In this chapter, we show how the use of hyperreal numbers can simplify our previous characterizations of Pareto optimality. In particular, we shall see that this approach will allow us to avoid the iterative procedures using partition sequence pairs, which involved a-maximization and b-maximization of convex combinations of measures in Chapter 7, and w-association in Chapter 10. Our new approach will also allow us to avoid the assumption (which we needed at times in Chapters 7 and 10) that each player receives a piece of cake that he or she believes to be of positive measure.

In Section 15A, we give the necessary background on hyperreals. In Section 15B, we illustrate the use of hyperreals by considering a two-player example. In Section 15C, we do the same for a three-player example. In Section 15D, we state and prove our new characterization. We make no general assumptions about absolute continuity in this chapter (although our examples in Sections 15B and 15C involve the failure of absolute continuity).

Introduction

Although the foundations of calculus evolved over many years, Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716) are generally considered to be the inventors of modern calculus. Their work involved two different but related notions, limits and infinitesimals.