Minimal usco and minimal cusco maps and compactness

K-Correspondences, USCOs, and fixed point problems arising in discounted stochastic games

“…For compact metric spaces (X, ρ X ) and (Y , ρ Y ), let U ρ Y -ρ X denote the collection of all upper semicontinuous correspondences (•) defined on Y taking nonempty, ρ X -closed (and hence ρ X -compact) values in X. Equip the product space Y × X with the sum metric, ρ Y ×X := ρ Y + ρ X . Following the literature, all such mappings are USCOs (e.g., see Hola and Holy [18]). We say that the USCO (…”

“…is convex-valued, we have that player d 's best strategy response correspondence B d (•, •) is a CUSCO * (i.e., is a convex valued USCO * ; see Hola and Holy [18]). Thus, the best strategy response cor-…”

“…c We will often refer to an m-player DSG having an uncountable state space and compact metric action spaces as an uncountable-compact DSG. d An USCO is an upper semicontinuous correspondence taking nonempty compact values (e.g., see Hola and Holy [18]). A weak * USCO, is an USCO where upper semicontinuity is with respect to weak * topologies (USCO * ).…”

“…is an USCO * (see Hola and Holy[18]) and takes decomposable values in L ∞ X . By Lemma 1 above E B (•) is a K -correspondence.…”

Ascoli-type theorems for locally bounded quasicontinuous functions, minimal usco and minimal cusco maps

“…Minimal usco and minimal cusco maps are used in many papers (see [5], [7], [13], [10], [11], [12]). Minimal usco maps are a very convenient tool in the theory of games (see [6]) or in functional analysis.…”