Two-Stage Bargaining with Reversible Coalitions: the Case of Apex Games

This paper studies coalition formation and payoff division in a class of majority games (apex games) assuming that payoff division can only be agreed upon after forming the coalition (two-stage bargaining) and that negotiations in the coalition can break down and a new coalition be formed (reversible coalitions). In contrast with the results of other two-stage models, all minimal winning coalitions may form and expected payoffs coincide with the per capita nucleolus. These results are robust to small changes in the bargaining procedure. Surprisingly, having a two-stage process (rather than a one-stage process with simultaneous coalition formation and payoff division) benefits the apex player.