 ) ¤ ∩ Gr ρZ ×w ∗f∗ = ∅.or equivalently,£Bδ 0(z 0 ) × B ε 0(x 0 ) ¤ ∩ Grη = ∅.Therefore, for some η 0 > 0, we have for all (z,x) ∈ B δ 0(z 0 ) × B ε 0(x 0 )Given that (z n ,f k n(z n ))inf (z,x)∈GrρZ ×w ∗f ∗ [ρ Z (z,z)+ρ w ∗(x, x)] > η 0 .→ρZ ×w ∗(z0 ,x 0 ), for some integer N 0 sufficiently large, wehave for all n ≥ N 0 , (z n ,f k n(z n )) ∈ B δ 0(z 0 ) × B ε 0(x 0 ), implying that for all n ≥ N 0and for all (z, x) ∈ Gr ρZ ×w ∗f ∗[ρ Z (z n ,z)+ρ w ∗(f k n(z n ),x)] > η 0 .35

In particular, for all n ≥ N 0 and for (z 0 ,l 0 ) ∈ Gr ρZ ×w ∗f ∗[ρ Z (z n ,z 0 )+ρ w ∗(f k n(z n ),l 0 )] > η 0 .Let δ n := ρ Z (z 0 ,z n )+ 1 n . Because zn →ρZz 0 , δ n > 0 and δ n ↓ 0, there is an integerN 1 ≥ N 0 so that for all n ≥ N 1 δ n < min{δ 0 , η02 } and B δ n (z n ) ⊂ B δ 0(z 0 ). Thus, ifn ≥ N 1 , then for all z 00 ∈ B δ n(z n ) ⊂ B δ 0(z 0 ), f k n(z 00 ) ∈ B ε 0(x 0 ), and we have for alln ≥ N 1 and for all z 00 ∈ B δ n(z n ),[ρ Z (z 00 ,z)+ρ w ∗(f kn (z 00 ),x)] > η 0 for all (z,x) ∈ Gr ρZ ×w ∗f ∗ .In particular, we have for all n ≥ N 1 and for all z 00 ∈ B δ n(z n ),ρ Z (z 00 ,z 0 )+ρ w ∗(f kn (z 00 ),l 0 ) > η 0 where (z 0 ,l 0 ) ∈ Gr ρZ ×w ∗f∗ . (61)Also note that for n ≥ N 1 ,wehavebothB δ n(z n ) ⊂ B δ 0(z 0 ) and z 0 ∈ B δ n(z n ).Thusfor n ≥ N 1 , B δ n(z n ) is a neighborhood of z 0 contained in B δ 0(z 0 ). Because B δ n(z n )is also a neighborhood of z n (contained in B δ 0(z 0 )), by the equi-quasicontinuity ofthe sequence, {f k n} n (see Lemma 20 in Appendix 1), we have for η06> 0, an integerN 2 ≥ N 1 , and a nonempty open set W n ⊂ B δ n(z n ) such that for all ν ≥ max{n, N 2 }and n ≥ N 1ρ w ∗(f kν (z 00 ),f kν (z n )) < η06 for all z00 ∈ W n ⊂ B δ n(z n ). (62)By pointwise convergence, there is an integer N 3 ≥ N 2 such that if ν ≥ max{n, N 3 },thenρ w ∗(f kν (z n ),f ∗ (z n )) < η06 . (63)Moreover, because f ∗ (z n ) → l 0 , there is an integer N 4 ≥ N 3 such that if n ≥ N 4 ,ρw ∗thenρ w ∗(f ∗ (z n ),l 0 ) < η06 . (64)We have, therefore, for all ν ≥ n ≥ N 4 and for all z 00 ∈ W n ⊂ B δ n(z n ) ⊂ B δ 0(z 0 ))ρ w ∗(f kν (z 00 ),l 0 )≤ ρ w ∗(f kν (z 00 ),f kν (z n )) + ρ w ∗(f kν (z n ),f ∗ (z n )) + ρ w ∗(f ∗ (z n ),l 0 ) < η02 , (65)for all z 00 ∈ W n ⊂ B δ n(z n ) ⊂ B δ 0(z 0 ).Finally, given that δ n < min{δ 0 , η02 }, there is an integer N 5 ≥ N 4 such that forall n ≥ N 5 and for all z 00 ∈ B δ n(z n ), ρ Z (z 00 ,z 0 ) < η02 .Thus,forn ≥ N 5 ,wehaveW n ⊂ B δ n(z n ) ⊂ B δ 0(z 0 ),and for all z 00 ∈ W n , ρ Z (z 00 ,z 0 ) < η02 , implying that for all ν ≥ n ≥ N 5 ,[ρ Z (z 00 ,z 0 )+ρ w ∗(f k ν(z 00 ),l 0 )] < η 0 for all z 00 ∈ W n ⊂ B δ n(z n ) ⊂ B δ 0(z 0 ). (66)36

