The desirability relation of simple games (Q1071662)

Let \(G=(N,W)\) be a simple game. A coalition S is at least as desirable as a coalition T if for every B contained in N such that \(B\cap (S\cup T)=\emptyset\), \(B\cup T\in W\Rightarrow B\cup S\in W\). The author investigates the properties of a simple game with a complete desirability relation and the relationships between the desirability relation of a simple game and the desirability of other simple games derived from it. A weighted majority game has a complete desirability relation the asymmetric part of which is acyclic. The author also shows that there exists a proper simple game that has these properties without being a weighted majority game.