The least core, nucleolus, and kernel of homogeneous weighted majority games (Q1195593)

A (generalized) weighted majority game on a set of players \(N=\{1,\dots,n\}\) is defined as follows: for every coalition \(S\subseteq N\), its weight \(M(S)\) is defined, and a minimal weight \(\lambda\) is given so that a coalition \(S\) is winning iff \(M(S)\geq \lambda\). For example, in simple voting, \(M(S)=| S|\), and \(\lambda=\lfloor n/2\rfloor+1\). In general, \(M(S)\) is not necessarily additive. Such a game is called homogeneous if, whenever \(M(S)>\lambda\), there exists a \(T\subset S\) such that \(M(T)=\lambda\). For some games, a simpler function \(M'\) leads to exactly the same set of winning coalitions. It is known that for every such game, there exists a representation in which the values \(M(S)\) are all integers, and among these, there exists a unique representation (called the minimal representation) for which \(M(N)\) is minimal. For constant-sum superadditive games, Peleg gave a complete description of the kernel and the nucleolus. These results are generalized to arbitrary homogeneous weighted majority games ``without steps'' (crudely speaking, a step is a player that cannot be substituted by several ``weaker'' ones). The least core, kernel, and nucleolus turn out to be directly related to the minimal representation.