From the time of its birth, the theory of collective choice and social welfare has led a curious life. Active investigations have come from quite variegated research fields, including economics, political science, moral philosophy, and decision theory, giving depth and breadth to this rapidly developing field. Nevertheless, there are still some who would deny its legitimacy as a branch of the social sciences in general and as a branch of economics in particular. Even the message of a central result thereof, that is, Arrow's celebrated general impossibility theorem, has not been made crystal clear. Indeed, Professor Arrow wrote in his Nobel Prize lecture delivered in 1972 that “[the] philosophical and distributive implications of the paradox of social choice are still not clear. Certainly, there is no simple way out. I hope that others will take this paradox as a challenge rather than as a discouraging barrier.” This book, which is an outgrowth of my doctoral dissertation submitted to Hitotsubashi University in 1979, represents a modest attempt to respond to this challenge.

There are four main topics that I discuss in this work. First, I examine the concept of a choice function and the rationalizability thereof in terms of the revealed preference and related axioms. Second, I examine the robustness of the Arrovian impossibility theorems with special emphasis on the choice-functional formulation of social choice theory. Third, I explore the extended sympathy approach in social choice theory, thereby clarifying such concepts as envy and impartiality in this extended conceptual framework.

Part II is concerned solely with the application of the methodology developed in Part I to selected theoretical discussions. The static, simultaneous relations model is used in Chapter 8 to distinguish between political structure and political system by thinking of the former as a system of simultaneous relations and the latter as a solution of it. In Chapter 9 a planner's view of the making of sequential decisions by society's decision-making units is modeled with a system of periodic relations. Time paths are examined with respect to the logic of their potential for manipulation by the planner. Chapter 10 again utilizes a system of simultaneous relations to describe the general structure of society. Society itself is defined as a solution of the system dependent on certain parameter values. Change over time also occurs as parameters modulate according to specific periodic relations.

It should be emphasized that in all three cases, claims of originality are strictly limited. No outstanding problem is solved, no conclusion is extended, no new proposition is proved, and no new question is raised in any of the fields that these chapters represent. In fact, practically all approaches, concepts, hypotheses, and propositions are lifted directly from the references cited. Thus the models presented are at least as deficient as the original sources. Nevertheless, the contribution of these chapters is to show how the methodology of Part I can be applied to existing ideas. The result is an infusion of fresh rigor, precision, and mathematical-type argument into work that has already been completed.

Economists do not yet have a satisfactory understanding of the means by which individuals choose among occupations. Current theory (see, e.g., Friedman, Scitovsky, and Stigler) is largely based on two assumptions: First, by balancing both monetary and nonmonetary components, individuals arrive at a preference ordering of all possible alternatives. Second, they choose the maximal preference occupation from those not eliminated by relevant nonpreferential constraints. This is fine as far as it goes. However, what are the nonmonetary elements and how are they evaluated? According to Friedman and Kuznets, they include “such subjective and intangible factors as the prestige value attached to the profession, the opportunity it offers for rendering service and making ‘social contacts’, the conditions under which professional work is performed,…” One purpose of this chapter is to obtain a better understanding of these nonpecuniary factors.

The fact that substantial numbers of nonmonetary ingredients may influence the individual's choice of occupation suggests that the theory explaining how choices are made ought to be broadened. If it is to provide more than a superficial understanding of real situations, it has to account for the role these inputs play. Thus the theory should view occupational decisions as choices between distinctive life-styles or “qualities of life,” encompassing internal components such as pride and sense of achievement, as well as external factors such as physical working conditions and the type of people to be associated with.

Having studied several specific systems in some detail, it is now worth considering systems and their properties in general. This will provide a firm foundation for the results of Chapter 5, and at the same time, suggest further directions in which techniques for analyzing nonquantifiable phenomena have been and can be developed.

Of the various approaches to the notion of system, that associated with the name of Mesarovic is most confluent with the viewpoint expressed here. Without any formal structure in mind, Mesarovic thinks of a system simply as a relation on abstract sets. To analyze particular problems, however, structure must be added – but only that necessary for the given purpose. This is the approach adopted below. After an initial presentation of the general definition of system, the specific systems of Chapter 5 are derived as special cases. Next various forms of systemic causality are examined. Subsequent considerations relate to connections between systems, feedback, and the problem of control. The chapter concludes with a classification scheme for systems based on the theory of categories. Because the discussion is primarily definitional and provides little in the way of formal proof, the reader who is interested in rigorous argument may wish to refer to Mesarovic and Takahara.

