Externalities, Nonconvexities, and Fixed Points

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