Theories of Coalition Formation

James P. Kahan, Amnon Rapoport

• Publisher: Psychology Press
• Year: 2014
• eBook ISBN: 9781317769187
• Topic: Psychology
• Subtopic: Cognitive Psychology & Cognition
• Index: Psychology

First published in 1984. In this book, the authors set forth the central ideas and results of the major theories of coalition forming behavior. These theories address situations of partial conflict of interest with the following aspects: (1) there are three or more players, (2) players may openly communicate with each other, and (3) players form coalitions by freely negotiating agreements on how to disburse the gains that result from the coalition members' joint coordinated efforts. These models arise from the two disciplines of mathematics, in the theory of cooperative n-person games with side payments, and social psychology, in theories of small group behavior in mixed-motive situations. The goal is to explore the various solution concepts that make up this body of theory, and in particular to examine the psychological premises that underlie the various theoretical models.

1	Prolegomena

1.1. ORIGINS

Whenever three or more parties get together to jointly decide an issue of substantive interest to all of them, it is likely that at least two of them will at some point in time combine forces to their mutual advantage. When this combining of forces is deliberate, done with the full awareness of all joining parties, and binding upon the joiners, we speak of a coalition being formed. This book is about the study of coalition formation and the bargaining processes that lead to it. The approach we take to investigate this phenomenon is grounded largely in the philosophy and methodology of psychology, and in particular experimental social psychology. The theoretical orientation we adopt, however, differs from that of the majority of our social psychological colleagues, and is based on the Theory of Games. We shall examine under a common format theories of coalition formation arising from game theoretic and social scientific bases, and then compare the derived theoretical statements to data from laboratory experiments. In order to provide a context in which to embed our approach, we begin with a brief, historically oriented overview of conflict of interest and the Theory of Games. Following this historical section, we look at the relationship of social psychology and game theory, in order to establish an integrative framework and to dispel several myths that have perpetuated a disregard on the part of social psychology for game theory. Finally, in the last section of this chapter, we outline the general format that later chapters follow.

1.1.1. Conflict of Interest

The study of conflict of interest is one of those few labors shared by all members of that amorphous collection of disciplines labeled the social sciences. Among several of these disciplines, conflict of interest is so central that the sages of these disciplines have occasionally felt the need to issue reminders that conflict does not exhaust the universe of worthy topics; such reminders only serve to underscore the importance and the pervasiveness of conflict.

Conflicts appear in many forms in the world. Political parties must come to terms with their differing ideologies and conflicting ambitions before they can enter a coalition. Partners in a small business may have diverging interests that result in different opinions on how to allocate the resources of their corporation. As a general statement, we may say that conflict of interest exists whenever two or more social beings interact in such a way that they affect each other’s worlds, and the nature of their interaction is such that it is not possible for all of these beings to simultaneously achieve their most desired ends. The specific nature of the social beings, the characterization of the interaction, and the specification and measurement of the goals will vary from one social scientific discipline to another, but the common “logic” of conflict of interest, which turns out to be intricate, often perplexing, and at times riddled with paradoxes, provides a common framework that allows for interdisciplinary understanding and crossfertilization (von Bertalanffy, 1968).

For example, there is a common structure underlying the study of such diverse topics as family therapy (Watzlawick, Weakland, & Fisch, 1974), industrial pollution (Dawes, Delay, & Chaplin, 1974), international disarmament (Howard, 1971), joining labor unions (Messick, 1973), littering city streets (Kelley & Grzelak, 1972), overpopulation (Kahan, 1974), and the provision of public goods (Olson, 1965). In each of these situations, there is a conflict of interest between what is beneficial for the social being, be it individual, family, firm, or nation, and what is beneficial for the collectivity of which the being is a member, be it family, society, nation, or the world.

Other, perhaps less grandiose examples can be easily summoned. Economic models of the marketplace (e.g., Shubik, 1975) bear resemblance to sociological models of mate selection (e.g., Blau, 1964) or psychological models of marital disputes (e.g., Satir, 1967). Each of these models concentrates on a bargaining process of successive compromises, although the currency that is the medium of exchange differs strikingly from model to model. Marxist economists, military strategists, and Freudian psychologists all too often view their respective worlds as broken down into irreconcilable opposing camps at war with each other, where every gain by one camp is necessarily at the cost of the other. That the respective camps are social classes, armies, and elements of the individual psyche does not belie the similarity of the analysis. Such examples are not at all restricted to the conflict between two parties. Consider the different issues of forming political and military coalitions among nations in order to achieve a balance of power, the formation of coalition governments in parliamentary democracies, the setting of prices by oil and other commodity cartels, the decision by a group of teenagers of what movie to attend on a particular Saturday evening, and the gentle skill by which children achieve their desires by setting one parent against the other. Each of these issues is analyzable, at least in principle, by the study of the formation and dissolution of coalitions of social beings who have partially opposed and partially coincident interests.