But expression (66) contradicts expression (61) stating that for all n ≥ N 1 ,ρ Z (z 00 ,z 0 )+ρ w ∗(f k n(z 00 ),l 0 ) > η 0 for all z 00 ∈ B δ n(z n ).Therefore, we must conclude thatF ∗ (z) =κ(η(z)) := ∩ {M ∈ C w ∗ f(X) :η(z) ⊆ M} for all z ∈ Z,implying that for all z ∈ Z, F ∗ (z) ∈ C w ∗ f(X) is the unique subcontinuum irreducibleabout η(z) ∈ P w ∗ f(X).It remains to show thatF ∗ (z) ⊆ N (z) for all z ∈ Z.To begin, define the limit point mapping L(·) as follows:L(z) :={x ∈ F ∗ (z) :(z,x) is a limit point of F ∗ \{(z,F ∗ (z))}} .L(·) is an USCO (i.e., L(·) ∈ U(Z, P w ∗ f(X))). Because F ∗ (·) is usc with connectedvalues, it follows from Hiriart-Urruty (1985) that F ∗ (= GrF ∗ (·)) is connected.Thus, GrL contains no isolated points, implying that L(·), in addition tobeing usc, automatically satisfies conditions (b-1) and (b-2) of Beer’s dense selectionTheorem above, further implying via Beer’s Theorem 4 (1983) that L(·) has adense selection, g ∗ . 17 But now by the Corollary 6 in Crannell, Franz, and LeMasurier(2005), because f ∗ (z) =g ∗ (z) for all z in the ρ Z -dense open G δ set D in Z where{f ∗ (z)} = η(z) =F ∗ (z),Grη = Grf ∗ = Grg ∗ = GrL.Because F ∗ has no isolated points, it follows from Beer’s Theorem on dense selectionsthat if (z 0 ,x 0 ) ∈ © (z 0 ,F ∗ (z 0 )) ª ©, then either (z 0 ,x 0 ) is a limit point of F ∗ \(z 0 ,F ∗ (z 0 )) ª or (z 0 ,x 0 ) is a limit point of © (z 0 ,F ∗ (z 0 )) ª only. Given the definitionof the limit point mapping, L(·), (z 0 ,x 0 ) is a limit points of F ∗ \ © (z 0 ,F ∗ (z 0 )) ª ifandonlyifx 0 ∈ L(z 0 ),and(z 0 ,x 0 ) is a limit point of © (z 0 ,F ∗ (z 0 )) ª only if andonly if x 0 ∈ F ∗ (z 0 )\L(z 0 ). Moreover, if x 0 ∈ F ∗ (z 0 )\L(z 0 ),thenz 0 ∈ X\D. Nowsuppose that for some z 0 ∈ Z, thereisapointx 0 ∈ F ∗ (z 0 ) not contained in N (z 0 ).Because η ∈ [N ] ρZ -w ∗ and Grη = GrL and because x0 ∈ F ∗ (z 0 )\L(z 0 ), implies thatz 0 ∈ X\D, F ∗ (z 0 ) has at least two values. Thus, we know by strong convexity thatthere is a unique segment x 1 x 2 ⊂ F ∗ (z 0 ) containing x 0 - and we have by argumentsgiven in the proof of the first part of (2) above that x 1 ∈ L(z 0 ) and x 2 ∈ L(z 0 ).Because F ∗ (z 0 ) is a dendrite and because x 0 ∈ x 1 x 2 ,notequaltox 1 or x 2 , we knowby Whyburn (1942, 1.1) (also see part (3) of Theorem 1.1 in Charatonik and Charatonik,1998) that x 0 is a cut point of F ∗ (z 0 ) (rather than an endpoint of F ∗ (z 0 ) -by part (3) of Theorem 1.1 in Charatonik and Charatonik, 1998, because F ∗ (z 0 ) isa dendrite, x 0 ∈ F ∗ (z 0 ) is either an endpoint or a cut point). Let {x 0 ,V 1 ,V 2 } be17 Recall that in a topological space a point z is isolated if {z} ∩ U z = {z} for all neighborhoodsU z of z. Thepointz is a limit point if for each neighborhood U z of z contains a point z 0 (6= z).37

the cutting defined by the cut point x 0 . Thus, V 1 and V 2 are nonempty, disjointρ w ∗-open sets such thatF ∗ (z 0 )\{x 0 } = V 1 ∪ V 2 ,and V 1 ∪ {x 0 } and V 2 ∪ {x 0 } are subcontinua of F ∗ (z 0 ) (see Willard, 1970, section28). Note that because x 0 /∈ Γ(z 0 ),forsomeε 0 > 0, the open ball of radius ε 0centered at x 0 is such that B ε 0(x 0 ) ∩ Γ(z 0 )=∅. With this in mind, define the USCOΓ ∗ (z) :=Γ(z) ∩ F ∗ (z),and observe that at z 0 Γ ∗ (z 0 ) is the union of two nonempty closed sets,whereΓ ∗ (z 0 )=F 1 ∪ F 2 ,F 1 ⊂ V 1 and F 2 ⊂ V 2 .Because η(z 0 ) ⊂ F ∗ (z 0 ) and F ∗ (z 0 )=κ(η(z 0 )) is the unique subcontinuum irreducibleabout η(z 0 ),wehaveatz 0 ,η(z 0 )=n(K(z 0 )) = n 1 (K(z 0 )) ∪ n 2 (K(z 0 )),where n 1 (K(z 0 )) ⊂ F 1 ⊂ V 1 and n 2 (K(z 0 )) ⊂ F 2 ⊂ V 2 . By Lemma 11 aboven(K(z 0 )) is minimally essential for N K(z 0 )(K(z 0 )) in S K(z 0 ). Therefore, because eachof the closed sets, n 1 (K(z 0 )) and n 2 (K(z 0 )), is a proper subset of n(K(z 0 )), neithern 1 (K(z 0 )) nor n 2 (K(z 0 )) are essential for N K(z 0 )(K(z 0 )) in S K(z 0 ). Therefore, thereare two nonempty, open sets G 1 and G 2 withn 1 (K(z 0 )) ⊂ G 1 and n 2 (K(z 0 )) ⊂ G 2such that for all δ > 0, there exists Ky Fan sets in S K(z 0 ), E δ1 and E δ2 inB hw ∗ ×w ∗ (δ,K(z 0 )) ∩ S K(z 0 ) such thatN(E δ1 ) ∩ G 1 = ∅ and N(E δ2 ) ∩ G 2 = ∅.Let U 1 = V 1 ∩ G 1 and U 2 = V 2 ∩ G 2 .WehaveU 1 and U 2 disjoint open sets suchthat n 1 (K(z 0 )) ⊂ U 1 and n 2 (K(z 0 )) ⊂ U 2 and for all δ > 0, thereexistE δ1 and E δ2in B hw ∗ ×w ∗ (δ,K(z 0 )) ∩ S K(z 0 ) such thatN(E δ1 ) ∩ U 1 = ∅ and N(E δ2 ) ∩ U 2 = ∅. (67)Given that the sets N(E δi ) are compact, under [A-1], there exists open sets W 1 andW 2 such that for i =1, 2,n i (K(z 0 )) ⊂ W i ⊂ W i ⊂ U i .Thus,wehaveforallδ > 0, E iδ ∈ B hw ∗ ×w ∗ (δ,K(z 0 )) ∩ S K(z 0 ) such thatN(E δ1 ) ∩ W 1 = ∅ and N(E δ2 ) ∩ W 2 = ∅. (68)38

