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 a weak version of Kohlberg's [SIAM J. Appl. Math. 20 (1971) 62]
balancedness result reinterpreting the balancing weights as probabilities in a mixed strategy …