A note on the group bargaining solution

The well known Nash's pure bargaining solution was generalized by the author [Zhang, X., 1995. The pure bargaining problem among coalitions. Asia-Pacific Journal of Operational Research 12, 1-15] for the bargaining problem with an exogenous coalition structure. Nash's axiom set was modified to develop a solution mapping for this new type of pure bargaining problem. Among other things, a similar solution concept for the same bargaining problem was given by [Chae, S., Heidhues, P., 2004. A group bargaining solution. Mathematical Social Sciences 48, 37-53]. Their solution's mathematical presentation is quite different from Zhang's solution. In this note, we present a proof to show that the two solutions are equivalent.