Now we have a contradiction: First, because n(K(z 0 )) is a minimal essential setfor N(K(z 0 )) in S K(z 0 ) and because n(K(z 0 )) ⊂ £ W 1 ∪ W 2¤ , there exists a positivenumber δ ∗ > 0 such that for all E ∈ B hw ∗ ×w ∗ (δ ∗ ,K(z 0 )) ∩ S K(z 0 ),N(E) ∩ £ W 1 ∪ W 2¤ 6= ∅. (69)But because δ > 0 can be chosen arbitrarily, choosing δ = δ∗ 3, we have by (68) andthe 3M property, the existence of a Ky Fan setE ∈ B hw ∗ ×w ∗ (3 δ∗ 3 ,K(z0 )) ∩ S K(z 0 ) = B hw ∗ ×w ∗ (δ ∗ ,K(z 0 )) ∩ S K(z 0 )such thathN(E) ∩ W 1 ∪ W 2i = ∅.Thus we must conclude that F ∗ (z) ⊆ N (z) for all z ∈ Z.(3) Finally, we will show that F ∗ (z) ∈ M N (z) (C w ∗ f(X)) for all z ∈ Z. Becauseη(z) ⊂ F ∗ (z) for all z ∈ Z and η ∈ [N ] ρZ -w ∗, we know that F ∗ (z) ∈ E N (z) (C w ∗ f(X))for all z ∈ Z.Let z 0 be any point in Z and Let C 0 ∈ C w ∗ f(X) be a proper subset of F ∗ (z 0 ) andconsider the nonempty set, F ∗ (z 0 )\C 0 . Because F ∗ (z 0 ) is the unique subcontinuumirreducible about η(z 0 ),η(z 0 ) ∩ [F ∗ (z 0 )\C 0 ] 6= ∅.Let x 0 ∈ η(z 0 )∩[F ∗ (z 0 )\C 0 ] and let B ε 0(x 0 ) be an open ball of radius ε 0 > 0 about x 0such that B ε 0(x 0 ) ∩ C 0 = ∅. Therefore, η(z 0 )\B ε 0(x 0 ) ⊂ η(z 0 ) and η(z 0 )\B ε 0(x 0 ) ∈P w ∗ f(X). Becauseη(z 0 ) ∈ M N (z 0 )(P w ∗ f(X))there exists some eε > 0 such that for all n, thereexistsz n ∈ B 1 (z 0 ) such thatnBut for some n 0 ,N (z n ) ∩ B w ∗(eε, η(z 0 )\B ε 0(x 0 )) = ∅. (70)N (z 0 ) ∩ B w ∗(eε, η(z 0 )) 6= ∅ for all z 0 ∈ B 1n 0 (z0 ). (71)Because B ε 0(x 0 ) ∩ C 0 = ∅, given (70) and (71), we must conclude that for all n, thereexists z n ∈ B 1 (z 0 ) such thatnwhereas, for some n 00 ,N (z n ) ∩ B w ∗(eε,C 0 )=∅,N (z 00 ) ∩ B w ∗(eε,F ∗ (z 0 )) 6= ∅ for all z 00 ∈ B 1 (z0 ).n 00Therefore, for any proper subset C 0 of F ∗ (z 0 ) in C w ∗ f(X), C 0 is not essential forN (z 0 ), implying therefore thatF ∗ (z) ∈ M N (z) (C w ∗ f(X)) for all z ∈ Z.39

According to our main result, under assumptions [A-1] all Nash USCOs are approximable.In particular, under assumptions [A-1] we have for any Nash USCO,N ∈ U ρZ −w∗, the existence of an approximating triple,(η,f ∗ , {f n } n ) N ,where η ∈ [N ] ρZ -w ∗, f ∗ ∈ QC ρZ -w ∗,and{f n } n ⊂ C ρZ -w∗(Z, X) are such that for allz ∈ Zf n (z) → f ∗ (z) ∈ η(z),w ∗andsuchthatf n →AKη ∈ [N ] ρZ -w ∗.By the h ρZ ×w ∗-compactness of P ρ Z ×w ∗ f(Z × X), wethenhaveGrf n k→ F ∗ forhρZ ×w ∗some F ∗ ∈ P ρZ ×w ∗ f(Z ×X) with induced CUSCO F ∗ (·). Bytheh ρZ ×w ∗-convergenceof {Grf n } n to F ∗ = GrF ∗ (·), wehaveforanyε > 0 the existence of a positive integerN ε such that for all n ≥ N ε ,h ρZ ×w ∗(Grf n ,F ∗ ):=max{e ρZ ×w ∗(Grf n ,F ∗ ),e ρZ ×w ∗(F ∗ ,Grf n )} < ε,implying via the excess of Grf n over F ∗ ,givenbye ρZ ×w ∗(Grf n ,F ∗ ), that for alln ≥ N ε ,Grf n ⊆ B ρZ ×w ∗(ε,F∗ ).But then, because F ∗ (z) ⊆ N (z) for all z ∈ Z, wehaveforalln ≥ N ε ,Grf n ⊆ B ρZ ×w ∗(ε,F∗ ) ⊆ B ρZ ×w∗(ε, N ).Before moving on to our fixed point result, we note that in our proof that for allz, F ∗ (z) is the unique, irreducible continuum about η(z) (see 2 in our proof above),the equi-quasicontinuity of the pointwise approximating sequence, {f n } n ,playedacritical role - without it, our proof does not work. Our Lemma 20 in Appendix 1,which allowed us to conclude that the sequence, {f n } n , is equi-quasicontinuous, issimple a partial restatement of a result due to Hola and Holy (2011) modified for thegame-theoretic setting considered here. Hola and Holy (2011) in fact show that if(Z, d X ) and (X, d X ) are metric spaces, then for any sequence, {f n } n ⊂ QM dZ -d X,ofquasicontinuous functions d X -converging pointwise to some quasicontinuous function,f ∗ ∈ QM dZ -d X, {f n } n is equi-quasicontinuous if and only if Z is a Baire space.Thus, the Baire space property of Z is critical to our approximation result - as arethe properties of compactness, connectedness, local connectedness, and hereditaryunicoherence in both the parameter space Z and the strategy profile space X.7 Fixed Points7.1 A Fixed Point Theorem for Nash USCOsGiven our approximation result, for cases where the parameter space and the spaceof strategy profiles are one and the same, it is a straightforward exercise to establish40

