Proportional payoffs in majority games

This paper extends the Baron-Ferejohn model of legislative bargaining to general weighted majority games with two modifications: first, payoff division can only be agreed upon after the coalition has formed (two-stage bargaining); second, negotiations in the coalition can break down, in which case a new coalition may be formed (reversible coalitions). Under the most natural bargaining protocol, both expected payoffs and actual payoff division are proportional to the voting weights provided that the set of winning coalitions of minimum weight is weakly balanced and that the breakdown probability tends to 0. Homogeneity of the voting weights is neither necessary nor sufficient for proportional payoffs. Intermediate values of the breakdown probability produce predictions consistent with the empirical evidence on portfolio allocation in Europe: a moderate propoper advantage and a linear relationship between weights and ex post payoffs for all coalition members other than the proposer.