Large symmetric games are characterized by completeness of the desirability relation

The paper presents a characterization of continuous cooperative games (set functions) which are monotonic functions of countably additive non-atomic measures. The characterization is done through a natural desirablity relation defined on the set of coalitions of players. A coalition S is at least as desirable as a coalition T (with respect to a given game υ (in colational form)), if for each coalition U that is disjoint from S ∪ T, υ(S ∪ U) ≥ υ(T ∪ U). The characterization asserts, that a game υ is of the form υ = f · μ, where μ is a non-atomic signed measure and f is a monotonic and continuous function on the range of μ, if, and only if, it is in pNA′ (i.e., it is a uniform limit of polynomials in non-atomic measures or equivalently it is uniformly continuous function in the NA-topology) and has a complete desirability relation.