a fixed point theorem for the Nash correspondence of a parameterized game satisfyingassumptions [A-1]. Our method of proof is similar in spirit to that of Cellina(1969), who proved the Glicksberg-Kakutani Fixed Point Theorem (1952) via graphicalapproximation by continuous functions. But here, rather than requiring that thecorrespondence be convex-valued in order to construct our approximating sequenceof continuous functions, we instead rely on the machinery developed in the sectionsabove to provide us with such a sequence. Thus, here we prove a proof of a new fixedpoint theorem without convexity - closely related to Ward’s Fixed Point Theorem(1958, 1961) - via graphical approximation by continuous functions. The problem ofgraphically approximating, by continuous functions, a nonconvex valued USCO hasbeen a long standing open question. As Cellina is to Glicksberg-Kakutani, our resultis to Ward-type of fixed point theorems.We will maintain the following assumptions [A-2] in our treatment the fixed pointproblem:(1) assumptions [A-1] remain in force;(2) the strategy profile space, X := X 1 ×···×X m , and the parameter space,Z, areoneandthesamespace.Note that under assumptions [A-2], the space Z (and hence the space X) satisfyour prior assumptions [A-1].We will take as our starting point the convex, dendrite continuum (Z, ρ w ∗) whereρ w ∗ is the convex metric compatible with the compact metrizable weak star topologyin Z. Our main result is the following:Theorem 17 (All Nash USCOs have fixed points)Let {G z : z ∈ Z} be a collection of parameterized strategic form games satisfyingassumptions [A-2] with Nash correspondence, z → N (z). ThenN has fixed points.Proof. Let(η,f ∗ , {f n } n ) N ,be an approximating triple for the Nash USCO N ∈ U w ∗ -w ∗,whereη ∈ [N ] w ∗ -w ∗, f ∗ isa w ∗ -s ∗ B 1 selector of η, and{f n } n ⊂ C w ∗ -s∗(Z, Z) is an USCO bounded sequence suchthat kf n (z) − f ∗ (z)k ∗ → 0 for all z. The existence of such an approximating triplefor N ∈ U w ∗ -w ∗ is guaranteed by assumptions [A-2], Proposition 4.3 in Drewnowskiand Labuda (1990), Theorem 3.3 in Jayne and Rogers (2002), and Theorem 1.2 inSpruny (2007).Let {Grf n k} k be an h w ∗ ×w ∗-convergent subsequence with h w ∗ ×w ∗-limit F ∗ ⊂GrN such that for all k,h hw ∗ ×w ∗ (Grf n k,F ∗ ) < 1 k .By h w ∗ ×w ∗-compactness of P w ∗ ×w ∗ f(Z ×Z), such a subsequence exists. Thus, for anyε > 0, there exists a K ε large enough so that for every z ∈ Z there exists (z 0 ,z 00 ) ∈ F ∗such thatρ w ∗(z,z 0 )+ρ w ∗(f n k(z),z 00 ) < 1 k . (72)41

By our main approximation result, for all z we haveη(z) ⊆ F ∗ (z) ⊆ N (z),and by the Tychonoff Fixed Point Theorem (see Theorem 17.56 in Aliprantis andBorder, 2006), each w ∗ -w ∗ -continuous approximating function, f n k(·), hasafixedpoint. Thus for k =1, 2,...,wehaveforsomez n k∈ Z, z n k= f n k(z n k). WLOG,assume that z n k→ z∗ .GiventhatF ∗ (·) is an CUSCO, by (72), we have (z ∗ ,z ∗ ) ∈ F ∗ .w ∗Thus,z ∗ ∈ F ∗ (z ∗ ) ⊂ N (z ∗ ).Rather than prove our fixed point result by continuous approximation, we couldinstead just apply Ward’s Fixed Point Theorem (see Theorem 3 in Ward, 1961). Inparticular, By our main approximation result, we have for any minimal Nash USCO ηof N the CUSCO, κ(η(·)), such that, κ(η(z)) ⊆ N (z). By Ward’s Theorem, becauseZ is a dendrite and κ(η(·)) has compact connected values in Z, κ(η(·)) has a fixedpoint. Thus,z ∗ ∈ κ(η(z ∗ )) ⊂ N (z ∗ ).For the sake of the reader we state Ward’s Theorem 3. First recall that a spacehas the fixed point property for a class of self-mappings if all self-mappings from thisclass and defined on the space in question have fixed points.Theorem 18 (Ward, 1961)If Z is a Peano continuum then the following statements are equivalent:(1) Z is a dendrite.(2) Z has the fixed point property for the class of upper semicontinuous, continuumvaluedmappings.(3) Z has the fixed point property for the class of continuous, closed-valued mappings.8 Fulfilled Expectations Nash Equilibria in Network FormationGames8.1 Consensus Beliefs and Strategic Network FormationReferring to the example in Subsection 2.6, example (2), the belief-parameterizedcollection of strategic form games over random networks is given byG := (P, {P i , Φ i (·),u i (·, (·, ·))} i∈N ) .We will assume that player i 0 s payoff function,u i (·, (·, ·)) : P×(P i ×P −i ) → R,42

is given byu i (μ e , (μ i , μ −i )) := R G + iRG + −iv i (g + i ,g+ −i )dμ i(g + i )dμe −i (g+ −i ),wherev i (·, ·) :G + i× G + −i → R,is continuous on G + i× G + −i . Note that because v i(·, ·) is h K -continuous on G + i× G + −i ,u i (·, (·, ·)) is w ∗ -continuous on P×(P i×P −i ),andforeach(μ e , μ −i ) ∈ P×P −i ,u i (μ e , (·, μ −i )) is affine on P i . Thus, under consensus beliefs μ e ∈ P, playeri 0 s payoffunder random sender network strategy profile μ := (μ i , μ −i ) ∈ P is u i (μ e , (μ i , μ −i )).8.2 Fulfilled Expectations Nash Equilibria in a Belief-ParameterizedCollection Network Formation GamesGiven consensus beliefs μ e , μ e -game, G μ e := {Φ i (μ e −i ),u i(μ e , (·, ·))} i∈N ,foreachμ eG μ e has a nonempty set of Nash equilibria given byoN (μ e ):=nμ ∗ ∈ P : ∀i, u i (μ e , (μ ∗ i , μ∗ −i )) := max μ i ∈Φ i (μ e −i ) u i (μ e , (μ i , μ ∗ −i )) .The parameterized collection of network formation games,G := (P, {P i , Φ i (·),u i (·, (·, ·))} i∈N ) ,has a fulfilled expectations Nash equilibria if the Nash USCO, N (·)), hasfixed points.8.3 ExistenceConsider the collection of μ e -games, {G μ e : μ e ∈ P}, where for each profile of consensusbeliefs, μ e ∈ P, G μ e is given byG μ e := {P i , Φ i (μ e ),u i (μ e , (·, ·))} i∈N.This game has Nikaido-Isoda function given byϕ(μ e , (σ, μ)):= U μ e(σ, μ) − U μ e(μ, μ):= P i u i(μ e , (σ i , μ −i )) − P i u i(μ e , (μ i , μ −i )).Our belief-parameterized collection network formation games, {G μ e : μ e ∈ P} withexternalities over random networks is easily seen to satisfy assumptions [A-1]. Thus,by our fixed point theorem, the Nash correspondence, μ → N (μ), hasafixed point,μ ∗ ∈ N (μ ∗ ),whereμ ∗ is a fulfilled expectations Nash equilibrium random network.43

