This paper studies non-cooperative bargaining with random proposers in proper simple games. A power index is called consistent if it can be obtained as equilibrium of the game with random proposers using the index itself as probability vector. Unlike the Shapley-Shubik and Banzhaf indices, the nucleolus has this property. The proof uses the balancedness result in Kohlberg (1971) reinterpreting the balancing weight as mixed strategies.