Introduction

This chapter represents the first of several putative papers on bargaining among a small number of players. The problem treated in the current paper may be thought of as the “three-player/three-cake” problem. Each pair of players exercises control over the division of a different cake, but only one of the cakes can be divided. Which of the cakes is divided and how much does each player receive? This problem is, of course, a paradigm for a much wider class of problems concerning the conditions under which coalitions will or will not form.

The general viewpoint is the same as that adopted in our previous papers on bargaining (e.g., [3], [4], and [5]). Briefly, we follow Nash ([15], [16], and [17]) in regarding “noncooperative games” as more fundamental than “cooperative games.” Operationally, this means that cooperative solution concepts need to be firmly rooted in noncooperative theory in the sense that the concept should be realizable as the solution of at least one interesting and relevant noncooperative bargaining game (and preferably of many such bargaining games).

The cooperative concept that we wish to defend in the context of the three-person/three-cake problem is a version of the “Nash bargaining solution.” A precise statement of the version required is given in Section 13.3. For the moment, we observe only that the notion can be thought of as synthesizing to some extent the different approaches of Nash and von Neumann and Morgenstern.