9 Appendix 1: USCO FundamentalsIn this Appendix we will present the fundamental ideas from the theory of USCOmappings that will be required for our analysis of Nash correspondences. We will uses ∗ to denote the relative strong topology (i.e., the norm or k·k ∗ -topology) on X andrecall that we will use w ∗ rather than ρ w ∗ to denote the relative weak star topologyon X.9.1 Basic DefinitionsLet Γ(·) be a ρ Z -w ∗ -upper semicontinuous set-valued mapping with nonempty, compactvalues, and denote byU ρZ -w ∗ := U(Z, P f(X)) (73)the collection of all such mappings, where as before P f (X) denotes the collection ofall nonempty w ∗ -closed subsets of X.The graph of Γ(·) is given byGrΓ := {(z,x) ∈ Z × X : x ∈ Γ(z)}. (74)Because (X, ρ w ∗) is a compact metric space, we know thatΓ(·) ∈ U ρZ -w ∗ifandonlyifGrΓ is a ρ Z ×w ∗ -closed subset of Z ×X. Conversely, if F is a nonempty,ρ Z × w ∗ -closed subset of Z × X, then the induced set-valued mapping, F (·), givenbyz → F (z) :={x ∈ X :(z,x) ∈ F },is an USCO (i.e., F (·) ∈ U ρZ -w ∗).We say that Λ(·) ∈ U ρZ -w∗ is a minimal USCO if for any other USCO, Ψ,GrΨ ⊆ GrΛ implies that GrΨ = GrΛ.We will denote byM ρZ -w ∗ := M(Z, P f(X)) (75)the collection of all minimal USCOs in U(Z, P f (X)).In general, given any USCO Γ ∈ U ρZ -w ∗ we say that Λ ∈ U ρ Z -w∗ is a minimalUSCO of Γ if Λ ∈ M ρZ -w ∗ and GrΛ ⊆ GrΓ. Wewilldenoteby[Γ] ρ Z -w∗ the collectionof all minimal USCOs belonging to Γ. Thus,forΓ ∈ U ρZ -w ∗[Γ] ρZ -w ∗ := {Λ(·) ∈ M ρ Z -w∗ : GrΛ ⊆ GrΓ}. (76)By Proposition 4.3 in Drewnowski and Labuda (1990), if Γ ∈ U ρZ -w∗,thenΓ possessesat least one minimal USCO Λ. Thus,viewing[·] ρZ -w ∗ as a mapping from U ρ Z -w ∗ intoM ρZ -w ∗, [·] ρ Z -w ∗ is nonempty-valued.Finally, we say that an USCO Γ ∈ U ρZ -w ∗ is quasiminimal if [Γ] ρ Z -w ∗ = {Λ}.Thus, USCO Γ is quasiminimal if it has (or contains) one and only one minimalUSCO.WewilldenotebyQM ρZ -w ∗ := QM(Z, P f(X))the collection of all quasiminimal USCOs in U(Z, P f (X)).44

9.2 Characterizing Minimal USCOsWe will need the notion of a quasi-continuous function in order to fully characterizeminimal USCOs.Definition 8 (ρ Z -w ∗ -Continuity and ρ Z -w ∗ -Quasi-Continuity)(1) (ρ Z -w ∗ -Continuity) f : Z → X is said to be ρ Z -w ∗ -continuous at z ∈ Z if forevery w ∗ -open subset G of X such that f(z) ∈ G there is a ρ Z -open set U z containingz such that f(U z ) ⊂ G. The function f is ρ Z -w ∗ -continuous if it is ρ Z -w ∗ -continuousat every z ∈ Z.(2) (ρ Z -w ∗ -Quasicontinuity) f : Z → X is said to be ρ Z -w ∗ -quasicontinuous atz ∈ Z if for every w ∗ -open subset G of X such that f(z) ∈ G and for every ρ Z -openset U z containing z there exists another ρ Z -open set W ⊂ U z such that f(W ) ⊂ G.The function f is ρ Z -w ∗ -quasi-continuous if it is ρ Z -w ∗ -quasicontinuous at everyz ∈ Z.We will denote by C ρZ -w ∗ := C ρ Z -w ∗(Z, X) the collection of all ρ Z-w ∗ -continuousfunctions defined on Z with values in X, and we will denote byQC ρZ -w ∗ := QC ρ Z -w∗(Z, X)the collection of all ρ Z -w ∗ -quasicontinuous functions defined on Z with values in X.Let F := F(Z, X) denote the collection of all functions defined on Z taking valuesin X. Recall that a function, f : Z → X, is a selection from USCO Γ ∈ U ρZ -w ∗ iff(z) ∈ Γ(z) for all z ∈ Z. Denote the collection of all selections of Γ by Σ Γ . Wesay that f ∈ Σ Γ is a ρ Z × w ∗ -dense selection of Γ if Gr ρZ ×w∗f = GrΓ (i.e., if theρ Z × w ∗ -graph closure of f is equal to the graph of Γ). When no confusion can occur,we will write Grf rather than Gr ρZ ×w ∗f.The following two part Lemma is due to Crannell, Franz, and LeMasurier (2005)and Hola and Holy (2009). In our statement of these results we take as given thefact that Z and X are compact metric spaces, equipped with convex metrics, ρ Z andρ w ∗,whereρ Z is compatible with the d Z metric topology on Z and ρ w ∗ is compatiblewith the weak star topology on X. In fact, Part (1) due to Crannell, Franz, andLeMasurier (2005) requires that Z and X be compact metric spaces.Lemma 19 (Characterizing Minimal USCOs)(1) (Crannell, Franz, and LeMasurier, 2005)For all g ∈ F there exists f ∈ QC ρZ -w ∗such that Grf ⊂ Gr ρ Z ×w ∗g.(2) (Hola-Holy, 2009)The following statements are equivalent.(i) Λ ∈ M ρZ -w ∗.(ii) There exists a function f ∈ QC ρZ -w ∗ ∩ Σ Λ such that Gr ρZ ×w∗f = GrΓ.(iii) Σ Λ ⊂ QC ρZ -w ∗and for all f ∈ Σ Λ, Gr ρZ ×w∗f = GrΓ.45