1.1.2. Enter the Theory of Games

Game theory is a branch of mathematics, which was created to study the structure and resolution of conflict. As such, it has the two goals of classifying types of conflicts according to systemic criteria and offering suggestions for optimal behavior to the parties engaged in conflict. The theory has been designed to treat rigorously individual, group, and institutional decision tasks in which there exist elements of conflict and cooperation among the parties, in the context of choices among decision alternatives whose resultant outcomes are well specified (Rapoport, 1959). The distinctive quality of game theoretic decision tasks is that the outcome to a particular participant, referred to by game theory as a player, depends not only on his own choices and the vagaries of chance elements, but also on the choices of one or more of the other players. The interests of the players may be diametrically opposed, totally coincident, or, more typically, partially opposed and partially coincident. The outcomes to each player, although well specified, are necessarily uncertain, because the choices of the other players, as well as the vicissitudes of chance, are not known with certainty. Although game theory easily treats those situations in which the interests of the players are totally coincident, the interesting aspects of the theory are in its applications to situations in which there is an element of conflict.

The relevance of game theory to the social scientific study of conflict of interest is self-evident, although the value of its contribution is not. The theory has been developed, presented, discussed, interpreted, criticized, and applied in myriads of articles and books. It has been saluted by some as one of the most outstanding scientific achievements of our century and has been condemned by others as behaviorally and socially of little or no relevance, or worse, as a pernicious influence (Bernard, 1965; Plon, 1976).

But heated arguments about relevance can be avoided if, in discussing a mathematical model of a situation, it is kept clear when the discussion treats the underlying “real” situation and when it addresses the model itself. The appropriateness of the assumptions may be judged to ascertain whether or not the model captures the relevant aspects of the situation. No conflict situation exists outside of a surrounding context, yet all formal models strip away large parts of that context in order to make the problem tractable. In this simplification, there is the inevitable choice of which sets of factors of the conflict to retain and which to discard. And any such decision is a common-sense one, which weighs the need for simplification in order to obtain concrete results against the risk of obtaining spurious results because the wrong factors were retained.

Before embarking on a short history of game theory, two caveats are warranted in order to dispel some common misperceptions. Both concern the very name “game theory.” First, the theory of games is a collection of formal models for studying decision making in conflict situations that are most easily exemplified as games of strategy. Hence its name. Although the analogy between games and other conflicts is often useful, the word “game” unfortunately carries with it undesirable connotations of frivolity and irrelevance (Shubik, 1964). But wars, political imbroglios, economic battles, and family disputes are not games in the recreational sense: they are not entered into for purposes of amusement, they are not easily and readily amenable to dispassionate investigation by the concerned parties, and their players do not always enter the game willingly. It is important, therefore, to understand the special meaning of “games of strategy.”

The first caveat having concerned possible misunderstandings about the word “game,” the second concerns the word “theory.” The term “game theory” is actually a misnomer, as there are a multiplicity of theories that properly fall under that rubric. “Game Theory” as first presented by von Neumann and Morgenstern in 1944 was based on a number of assumptions about the nature of the logic pursued by players, both individually and collectively. Later, these original assumptions were amended by alternative formations leading to new theories. The term “game theory” is assigned to the collectivity, which is the entire body of mathematical research within the general context laid down by von Neumann and Morgenstern. In addition, applications of the abstract mathematical formulations to the behavioral and social sciences have been frequently subsumed under the general rubric.

This variety of theories under a single title is not a weakness; rather it is precisely this variety that supplies much of the interest of what is inherently an exercise in mathematical logic for the student of human behavior. As Lucas (1972) puts it,

… this multiplicity of solutions is for the most part inherent to the problems, and shows the richness of the theory due to the diversity of possible human behavior in a multipolar world in which concepts like ‘fairness’ and ‘most preferred’ mean different things to different people. It is very unlikely that all groups of people will accept any one well-defined solution concept as being universally applicable [p. 16].

