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.

It is a pleasure to write a preface for the second edition of Mathematical Analysis: A Straightforward Approach. The first edition was well-received and I have therefore thought it wise to leave its text substantially unaltered except for one or two minor points of clarification and the correction of misprints. The major change is the addition of two long chapters on analysis in vector spaces for which there has been a considerable demand. These get as far as the idea of a derivative as a matrix and the use of the second order derivative of a real-valued function in classifying stationary points. More advanced material than this would seem to me better delayed until after the basic topological notions have been mastered. As far as the material covered is concerned, it does not involve the proof of many theorems and the necessary proofs involve no new analytic ideas. However, the material does require a certain facility with algebraic and geometric ideas and students with only a very limited knowledge of linear algebra may find it heavy going in spite of the fact that some discussion of the necessary concepts from linear algebra is included where appropriate. Another innovation is the inclusion of a collection of further problems for which the solutions are not given. I am grateful to John Erdos for some of these as well as other helpful suggestions.