9.3 Equi-QuasicontinuityIn order to prove our main result on the approximability of USCOs with the 3Mproperty, we will need the notion of equi-quasicontinuity. We begin with the definitionof equi-quasicontinuity.Definition 9 (Equi-Quasicontinuity)A sequence of functions, {f n (·)} n , f n : Z → X, is equi-quasicontinuous at z ∈ Z,if for every ε > 0, and every open neighborhood U z of z, there is an integer, N ε anda nonempty open set W ⊂ U z such that for every n ≥ N εd ρX (f n (z),f n (z)) < ε for all z ∈ W .Thesequenceoffunctions,{f n (·)} n , is equi-quasi-continuous if it is equi-quasicontinuousat all z ∈ Z.The following result due to Hola and Holy (2011) gives necessary and sufficientconditions for a sequence of quasi-continuous functions (or continuous functions),{f n } n , converging pointwise to some function f to be equi-quasi-continuous.Lemma 20 (Quasi-Continuity and Equi-Quasi-Continuity)Suppose assumptions [A-1] hold. Let {f n (·)} n ⊂ QC(Z, X) be a sequence of quasicontinuousfunctions converging pointwise to a function f. The following statementsare equivalent.(i) f is quasi-continuous (i.e., f ∈ QC(Z, X)).(ii) The sequence {f n (·)} n is equi-quasi-continuous.9.4 Densely Continuous FormsFor each function f : Z → X let C ρZ -w ∗(f) denote the set of ρ Z-w ∗ -continuity pointsof f in Z. Thus,C ρZ -w∗(f) is given byC ρZ -w ∗(f) :={z ∈ Z : f is ρ Z-w ∗ -continuous at z} . (77)If f is a ρ Z -w ∗ -continuous function, then C ρZ -w ∗(f) =X. IfC ρ Z -w∗(f) is dense (i.e.,ρ Z -dense) in Z, thenf is said to be a densely ρ Z -w ∗ -continuous function. We willlet DC ρZ -w ∗ := DC ρ Z -w ∗(Z, X) denote the collection of all densely ρ Z-w ∗ -continuousfunctions.For f ∈ DC ρZ -w ∗,letf| C ρZ -w ∗ (f) denote the function f restricted to C ρZ -w ∗(f)and note that Grf| CρZ -w ∗(f) is a subset of Z × X with ρ Z × w ∗ -graph closure,Gr ρZ ×w ∗f| C ρZ -w ∗(f) , contained in P f (Z × X). WewillletDCF f := DCF f (Z × X)46

denote the collection of all densely continuous forms. Thus, DCF f (Z × X) is givenbyDCF f (Z × X)o:=nE ∈ P ρZ ×w ∗ f(Z × X) :E = Gr ρZ ×w ∗f| C ρZ -w ∗ (f) for some f ∈ DC ρZ -w ∗ .(78)The following two part Lemma summarizes what we need to know about denselycontinuous forms. Part (1) of the Lemma, due to Crannell, Franz, and LeMasurier(2005), tells us that all selections from a densely continuous form share the same setof continuity points, and moreover, that all such selections are equal on this set ofcontinuity points. Part (2) is due to Hola and Holy (2009) and tells us that for anyselection from a densely continuous form, the graph closure of the selection restrictedto its set of continuity points is equal to the graph closure of the selection itself.Thus, together Parts (1) and (2) tell us that the graph closures of all selections froma densely continuous form are equal.In our statement of these results we take as given the fact that Z and X arecompact metric spaces, equipped with convex metrics, ρ Z and ρ w ∗, compatible withthe d Z topology in Z and with the weak star topology in X. In fact, Part (1) due toCrannell, Franz, and LeMasurier (2005) requires that Z and X be compact metricspaces. Part (2) continues to hold if Z and X are not compact - but requires that Zbe a metric Baire space and X a metric space.Lemma 21 (Minimal USCOs and Densely Continuous Forms)(1) (Crannell, Franz, and LeMasurier, 2005)If GrΛ ∈ DCF f (Z × X) and f ∈ Σ Λ ∩ QC ρZ -w ∗, then for all g ∈ Σ Λ∩ QC ρZ -w ∗,C ρZ -w ∗(f) =C ρ Z -w ∗(g) and Grf| C ρZ -w ∗(f) = Grg| CρZ -w ∗ (g) .(1) (Hola and Holy, 2009)The following statements are equivalent.(i) Λ ∈ M ρZ -w∗(Z, X).(ii) GrΛ ∈ DCF f (Z × X).(iii) If f ∈ Σ Λ ,thenGrf| CρZ -w ∗(f) = Grf.(iv) If f ∈ Σ Λ , then for every z ∈ Z and every neighborhood U of (z, f(z)) thereexists z 0 ∈ C ρZ -w ∗(f) such that (z0 ,f(z 0 )) ∈ U.47

9.5 Dense SelectionsFor each F ∈ P ρZ ×w ∗ f(Z × X) with induced USCOthere is a single-valued set given byz → F (z) :={x ∈ X :(z,x) ∈ F } , (79)S F := {z ∈ Z : F (z) is a singleton} . (80)We will denote by DSF f (Z × X) the collection of all densely single-valued forms,that is, the collection of all sets F ∈ P ρZ ×w ∗ f(Z × X) such that S F is ρ Z -dense in Z.Thus, the collection DSF f := DSF f (Z × X) is given byDSF f (Z × X) := © F ∈ P ρZ ×w ∗ f(Z × X) :S F is ρ Z -dense in Z ª . (81)It follows from Proposition 1 in Crannell, Frantz, and LeMasurier (2005) that if a setF ∈ P ρZ ×w ∗ f(Z × X) is such that F = Gr ρZ ×w∗f for some function f : Z → X, thenS F = C ρZ -w ∗(f) =S Gr ∗f. Thus, the single-valued set, S ρZ ×w Gr ρZ ×w∗f, correspondingto any function f : Z → X, or the single-valued set, S F , corresponding to any setF ∈ P ρZ ×w ∗ f(Z ×X) such that F = Gr ρZ ×w ∗f is equal to the set of ρ Z-w ∗ -continuitypoints, C ρZ -w∗(f), of the function f generating that set via graph closure. Moreover,given F ∈ P ρZ ×w ∗ f(Z × X), iff(·) is a selection of F (·), the closure of whose graph isstrictly contained in F ,thenS F ⊂ C ρZ -w ∗(f), andifg is a selection of Gr ρ Z ×w ∗f(·),then C ρZ -w ∗(f) ⊂ C ρ Z -w ∗(g).Let F ∈ P f (Z × X) with induced USCO F (·) ∈ U ρZ -w ∗ := U(Z, P w ∗ f(X)). Wesay that a function, f : Z → X, is a selection of F (·) if f(z) ∈ F (z) for all z ∈ Z.Definition 10 (Dense Selections)We say that f is a dense selection of USCO F (·) ∈ U ρZ -w ∗F (·) andGr ρZ ×w∗f = GrF.if f is a selection ofWe close our brief discussion of USCO mappings with a result by Beer (1983)characterizing dense selections. Recall that in a topological space a point z is isolatedif {z}∩U z = {z} for all neighborhoods U z of z. Thepointz is a limit point if for eachneighborhood U z of z contains a point z 0 (6= z). The following result, characterizingUSCOs with dense selections, is an immediate consequence of Theorem 1 in Beer(1983). In our statement of Beer’s result we take as given the fact that Z and Xare compact metric spaces, equipped with convex metrics, ρ Z on Z compatible withthe metric d Z and ρ w ∗ on X compatible with the weak star topology in X. Infact,Beer’s result requires only that Z be a complete separable metric space (i.e., a Polishspace) and that X be a sigma compact complete, separable metric space.48

