Simple games on closure spaces (Q1592627)

A game on a closure space \(L\subseteq 2^{N}\) where \(N\) is a finite set is a real function \(v:L\rightarrow R\) such that \(v(\emptyset)=0.\) The authors assume that the games are simple and monotonic. After defining the closure spaces and convex geometries, some of their properties are described. Let \(\Gamma(L)\) denotes the vector space of all games on \(L\) and \(v\in L\) be a simple game. Under the appropriate assumption on \(L,\) the authors show that \(v\) is supermodular if and only if the game \(v\) is a unanimity game and the core\((L,v)\) is a stable set if and only if the game \(v\) is a unanimity game. Further, it is also shown that \(v\) is weak if and only if \(\text{core}(L,v)\neq \emptyset\) and each imputation that satisfies the dummy axiom is an element of the Weber set.