Analogous circumstances exist in many areas of psychology that have proved amenable to mathematical formalization, for example, probability learning, paired-associate learning, concept formation (Atkinson, Bower, & Crothers, 1965), individual decision making under uncertainty (Luce & Suppes, 1965), and choice in the 2-person Prisoner’s Dilemma game (Rapoport & Chammah, 1965). In all of these examples, behavior in a single relatively simple experimental paradigm may be abstracted by models that differ from one another in their assumptions about information processing, response mechanisms, or the subject’s objectives. With respect to game theory, the multiplicity of independently-minded decision makers, various possible norms of fairness and reasonableness, and difficulties in capturing the various aspects of rationality within a single definition lead to this proliferation of theories.

The history of game theory might have commenced at the beginning of the Eighteenth Century. At that time, a new political order was established in Europe, founded on the balance of power between the leading European states. The new concepts of “balance of power” and “political equilibrium” spread rapidly in the community of intellectuals, became common phrases, and were invoked to explain and justify many different political arrangements, in particular, the formation and dissolution of alliances among nations. Moreover, Lossky (1970) states: “the early eighteenth century certainly tended to calculate the balance of power in precise mathematical terms, for which the new science of ‘political arithmetick’ already supplied a warrant [p. 156].” It is not surprising then that the origins of game theory have been attributed (Morgenstern, 1968) to Leibnitz, who as early as 1710 foresaw the need and possibility of a theory of games of strategy, and to Waldegrave, who two years after Leibnitz first formulated a primitive form of the minimax strategy. From the end of the Nineteenth Century, contributions to game theory became more frequent. Edgeworth (1881) noted the similarity between games of strategy and certain economic processes, and Zermelo (1913) stated specialized theorems for certain games. Borel (1921) provided a clear statement of an important class of game theoretic problems, and introduced some of the major concepts of game theory, but conjectured that the minimax theorem was false in the general case. But von Neumann (1928) proved the fundamental minimax theorem and thereby laid the cornerstone of the edifice of game theory as we know it.

However, it was the publication of the monumental work by von Neumann and Morgenstern, entitled Theory of Games and Economic Behavior (1st edition 1944; 2nd enlarged edition 1947) that provided the starting point for the impact that this mathematical discipline has had on the scientific world. The book developed the theory of games beyond the minimax theorem, especially to games involving more than two players, and extensively discussed the assumptions, reasoning, and conclusions of the theory in their full generality. In so doing, it provided a logical classification system for conflict situations and a language for describing such situations that has been incorporated into the argot of the social sciences.

In the years following the appearance of von Neumann and Morgenstern’s work, game theory has taken on all of the trappings of a full-fledged scientific paradigm (Kuhn, 1970). A literature has arisen in which new concepts and ideas are introduced in elaboration of the original formulation. Textbooks (e.g., Blackwell & Girshick, 1954; McKinsey, 1952; Owen, 1968), volumes of “advances,” and Festschriften for the grand old men of the field have been published, and since 1972 there has been an International Journal of Game Theory. Considered as a new mathematical discipline, game theory has been eminently successful. Additionally, it has been used in contributions to theoretical and methodological developments in economics, political science, operations research, sociology, and psychology for the past 25 years.

This success has not been easily obtained. Most of the mathematical work, both past and present, is in a form not readily digestible by the mathematically unsophisticated social scientist. But, fortunately, intermediate catalysts have enabled social scientists equipped with the appropriate appetite to savor the flavor of the theory. Starting with Luce and Raiffa (1957), introductory books on game theory (e.g., Brams, 1975; Davis, 1970; Dresher, 1961; Rapoport, 1959, 1960, 1966, 1970a; Shubik, 1964) have presented game theory in a form incorporate by social scientists. Moreover, some of these books provided psychological and sociological interpretations of the underpinnings of game theory and criticisms of its concepts. In particular, Luce and Raiffa’s (1957) Games and Decisions: Introduction and Critical Survey, in spite of being outdated in its discussion of n-person games, remains one of the masterpieces of the field, still valuable for student and professional alike as a major reference.