Lemma 22 (A characterization of USCO correspondences with dense selections,Beer, 1983)Suppose [A-1] holds. Let Γ ∈ U ρZ -w ∗ := U(Z, P w ∗ f(X)). The following statementsare equivalent.(a) Γ has a dense selection.(b) Γ has the following properties:(b-1) For each z ∈ Z the set {(z, Γ(z))} := {(z,x) :x ∈ Γ(z)} includes at most oneisolated point of GrΓ;(b-2) For each (z, x) ∈ {(z,Γ(z))}, (z,x) is not a limit point of GrΓ\{(z, Γ(z))} ifand only if (z, x) is an isolated point of GrΓ.Let X = Y =[0, 2] and consider the USCO, Λ ∈ U := U(X, P f (Y )), givenbyΛ(x) =⎧⎨⎩{0} 0 ≤ x

10 Appendix 2: The Proof of Lemma 8(1) Let {E n } n ⊂ S be a sequence of Ky Fan sets such that h w ∗ ×w ∗(En ,E 0 ) → 0. Wemust show that E 0 ∈ S. Becauseh w ∗(D(E n ), D(E 0 )) → 0and because each D(E n )=R(E n ) is convex, we have that D(E 0 ) ∈ P w ∗ fc(X) andD(E 0 )=R(E 0 ). Thus, E 0 satisfies (Z1). Also, note that E 0 satisfies (Z2). Theproof will be complete if we can show that for all x ∈ R(E),{y ∈ D(E) :(y, x) /∈ E}is convex and possibly empty (i.e., that E 0 satisfies (Z3)). Suppose not. Then forsome y 1 , y 2 ,andx 0 in R(E 0 ),wehavey i ∈ © y ∈ D(E 0 ):(y,x 0 ) /∈ E 0ª , i =1, 2, (*)but for some λ 0 ∈ (0, 1), y 0 = λ 0 y 1 +(1− λ 0 )y 2 ∈ D(E 0 ) but(y 0 ,x 0 ) ∈ E 0 . (**)Because D(·) is continuous (and in particular, lower semicontinuous) there exist sequences,{y 1n } n and {y 2n } n , such that y in ∈ D(E n ) for all n, i =1, 2, andy 1n → y1 ,w ∗y 2n → y2 .w ∗Therefore,y 0n = λ 0 y 1n +(1− λ 0 )y 2n →w ∗ y0 ∈ D(E n )We have (y 1 ,x 0 ) /∈ E 0 , (y 2 ,x 0 ) /∈ E 0 , but (y 0 ,x 0 ) ∈ E 0 .Becausewe have for all n sufficiently large,(y in ,x 0 ) →w ∗ ×w ∗ (yi ,x 0 ) /∈ E 0 , i =1, 2,andh w ∗ ×w ∗(En ,E 0 ) → 0,(y in ,x 0 ) /∈ E n , i =1, 2.Because E n ∈ S for all n, wehaveforalln sufficiently large,(y 0n ,x 0 )=(λ 0 y 1n +(1− λ 0 )y 2n| {z }y 0n ,x 0 ) /∈ E n .Thus,(y 0n ,x 0 ) ∈ [D(E n ) × X] ∩ [(X × X)\E n ] for all n.50

Because E n → E 0 , we have by Proposition 3.2.2 in Klien and Thompson (1984)hw ∗ ×w ∗that(X × X)\E 0 = Li w ∗ ×w ∗{(X × X)\En }. (82)Given (82), we have, therefore, that£D(E 0 ) × X ¤ ∩ £ (X × X)\E 0¤ = £ D(E 0 ) × X ¤ ∩ Li w ∗ ×w ∗{(X × X)\En }. (83)But now we have a contradiction. We haveThus,(y 0n ,x 0 ) −→w ∗ ×w ∗ (y0 ,x 0 ) ∈ £ D(E 0 ) × X ¤ ∩ Li w ∗ ×w ∗{(X × X)\En }.(y 0 ,x 0 ) ∈ £ D(E 0 ) × X ¤ ∩ £ (X × X)\E 0¤ .But by assumption,(y 0 ,x 0 ) ∈ £ D(E 0 ) × X ¤ ∩ £ E 0¤ ,a contradiction.(2) It is easy to see that D(K(z)) = R(K(z)) = Φ(z). Thus (Z1) holds. Letz ∈ Z be any parameter vector. We must show that K(z) ∈ S. RecallthatK(z) :={(y, x) ∈ Φ(z) × Φ(z) :ϕ(z,(y, x)) ≤ 0},and note that for all y ∈ Φ(z), ϕ(z, (y, y))=0. Thus, (Z2) holds.Toseethat(Z3)holdsobservethatbecauseϕ(z, (·,x)) is quasiconcave, y ∈ Φ(z)such that(y, x) /∈ K(z)is given by the set,{y ∈ Φ(z) :ϕ(z,(y, x)) > 0},and this set is convex (or empty).11 Appendix 3: The Proof of Lemma 11Because S is an h w ∗ ×w∗-compact metric space and hence a Baire space and becausen(·) is a minimal USCO, there is an h w ∗ ×w∗-dense subset, D, ofS such n(E) is asingleton for all E ∈ D (see Anguelov and Kalenda, 2009, Lemma 7). Moreover, forany E e ∈ S, D eE := D ∩ S eE is dense in S eE .Suppose n(·) ∈ [N(·)] but that for some E e ∈ S, n eE (·) is not a minimal USCO ofNE e (·) on S eE . Then, there is a minimal USCO m(·) of n eE (·) (i.e., m(·) ∈ [n (·)]) sucheEthat for some E 0 ∈ SE e , m(E0 ) is a proper subset of nE e (E0 ). By Lemma 11, becausem(·) ∈ [n eE (·)], wehaveforanyE 0 ∈ S eE , m(E 0 ) ∈ M n eE (E 0 )(S eE ),implyingthatforall ε 0 > 0 there exists δ 0 > 0 such that for allE ∈ B hw ∗ ×w ∗ (δ 0 ,E 0 ) ∩ SE e , ⎫⎬(84)n eE (E) ∩ B w ∗(ε 0 ,m(E 0 ⎭)) 6= ∅.51

