Noncooperative foundations of the nucleolus in majority games

This paper studies coalition formation, payoff division and expected payoffs in a ”divide the dollar by majority rule” game with random proposers. A power index is called self-confirming if it can be obtained as an equilibrium of the game using the index itself as probability vector. Unlike the Shapley value and other commonly used power indices, the nucleolus has this property. The proof uses Kohlberg’s (1971) characterization of the nucleolus, reinterpreting the balancing weights as probabilities in a mixed strategy equilibrium. As for arbitrary probability vectors, the nucleolus seems a more likely equilibrium than the Shapley value because it assigns competitive prices to the players.