Purpose

Although the modern trend of analysis in areas such as the social sciences is to favor those problems and techniques that can be expressed and explored through the use of numerical representation, it is still readily apparent that many of the important issues of our day involve phenomena exceedingly difficult, if not impossible, to measure. Examples abound. Human behavior would seem to involve a complex of interrelated, nonquantifiable elements. Economic (e.g., labor-management) and diplomatic-political bargaining illustrate the point. At times the significance of numerical information pales in comparison to the many hard-to-scale political, social, and psychological pressures under which negotiators operate. And the outcome is likely to depend further on the personalities involved. In another context, nonquantifiability turns up in many “resistances” to economic development (e.g., cultural patterns) that have been discovered in so-called underdeveloped countries. Kuznets has also suggested that there are institutional and structural changes (urbanization, etc.) accompanying growth that are equally tough to gauge. It follows that purely quantitative analyses of economic development are of limited usefulness because important inputs and outputs of the growth process cannot, at present, be calibrated. Recent concern and interest in the “quality of life” also runs into a similar barrier. Attempts to measure it (see, e.g., Liu and Ontell) have not yet and may never be able to capture its essence fully. Hence the concept itself cannot be defined and understood solely in numerical terms.

Although, as clearly apparent by now, there are a considerable number of set-theoretic and topological concepts and propositions available for use, even in the absence of measurement, it is still not enough. The inclusion of nonquantifiable objects in an analytical framework obviously requires that addition, subtraction, multiplication, division, and so forth be dropped. These must be replaced with other manipulative tools before proceeding. In doing so, one naturally turns to algebraic operations on functions, several of which may be performed regardless of whether their variables can be measured.

This chapter develops an algebraic structure to serve as the basis for subsequent analysis. The general theory of semigroups of partial transformations is outlined first, followed by a discussion of the specific circumstance in which it is employed later on. Once again, because all results are, at worst, relatively minor deviants of well-known propositions, no proofs are given. They may be found in Birkhoff and MacLane and Ljapin. A more general and comprehensive development is provided by Schweizer and Sklar.

Semigroups of partial transformations

It is appropriate to begin with a few preliminaries. Let X and Z be sets. As suggested in Section 2.2, an operation on X is a function g: X × X → Z. Function values under g are often written as, say, x′·x″ instead of g(x′, x″), where x′ and x″ are in X and the “·” replaces g as the operation symbol.

Political life, according to Easton, is a “system of behavior.” At the same time that its elements are interacting with each other, the system as a whole is influenced by and exerts pressure on its environment. Stress is constantly arising within the system and out of its surroundings, and efforts to deal with it result in parametric and even structural variation. The ability of the system to survive depends on the kind of information feedback it receives from its environment, for this is the only channel through which political decision makers can discover which parametric and structural changes are necessary.

More precisely, a political system is a collection of certain kinds of human interactions. These are social in nature and involve both individuals and groups. The distinguishing feature of political interactions is that they are directed toward the authoritative allocation of things that are considered of value within society. Authoritative allocations are accomplished, for example, by physically taking away valued possessions, erecting barriers to prevent their procurement, and providing some individuals or groups with the means for their acquisition. The allocations are authoritative in that individuals and groups consider themselves bound by them. Observing that political interactions as described here appear in many parapolitical groups (such as the family, business firms, and religious and social groups), Easton reserves the term “political system” only for those sets of interactions that are relevant for allocating valued things to a society as a whole.

Neoclassical theory has traditionally taken the production function as an exogenous datum reflecting the technical structure of production (Samuelson, Ferguson). Hence little effort has been expended on the analysis of the production process itself. (Two notable exceptions, Coase and Simon, are discussed subsequently.) Yet surprisingly powerful theorems follow from this hypothesis according to the microeconomic logic of profit maximization. Certainly the most central of these theorems is that the profit-maximizing firm will choose Paretooptimal production under normal conditions.

But in general, there is no reason to suppose that profit-maximizing entrepreneurs will make profit-maximizing decisions that are also in the best interests of their workers. In any given circumstance, therefore, it may be possible to make workers better off without reducing profits. Such a situation cannot be Pareto optimal, and there is no obvious mechanism, such as competition, operating within the firm that automatically leads it to optimality. Indeed, by modeling the firm as a complex collection of social relationships, we shall suggest that the neoclassical assertions concerning optimality in the firm hold only under restrictive and highly unrealistic assumptions. Specifically, profit maximization need not always result in the optimal allocation of human resources.

This observation calls into question a number of derivative but socially critical propositions often expressed as follows. First, the structure of authoritarian relations in the firm will be Pareto optimal. Secondly, wage differentials will reflect differences in the marginal productivity of workers in specified jobs. Thirdly, worker sovereignty will obtain in the same sense, and under the same conditions, as the more traditional consumer sovereignty (Gintis).

Planning, as defined by Ozbekhan, is an activity designed to operate on the present environment for the purpose of changing it into a more desirable state. Individual values determine which states are preferable. Given a particular environment and the configuration of values perpetuating it, fundamental change (i.e., that which creates a distinct future as opposed to an extension of the present) is introduced only through alterations of values. The latter can be varied by individuals alone. If and when specific value changes become socialized and widely accepted, movement from the original state to that implied by the new values is thought of as “progress.” Stated in these terms, planning is the “organization of progress.”