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) andwehaveforeachE ∈ 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 and 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) ∩ Dand 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 thatand 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 HE0and choose k γ so thatk 1γ< ε∗ Xγ. Second, for each ν >k γ ,chooseE ν ∈ B hw ∗ ×w ∗ ( 1 ν ,E0 ) ∩ D eE and H ν ∈ B hw ∗ ×w ∗ ( 1 ν ,E0 ) ∩ Sand for each ν and 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 →∞, and 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 →∞and 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 ,andh 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) and the continuity of the excess functions, e w ∗(N(·),N(·))on S × S that there exists a δ 4 > 0 and positive integers, k ν 3 and ν 3 , such that forν ≥ ν 3 ,andfork ≥ 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 ,andk ≥ 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 ,andk ≥ k ν 3z vk ∈ B w ∗(ε 2 ,n eE (E 0 ) ∩ B w ∗(ε 0 ,m(E 0 ))). (90)Together, (89) and (90) contradict (87).53

12 Appendix 4: The Proof That All KFCs Have the 3MPropertySuppose not. Then for some E e ∈ S and E 0 ∈ SE e ,theD-restricted KFCN eE (·) :S eE → P w ∗ f(X)is such that there exists a pair of disjoint closed sets, F 1 and F 2 in D( e E), andanopen ball, B δ 0(E 0 ) ∩ S eE , δ 0 > 0, containingtwoD-equivalent Ky Fan sets, E 1 and E 2in S e E , such that N e E (E1 ) ∩ F 1 = ∅ and N e E (E2 ) ∩ F 2 = ∅,but such that for all E 3 ∈ B hw ∗ ×w ∗ (3δ 0 ,E 0 ) ∩ S e EN eE (E 3 ) ∩ [F 1 ∪ F 2 ] 6= ∅.First, given that NE e (·) is an USCO, under [A-1] there are disjoint open sets Uisuch that F i ⊂ U i and NE e (Ei ) ∩ U i = ∅, i =1, 2. Thus,N eE (E 3 ) ∩ [F 1 ∪ F 2 ] 6= ∅ for all E 3 ∈ B hw ∗ ×w ∗ (3δ 0 ,E 0 ) ∩ S eE ,implies thatN e E (E3 ) ∩ [U 1 ∪ U 2 ] 6= ∅ for all E 3 ∈ B hw ∗ ×w ∗ (3δ 0 ,E 0 ) ∩ S e E . ⎫⎬⎭(91)We will show that (91) leads to a contradiction by constructing a Ky Fan set,E ∗ ∈ S eE with E ∗ ∈ B hw ∗ ×w ∗ (3δ 0 ,E 0 ) such thatN eE (E ∗ ) ∩ [U 1 ∪ U 2 ] 6= ∅ (*),and such that (*) implies that N eE (E i ) ∩ U i 6= ∅ for some i =1and/or 2.candidate for such a set is given byOurE ∗ := [E 1 ∩ (X × U 2 ) c ] ∪ [E 2 ∩ (X × U 1 ) c ] (92)where(X × U i ) c := © (y, x) ∈ X × X : x/∈ U iª .To complete the proof we must show that, (1) E ∗ ∈ SE e ,(2)E∗ ∈ B hw ∗ ×w ∗ (3δ 0 ,E 0 ),and (3) NE e (E∗ ) ∩ £ U 1 ∪ U 2¤ 6= ∅⇒NE e (Ei ) ∩ U i 6= ∅ for some i =1and/or 2.(1) E ∗ ∈ SE e : It is easy to see that E∗ ∈ P w ∗ w ∗ f(X × X). Moreover, becauseE i ∈ S eE i =1, 2, it is easy to see that (Z1) holds for E ∗ .Thus,D(E ∗ ) is nonempty,w ∗ -closed, and convex andD(E ∗ )=R(E ∗ )=D( e E).Also, it is easy to see that (Z2) holds for E ∗ .Thus,(y, y) ∈ E ∗ for all y ∈ D(E ∗ ).It remains to show that for all x ∈ R(E),{y ∈ D(E ∗ ):(y, x) /∈ E ∗ }54

is convex or empty.Let x ∈ U 1 , then because U 1 and U 2 are disjoint,{y ∈ D(E ∗ ):(y, x) /∈ E ∗ } = © y ∈ D(E ∗ ):(y,x) /∈ E 1ª ,a convex or empty set because E 1 ∈ S eE .Let x ∈ U 2 , then because U 1 and U 2 are disjoint,{y ∈ D(E ∗ ):(y, x) /∈ E ∗ } = © y ∈ D(E ∗ ):(y,x) /∈ E 2ª ,a convex or empty set because E 2 ∈ S e E .Let x ∈ D(E ∗ )\U 1 ∪ U 2 .Then{y ∈ D(E ∗ ):(y, x) /∈ E ∗ }= © y ∈ D(E ∗ ):(y, x) /∈ E 1ª ∩ © y ∈ D(E ∗ ):(y, x) /∈ E 2ª ,the later being the intersection of convex or empty sets. Therefore,is convex or empty.(2) E ∗ ∈ B hw ∗ ×w ∗ (3δ 0 ,E 0 ):Wehaveand by the triangle inequality,{y ∈ D(E ∗ ):(y, x) /∈ E ∗ }E ∗ =[E 1 ∩ (X × U 2 ) c ] ∪ [E 2 ∩ (X × U 1 ) c ] (93)h w ∗ ×w ∗(E1 ,E 2 ) ≤ h w ∗ ×w ∗(E1 ,E 0 )+h w ∗ ×w ∗(E2 ,E 0 ) < 2δ 0 ,andh w ∗ ×w ∗(E∗ ,E 0 ) ≤ h w ∗ ×w ∗(E∗ ,E 1 )+h w ∗ ×w ∗(E1 ,E 0 ).(94)We know already that h w ∗ ×w ∗(E1 ,E 0 ) < δ 0 .Considerh w ∗ ×w ∗(E∗ ,E 1 ).Wehaveh w ∗ ×w ∗(E∗ ,E 1 ):=max © e w ∗ ×w ∗(E∗ ,E 1 ),e w ∗ ×w ∗(E1 ,E ∗ ) ª .It is easy to check that,e w ∗ ×w ∗(E∗ ,E 1 )=sup (y,x)∈E ∗ ρ w ∗ ×w ∗((y, x),E1 )=sup (y,x)∈[E 2 ∩(X×U 1 ) c ] ρ w ∗ ×w ∗((y,x),E1 )≤ sup (y,x)∈E 2 ρ w ∗ ×w ∗((y, x),E1 )=e w ∗ ×w ∗(E2 ,E 1 ).To show that e w ∗ ×w ∗(E1 ,E ∗ ) ≤ e w ∗ ×w ∗(E1 ,E 2 ) observe thate w ∗ ×w ∗(E1 ,E ∗ )=sup (y,x)∈E 1 ρ w ∗ ×w ∗((y, x),E∗ )=sup (y,x)∈E 1 ρ w ∗ ×w ∗((y, x), [E1 \(X × U 2 )] ∪ [E 2 \(X × U 1 )]).55

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

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