The Mathematics of Collective Action

James Coleman

• Publisher: Routledge
• Year: 2017
• Print ISBN: 9781138536722
• eBook ISBN: 9781351479707
• Topic: Social Sciences
• Subtopic: Sociology
• Index: Social Sciences

Philosophers, social scientists, and laymen have used two perspectives in analyzing social action. One sees man's action as the result of causal forces, and the other sees action as purposive and goal directed. Mathematical treatment of social action has shown this same dichotomy. Some models of behavior describe a causal process, in which there is no place for intention or purpose. Most stochastic models of behavior, whether individual or group, are like this. Another body of work, however, employs purpose, anticipation of some future state, and action designed to maximize the proximity to some goal. Classical microeconomic theory, statistical decision theory, and game theory exemplify this direction.This book examines these two directions of work, and makes original contributions to the second. An introductory chapter outlines these two bodies of work, and casts them in a common frame, to display their similarities and differences. Chapter 2 reviews at length recent work in stochastic processes that makes up the first body of work, which sees social action as the resultant of causal forces. The remaining chapters develop a mathematical framework for the study of systems of social action using a purposive theoretical base. These chapters are designed particularly to contribute to the study of collective decisions, a form of social action that has proved particularly challenging to theoretical analysis. First published in 1973, this became a significant work both in problem solving and in the future career of the author. It is of continuing importance to researchers and students interested in statistical analysis.

1 Mathematics of Social Action

There are two quite different streams of work in the study of social action, both of which begin at the level of the individual.¹ The two streams of work represent fundamentally different conceptions of man. Where the ordinary lay conception of man is as a person responding to his environment in pursuit of some goal, these conceptions each recognize half of that description. The first conception explains man’s behaviour as response to his environment; the second explains his behaviour as pursuit of a goal. The first searches for causal processes and determinants of behaviour, and often uses a mechanistic explanatory frame, which employs the concepts of ‘forces’ and ‘resultant action’. It has been the implicit basis of much of the best empirical work in sociology, from Durkheim’s analysis of the causes of suicide to the present.

The second conception sees man’s action as goal-directed, and focuses attention less on present environmental conditions than on future desired states. Man is conceived less as a product of his environment than as the source of preferences which lead to action. This conception has been the basis of much common-sense explanation of behaviour, of some theoretical work in sociology, represented by Weber and Parsons, and of one major theoretical structure: economic theory, based upon a conception of rational economic man.

The two streams have foundations in psychological theory, where they may be termed ‘stimulus-response’ theories and ‘purposive action’ theories, respectively. A decade ago, one might have pointed to Clark Hull’s work as the archetype of stimulus-response theories, and E. C. Tolman’s work as the archetype of purposive action theories.

In philosophy and religion, these two perspectives sharply divide two sets of beliefs. The first is of man as a creature of his fate, whose life is determined by the forces that surround him. The second is of man as the architect of his future.

In recent years, both of these streams of work have been extended through mathematical treatment. Two rather distinct bodies of mathematical applications have developed, each of which has a certain mathematical coherence or unity. As will be evident, these two theoretical approaches do not imply wholly distinct mathematical treatments. Nevertheless, the development of mathematical models for these two conceptions of behaviour has generated distinctive types of mathematical approaches.

Although these approaches are distinct, and different mathematics is used to mirror them, there are certain similarities that flow from the fact that both approaches involve the study of social action.

Causal processes

In the mathematical treatment of causal theories, an action is described as an event with particular outcomes. Ordinarily, the outcome of an event is seen as putting an individual into a given state. Thus the result of an action is ordinarily for the individual to be in a particular state, which may differ from his prior state. For example, in the simple case, an action like voting is assumed to be an event with n possible outcomes, where n is the number of candidates. The outcome is assumed to be governed by a set of probabilities, p₁, …, p_n, that he will cast his vote for candidate 1, …, n.

A system of action can consist either of a set of events all representing actions of a single individual, a set of events each representing an action of a different individual, or a set of events representing multiple actions of different individuals. What is problematic about an event, and requires explanation, is which outcome of the event will occur. As a consequence, many of the variations in these models concern the dependence of the event’s outcome on various possible causes or determinants. Two major types of causes have been examined in the literature. First of all, events may be dependent on the outcome of prior events. Since events may represent one individual’s actions, or those of many individuals, dependence of an event on the outcome of other events may be of two types: dependence on the state of the same individual, or dependence on the state of other individuals. It is sometimes the case, in fact, that the same mathematical models may be used to represent both intra-individual dependence and inter-individual dependence.

The second major type of cause of an event’s outcome is a state of the individual or the environment that does not change, or changes independently of the other events included in the system. Some of these states are the attributes or variables commonly used to account for behaviour: the individual’s age, level of education, attitude, sex, family size, occupation, etc. Because these states either do not change, or change independently of events in the system, the mathematical models necessary to represent these causal processes are different from those in which events are dependent on outcomes of other events internal to the system.

Purposive action

When we turn to mathematical treatment of purposive theories, there is a somewhat different conception of action. There is still a system of events, each event with a set of possible outcomes; but the events are no longer identical to the actions. Actions of the actors in the system control the outcomes of events, either wholly or partially; and the action is selected through the conscious choice of the actor, choosing that action which he believes will lead to the outcome most beneficial to him. Thus the outcome of a future event does not depend directly upon the outcome of previous events, through a causal process, nor upon attributes of the individual. There is the interposition of a conscious, rational agent (or agents), whose choice determines, partly or wholly, the outcome of the subsequent event. The actor makes his choice of actions through his perception of the consequences that particular outcomes will have for him, and his perception of the dependence of outcomes upon his actions.

Comparison between the causal theories and the purposive theories may be made more systematic by setting down the sequence of elements that comprise the conceptual framework of each. In the causal theories, the sequence is

cause→event outcome

The cause may be other events internal to the system of events under consideration, or external. The outcome is ordinarily conceived as a state of an individual or a collectivity.

In the purposive theories, the sequence is still built around event and outcome, but contains the concepts of action and consequence:

action→eventoutcome→consequenceforactoe

In purposive theories, there is an implicit or explicit ‘look-ahead’ feature, in which the actor ‘looks ahead’ at the expected consequences of different outcomes for him, and adjusts his action to these possible consequences. This gives rise to the essential behaviour principle of purposive action theories, the principle of utility-maximization. This principle states nothing more than that the actor will choose that action which according to his estimate will lead to an expectation of the most beneficial consequences. As such, it constitutes a ‘functional’ theory, in which an action is conceived to be shaped by its future consequences, rather than by prior causal factors. Further, as in other functional theories, there is conceived to be an organism that is acting homeostatically, that is, toward self-maintenance. In this case, the organism is the individual, and the homeostatic mechanism devised by economic theory is a principle of utility-maximization, or satisfaction-maximization, a principle that generates the concept of ‘utility’ in such theories.

Functional theories differ sharpl...

Table of contents

• Cover Page
• Halftitle Page
• Title Page
• Copyright Page
• AldineTransaction Introduction
• Content Page
• Preface
• 1 Mathematics of Social Action
• 2 Concepts of Rational Action
• 3 Collective Actions
• 4 Further Concepts and Applications
• 5 The Dynamic System, and Other Elaborations
• References
• Computer Program and Output
• Index