Individual and collective opportunities

We consider the bargaining problem in the context of a variable number of agents. When new agents enter the scene but the opportunities open to the enlarged group do not expand, some solutions paradoxically may recommend that some of the agents originally present gain. We propose a quantitative measure of the extent to which a solution allows this phenomenon to occur and we rank the major solutions on that basis. The Kalai-Smorodinsky solution performs better than all weakly pareto-optimal and anonymous solution, and in particular strictly better than the Nash solution. However, the two solutions are equivalent when it is the opportunities for gains offered to initial groups, instead of individuals, that are being compared.