The Expected Utility Rule and the Expected Utility Hypothesis

In this chapter, we consider for the first time a situation of decisionmaking under uncertainty. We allow for the possibility that one of a set of uncertain events may occur, each with a certain probability, and each with a known consequence. We face such decisionmaking problems often in our lives. For example, when we consider whether or not to buy insurance before we are 30 years old, we consider the possible but uncertain event of death. A doctor often faces a choice among alternative treatments, with different uncertain side effects. A government must spend money on one of several technologies designed to solve a problem, each of which has only a certain probability of providing the desired results. (For example: What design should we use for a rapid transit system? Should we invest large amounts of money on breeder reactor research, rather than on solar power? And so on.)

To motivate our discussion of such decisionmaking problems, let us consider a simple gambling situation. Suppose you are attending a business meeting and have a choice between parking your car at a meter or putting it in a parking lot. The meter can take coins for up to 2 hours–the maximum would require $1. The meter is monitored almost constantly, and the fine for overtime parking is $25. The lot would be $5, a flat fee. You think that the chance the meeting will be over within 2 hours is 80%.

Measurement Theory

There is a large body of research work in a gray area which seems to have no disciplinary home; it can be called measurement theory. This work has been performed by philosophers of science, physicists, psychologists, economists, mathematicians, and others. In the past several decades, much of this work has been stimulated by the need to put measurement in the social sciences on a firm foundation. As well as being closely tied to applications, measurement theory has a very interesting and serious mathematical component, which, surprisingly, has escaped the attention of most of the mathematical community.

This book presents an introduction to measurement theory from a representational point of view. The emphasis is on putting measurement in the social and behavioral sciences on a firm foundation, and the applications will be chosen from a variety of problems in decision theory, economics, psychophysics, policy science, etc. The purpose of this book is to present an introduction to the theory of measurement in a form appropriate for the nonspecialist. I hope that both mathematics students and practicing mathematicians with no prior exposure to the subject will find this material interesting, both as mathematics in its own right and because of its applications. I hope, indeed, that a number of mathematicians will find this subject interesting enough to solve some of the open problems posed in the text. I also hope that nonmathematicians with sufficient mathematical background will find the work thought-provoking and useful.

Let us begin by acknowledging that the word “partition” has numerous meanings in mathematics. Any time a division of some object into subobjects is undertaken, the word partition is likely to pop up. For the purposes of this book a “oartition of n” is a nonincreasing finite sequence of positive integers whose sum is n. We shall extend this definition in Chapters 11, 12, and 13 when we consider higher-dimensional partitions, partitions of n-tuples, and partitions of sets, respectively. Compositions or ordered partitions (merely finite sequences of positive integers) will be considered in Chapter 4.

The theory of partitions has an interesting history. Certain special problems in partitions certainly date back to the Middle Ages; however, the first discoveries of any depth were made in the eighteenth century when L. Euler proved many beautiful and significant partition theorems. Euler indeed laid the foundations of the theory of partitions. Many of the other great mathematicians – Cayley, Gauss, Hardy, Jacobi, Lagrange, Legendre, Littlewood, Rademacher, Ramanujan, Schur, and Sylvester-have contributed to the development of the theory.

There have been almost no books devoted entirely to partitions. Generally the combinatorial and formal power series aspects of partitions have found a place in older books on elementary analysis (Introductio in Analysin Infinitorum by Euler, Textbook of Algebra by Chrystal), in encyclopedic surveys of number theory (Niedere Zahlentheorie by Bachman, Introduction to the Theory of Numbers by Hardy and Wright), and in combinatorial analysis books (Combinatory Analysis by MacMahon, Introduction to Combinatorial Analysis by Riordan, Combinatorial Methods by Percus, Advanced Combinatorics by Comtet).

The Theory of Fundamental Measurement

Formalization of Measurement

In this chapter, we introduce the theory of fundamental measurement and the theory of derived measurement, and study the uniqueness of fundamental and derived measures. Fundamental measurement deals with the measurement process that takes place at an early stage of scientific development, when some fundamental measures are first defined. Derived measurement takes place later, when new measures are defined in terms of others previously developed. In this section, we shall' begin with fundamental measurement. Derived measurement will be treated in Section 2.5. Our approach to measurement follows those of Scott and Suppes [1958], Suppes and Zinnes [1963], Pfanzagl [1968], and Krantz et al. [1971].

Russell [1938, p. 176] defines measurement as follows: “Measurement of magnitudes is, in its most general sense, any method by which a unique and reciprocal correspondence is established between all or some of the magnitudes of a kind and all or some of the numbers, integral, rational, or real as the case may be.” Campbell [1938, p. 126] says that measurement is “the assignment of numerals to represent properties of material systems other than number, in virtue of the laws governing these properties.” To Stevens [1951, p. 22], “measurement is the assignment of numerals to objects or events according to rules.” Torgerson [1958, p. 14] says that “measurement of a property … involves the assignment of numbers to systems to represent that property.”

Introduction

The results in Chapter 7 indicate that a general study of partition identities such as the Rogers–Ramanujan identities or the Göllnitz–Gordon identities should help to illuminate these appealing but seemingly unmotivated theorems. In this chapter we shall undertake the foundations of this study. As will become abundantly clear, there are very few truly satisfactory answers to the questions that we shall examine. We shall instead have to settle for partial answers. After presenting the fundamental structure of such problems in the next section, we devote Section 8.3 to “partition ideals of order 1,” a topic which we can handle adequately and which suggests the type of answers we would like for our general questions of Section 8.2. The final section of the chapter describes a large class of partition problems wherein the related generating function satisfies a linear homogeneous q-difference equation with polynomial coefficients. In some ways this final section is unsatisfactory, in that the theory of q-difference equations has not been adequately developed to provide answers generally to questions about partition identities and partition asymptotics; however, the theorems of Section 8.4 do suggest that q-difference equations are indeed worthy of future research.

Foundations

We begin with a simple intuitive observation which forms the basis of our work here. In all the partition identities considered in Chapter 7 (Theorem 7.5, Corollaries 7.6 and 7.7, Theorem 7.11) partition functions were considered that enumerated partitions lying in some subset C of the set of all partitions.

Incidence Matrices

Matrices all of whose entries are either 0 or 1—that is, (0, 1)-matrices—play an important part in linear algebra, combinatorics, and graph theory. In some of these applications it is at times preferable to consider 1 as the “all” element in a Boolean algebra, or the identity element in a field of two elements. In what follows, however, the symbol 1 will represent the positive integer 1, since we shall be mainly concerned with enumerations of systems of distinct representatives and with related problems in the theory of permanents.

Many problems in the theory of nonnegative matrices depend only on the distribution of zero entries. In such cases the relevant property of each entry is whether it is zero or nonzero, and the problem can be often simplified by substituting for the given matrix the (0, 1)-matrix with exactly the same zero pattern.

Definition 1.1. Two m×n matrices A=(aij) and B = (bij) are said to have the same zero pattern if aij = 0 implies bij = 0, and vice versa.

Suppose that A, B, C, and D are nonnegative n-square matrices, and that A has the same zero pattern as B, and C has the same zero pattern as D. Then clearly A + C has the same zero pattern as B + D, and AC has the same zero pattern as BC.