Introduction

In the traditional formulation of the bargaining problem, it is typically assumed that a fixed number of agents are involved. The possibility that their number varies has recently been the object of a number of studies, which it is the purpose of the present chapter to review in a unified way, with emphasis on the main results and on the main lines of their proofs.

I propose to evaluate solutions by focusing on their behavior in circumstances where new agents come in without their entry being accompanied by an expansion of opportunities. The standard economic problem that motivated much of the work presented here is that of dividing fairly a bundle of goods among a group of agents. The number of agents involved in the division is allowed to vary while the resources at their disposal remain fixed. (Technically, this implies that the intersection in utility space of the set of alternatives available to the enlarged group with the coordinate subspace corresponding to the original group coincides with the set of alternatives initially available to that group.) Of course, this does not mean that new agents are required never to bring in additional resources nor that their presence itself may not affect the alternatives available to the original group. I simply want to allow for the case of fixed resources, and I claim that a study of this special situation yields important insights into the relative merits of solutions.