Ozbekhan also conceives of planning as a system that interacts in relation to another system – the environment. The environmental system takes the output of the planning operation as one of many inputs and produces environmental states. The latter also serve as inputs for planning. In this context the “planning system” is a relation as defined in Chapter 6. It may be considered in a static or dynamic framework and is capable of exhibiting various forms of causality, control, and feedback mechanisms. Ozbekhan endows it with a three-level structure: at the highest level, norms and guidelines are sought and general policy decisions are made; on the middle tier, specific goals are set and strategies devised; at the lowest level, decisions taken on the higher two are made operational and implemented. An example of such a system, in which it is assumed that inputs from the environment enter at each level, is diagrammed in Figure 9–1.

The preceding pages have considered the problem of thinking – in an intellectually sophisticated way – about phenomena that are difficult, if not impossible, to scale. (A detailed summary has been provided in Section 1.2.) In so doing, the objectives set out at the beginning of Chapter 1 have largely been met. At this point, however, it is interesting to observe briefly a few ways in which life structures emerging out of differing cultural and historical circumstances have cohered quite independently of abilities to calibrate. As a practical matter, it turns out that human beings are able to get on surprisingly well without measures or even numbers.

First of all, it is clear that numbers themselves are not needed to count. Certain orthodox Jews, for example, require that at least ten men, a minyan, be present to conduct particular religious services. Before they start, a ten-word sentence is recited in which each man is identified with one word. If the sentence is completed, then the minyan has been constituted and the service can begin (Zaslavsky). As a second illustration, the Kpelle people of Liberia have no independent abstract numbers in their language. Objects are still “counted,” however, and the results of any particular count appear in number-words that always must modify a noun or pronoun (Gay and Cole).

The last chapter considered, in part, statistical tests for the existence of relations among variables. These tests are based on how observations of the variables in question fit together. But in many situations (e.g., prediction) merely to know that statistical evidence supports its existence is not enough: Some knowledge of the nature of the relation itself is essential. The only empirical way of approaching such a problem is to ask what sorts of relations best fit the observations at hand. When meaningful numbers are available the usual technique is to assume that the relation has a certain form, say, linearity, and then estimate its parameters. If according to some acceptable criterion the fit turns out to be “good,” the assumed form is taken as a reasonable approximation of reality. Parallel procedures in the absence of ratio, interval, and ordinal measures are of interest here.

As remarked earlier, empirical relations are obviously capable of detection even in the absence of measurement. It is only necessary to observe that, under appropriate conditions, certain values of, say, (x1,…,xK) arise with certain values of (p1,…,pM). Once such information is obtained, subsequent investigation might include the drawing of structural inferences and the formulation of predictions. This chapter explores these possibilities.

Even without ratio, interval, and ordinal scales, introduction of numbers in the form of dummy variables is often legitimate in order to facilitate analysis. Recall (Section 2.2), any typology or classification scheme permits nominal measurement. That is, given a set A” and some decomposition of X into mutually exclusive and exhaustive subsets, Sv, the objects of X can be assigned real numbers such that within each Sv, all elements receive the same numerical value.

In much the same way as ordinary investigation of quantifiable experience rests on concepts such as set, relation, function, topology, and a multitude of others, so techniques for handling nonquantifiable phenomena may be built upon a similar foundation. To avoid confusion later, it is well to begin by specifying these fundamentals. The following discussion presents a brief outline of the basic set-theoretic and topological concepts and propositions that are required for subsequent use. It also illustrates the fact that a good deal of mathematical analysis does not depend on an ability to measure. Proofs that are well known and readily available are not repeated. The material is based, for the most part, on Halmos and Kelley.

The results to be developed here have immediate application in social science. Consider, as one example, the dispute over whether the U.S. and Soviet economic systems are becoming more alike as time passes (see, e.g., Prybyla). Many factors involved in the argument do not appear capable of measurement. To resolve this issue one clearly must know what it means for two economies to “come together.” Provided general agreement can be reached concerning the relevant properties on which comparisons are to be based, Section 3.2 shows that a definition of closeness is indeed possible. This, then, furnishes a framework for probing the question of U.S.-Soviet similarity.

The purpose of this chapter is to outline briefly some of the important ideas relating to various forms of measurement. A complete survey is not intended. Only those notions relevant to determining the sorts of things that are required to have measurement (and hence must be absent without it) are discussed. The chapter begins with the most primitive form of measurement, namely, verbal classification by type or property. It then proceeds to the more precise and systematic characterization of properties on ordinal, interval (cardinal), and ratio scales. Problems arising from treating ordinal data as if they were interval data are also considered.

Types

Our world is so complex and diverse that raw, untempered observation of it produces only massive and hopeless confusion. To make any sense of the chaos at all requires abstracting from differences (based on properties or attributes) between things or objects that are seen. This furnishes a basis for making comparisons and hence for determining when two things are distinct or identical. Collections of identical things define groups. The process is one of categorization or delineation of type. Once categories are established, identification of a fresh object, that is, its assignment to one of the groups, becomes possible. Classification therefore emerges at the very core of human understanding. Implicitly, if not explicitly, it is a prerequisite to the modeling of any real phenomenon.