Further topological aspects of vector-valued optimization in multi-player quantitative games

First, let us define a N-player quantitative game as in Ref. (1). Let there be given a set G, whose members will be denoted by x, ordered by a reflexive relation >, such that x\´ > x" or x" > x\´ whenever x\´ and x" are distinct members of the union of the domaine and the range of >. Let there be given prescribed sets S??, ??=1,2,...N, whose members will be denoted by S??, ?? - 1,2,...N, respectively; and let there be given a relation RCD ?? P(G), where D = G ?? S1 ??... ?? SN and P(G) is the collection of all non empty subsets of G. We shall suppose that (xi, s) R??, where s = (s1, s2,... sN), (xi, s) ?? D, ?? ?? P(G), implies that (a) xi ????, and (b) x > xi, ??V ?? ?? ??