The nucleolus and kernel for simple games, or special valid inequalities for 0-1 linear integer programs

An equivalence between simplen-person cooperative games and linear integer programs in 0–1 variables is presented and in particular the nucleolus and kernel are shown to be special valid inequalities of the corresponding 0–1 program. In the special case of weighted majority games, corresponding to knapsack inequalities, we show a further class of games for which the nucleolus is a representation of the game, and develop a single test to show when payoff vectors giving identical amounts or zero to each player are in the kernel. Finally we give an algorithm for computing the nucleolus which has been used successfully on weighted majority games with over twenty players