1.1.3. n-Person Games

The distinctiveness of n-person game theory, where “n-” refers to three or more players, requires elaboration. As Shubik (cited in Archibald, 1966; Shubik, 1968) has observed, von Neumann and Morgenstern’s book could have been written as three separate entities: one presenting the concept of individual utility, one presenting 2-person games, and one presenting n-person theory for n ≥ 3, although, as Morgenstern (1968) notes, interesting differences and distinctions continue to appear as the number of players increases.

In 1-person games, better characterized as individual decision making tasks under risk (e.g., Lee, 1971; Luce & Suppes, 1965), the player faces Nature, who may have some degree of control over the player’s fate, but who receives no reward and has no interest in the outcome. The player is therefore master of his own fate; his decisions, within the probabilistic bounds dictated by chance, determine his outcome. When moving from one to two players, game theory proper begins, for with two players there is for the first time interdependence between players. Now, the player must consider not only his own actions and those of chance, but also those of another interested party whose interests may be either aligned with, opposed to, or some mixture of his own interests (Schelling, 1958).

When moving from n = 2 to n ≥ 3 (Rapoport, 1971), there is a basic qualitative difference in the nature of the conflict situation, a difference which has long been recognized in the various disciplines of the social sciences. This difference concerns the possibility of coalition formation, by which a group may control the outcomes of individuals or other groups. Simmel (1902b) wrote about triadic interaction, “The essential point is that within a dyad, there can be no majority which could outvote the individual. This majority, however, is made possible by the mere addition of a third member [p. 158].” Subsequent writers, e.g., Caplow (1968) in sociology and Haley (1976) in clinical psychology, have carried the argument a step further in arguing that the essence of social interaction is triadic rather than dyadic.

Caplow is a devoted student of Simmel, and the title of his 1968 book Two against one: Coalitions in triads sums up his view of the triadic world. Even when only two actors are physically present, there is, he argues, always a nearby audience, which whether friends, relatives, allies, or enemies, serves as the social context within which the dyad behaves, and may become an involved participant at any moment. Haley (1976) views social interaction as built on a system of triangles, that most stable geometric form. The sides of the triangle for Haley are the three dyadic interactions of a triad. Haley’s psychotherapeutic strategy is to not attack directly a dyadic interaction that he regards as problematical, but instead to alter the two dyadic relationships that the couple have with the same third party in such a way that the original dyadic interaction is altered in the desired direction so as to produce a new triangular stability.

Caplow’s Orwellian triads and Haley’s family triangles both capture the basic difference between dyads and triads, namely the possibility of coalitions. Within the formal framework of game theory, this same difference is captured by the shift from n = 2 to n ≥ 3. Although there have been several attempts to extend 2-person game theory to the general n-person case (e.g., Harsanyi, 1963; von Neumann & Morgenstern, 1947), theories of coalitions are in general markedly different from theories of 2-person conflicts. This is evidenced by, for example, the two expository books by Rapoport (1966, 1970a) on 2- and n-person games, respectively, which can be read independently one from the other.

Social scientists are generally familiar with individual decision making and 2-person games. They are considerably less familiar with the n-person branch of game theory, which has been steadily growing. One of the purposes of this volume is to correct this imbalance. We are in complete agreement with Luce and Raiffa’s (1957) conclusion,

… it is the n-person theory which must be of greater interest in sociology and economics. It is here, more than in two-person theory, that game theory as a part of social science, though not as a part of mathematics, will stand or fall [p. 157]”

1.2. SOCIAL PSYCHOLOGY AND n-PERSON GAME THEORY

Investigators of n-person conflict of interest who adopt a gam...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1. PROLEGOMENA
• 2. THE LANGUAGE OF COOPERATIVE n-PERSON GAMES
• 3. TWO CLASSICAL THEORIES: THE CORE AND THE STABLE SET
• 4. BARGAINING SETS
• 5. POWER BARGAINING SETS
• 6. EXCESS THEORIES
• 7. THE SHAPLEY VALUE
• 8. SIMPLE GAMES (I): INTERPERSONAL CONTROL THEORIES
• 9. SIMPLE GAMES (II): EQUITY THEORIES
• 10. BARGAINING PROCESS MODELS
• 11. PARADIGMS FOR EXPERIMENTAL GAMES
• 12. EXPERIMENTAL GAMES: 3-PERSON QUOTA GAMES
• 13. OTHER EXPERIMENTAL GAMES
• 14. CONCLUDING REMARKS
• REFERENCES
• AUTHOR INDEX
• SUBJECT INDEX