Coalitions
Coalitions
I. The Study of CoalitionsWilliam H. Riker
II. Coalition FormationWilliam A. Gamson
I THE STUDY OF COALITIONS
The word “coalition” has long been used in ordinary English to refer to a group of people who come together (usually on a temporary basis) to obtain some end. Typically, a coalition has been regarded as a parliamentary or political grouping less permanent than a party or a faction or an interest group [seeParliamentary government]. Recently, however, the word has acquired a technical significance in social science theories with the elaboration (in the last two decades) of the theory of n-person games. The notion of coalition formation is central to this theory, since coalitions are the characteristic form of social organization by which the outcomes of such games are determined. To the degree that the theory provides a model for the study of national decision making in elections, parliaments, committees, cabinets, etc., or of international decision making in wars, diplomatic maneuvers, and international organizations, to that degree, coalitions are the characteristic form of social organization for political decision making generally.
The originators of the theory of n-person games, John von Neumann and Oskar Morgenstern, observed a fundamental difference, with respect to discovering the best way to win, between two-person games and games involving more than two persons: In two-person games, the problem for each player is to select the best strategy against his opponent; but in three-person or larger games, the problem for each player is to select partner(s) who can collectively win. They called the artifact resulting from the mutual selection of partners a coalition, and they constructed the whole theory of n-person games about the process of forming coalitions. Since politics is often defined as the authoritative allocation of values and since, in all but dictatorial or duopolistic situations, allocation is a process of coalition formation (Riker 1962, chapter 1), it is apparent that a theory of coalitions is a central part of a theory of politics.
Main problems
Three main questions have been dealt with in the theory: (1) How should winnings be divided to ensure victory for a player or a coalition? (2) Given a particular set of rules, what chance does a particular player have to be in a crucial position in a winning coalition? (3) Which potential partners should come together in particular play? The theories relating to each of these questions will be summarized and then attempts at verification and use of the theories will be discussed.
The division of winnings
The Von Neumann—Morgenstern notion of a solution is the main contribution here. The substance of this notion is that, while it is not possible to specify a uniquely preferable coalition, it is possible to specify a set of preferable imputations, that is, a set of preferable ways to distribute gains and losses. To explain this notion the following vocabulary is required.
Let there be a set of players, I, where I is given by {1, 2, … n}, and let the subsets of I, which are coalitions, be designated by S, T, …. Let the payment to each player at the end of the game be designated by xt, where i = 1, 2, …, n, so that the totality of payments, which is an imputation is a vector, x = (xl,x2 …, xn). Let the payment to a coalition, S, be designated by a function, v, which is a real valued set function with at least the following properties:
(1) v(?) =0, where ? is the empty coalition; and (2) v(S È T) ?v(S) + v(T), where S and T are disjoint subsets of 1.
The second property, superadditivity, records the fact that for at least some players in some games there is an increment in payoff from the very act of joining together. A coalition, S, is said to be effective for an imputation, x, if
That is, a coalition is effective if it can win as much or more than the sum of the payoffs to its members. An imputation, x, is said to dominate an imputation, y, if
( 1 ) S is not empty
(2) S is effective for x
(3) Xi> yi, for alH in S.
A solution, V , is a set of imputations such that (1) no y in V is dominated by an x in V ; and (2) every y not in V is dominated by an x in V. To illustrate, for the zero-sum three-person game in normal form (for definitions of “zero-sum” and “normal form” see, e.g., Riker & Niemi 1964) the set V is The essence of this definition is that, regardless of which winning coalition forms [i.e., (1, 2), (1, 3), or (2, 3)], still any one that does form ought to adopt a split-the-winnings-equally kind of imputation. The heuristic rationale leading to this definition is the observation that a player, i, who seeks to obtain is creating a situation in which he may receive the worst possible payoff, i.e., Xi = –l. If, in the course of negotiations, a coalition of players 1 and 2 has arrived at a tentative imputation y, where player 2 is likely to be especially receptive to offers from player 3 to form (2,3) with imputation and where, of course, y is now dominated by x. Thus player 1 is undone by his greed. On the other hand, none of the imputations in the solution are susceptible to that kind of renegotiation of coalitions, for no partner in a winning coalition is likely to feel disadvantaged by the imputation adopted. Hence, if an imputation in the solution is adopted, it is likely to be stable as are also the coalitions associated with it.
In short, a solution specifies a set of preferable imputations, but not preferable coalitions. Since, however, Von Neumann and Morgenstern had heuristically stated the problem of n-person games to be that of finding partners and since, in contrast, their mathematical “solution” specified not partners but rather the division of spoils among unspecified partners, much dissatisfaction has been expressed with the notion of a solution, and a number of alternatives have been offered (Von Neumann & Morgenstern 1944). The spirit of these alternatives, like the spirit of the Von Neumann-Morgenstern solution, is to pick out a limited number of acceptable imputations from the (usually) infinite number of imputations possible.
?-stability: The most interesting of these alternatives is the notion of Y-stability, which can be explained with the following vocabulary: Let a partition of I into disjoint subsets be defined as a coalition structure, t . For example, if n = 3, the partitions ({1, 2}, {3}) and ({1, 3}, {2}) are possible coalition structures. Let ? denote a rule of admissible changes from t , and let ? (t) denote the set of coalition structures resulting from the application of ? to t. For example, if for n = 3, t is the partition ({1, 2}, {3}), and if ? is a rule that permits any coalition formed by the addition of a single player to an existent coalition, then ?(t) is the following set of alternative partitions:
[({1,2,3}), ({1,3}, {2}), ({2,3}, {1})].
Since the point of this argument is to set some limit on possible imputations and coalitions, one wishes to find pairs, (x, t ), of an imputation and a coalition structure such that players would not wish to depart from the pair (once they have reached it) by means of the changes permitted by ?. Such pairs, (x, t ), are in a kind of equilibrium, which Luce called ?-stability (1954). Formally defined, a pair, (x, t ), is ?-stable if:
(1) , for all S in ?(?), and
(2) if xi = v({i}), then {i} is in ?.
The first condition states that, for any coalition, S, that might be formed by the application of Y to t , the players in this potential coalition cannot better themselves. The second condition states that if a player receives no more than he can get in the worst possible circumstances (i.e., when he is in a single-member coalition), then he must in fact be in such a condition in t , that is to say for a member, i, of a multimember coalition in t, xi> v({i}).
The advantage of this definition is that it permits some discussion of actual partners along with a discussion of imputations. Its disadvantage is that it requires a precise specification of ? for a particular game— and in social situations this is usually difficult to specify. Furthermore, there is an embarrassingly ad hoc flavor to the whole definition inasmuch as the specification of ? must be in terms of standards of behavior prevailing among particular players of a game.
Other alternatives to solution theory have also been offered, but they are no more satisfactory than what they purport to supplant; so instead of summarizing them here, the reader is referred to Luce and Raiff a, where ? -stability is also admirably discussed (1957, pp. 220–245).
The chance of obtaining a crucial position
Here the main contribution is Shapley’s notion of a value for n-person games, which is an a priori method for estimating whether or not, for a particular player, a game is worth playing (Shapley 1953; Shapley & Shubik 1954). Suppose, in a game of n players, a coalition of k players, where k?n, is necessary to win. Let coalitions of k players be constructed by permuting the n players so that the first k players in any permutation are the minimally winning coalition, which is defined as a winning coalition that ceases to be winning if the kth player is subtracted. Let the kth position be designated as the pivot and let the number of times a player, i, occupies the pivot position be designated by pi. Since there are n! permutations of n, the chance, v, that a player, i, occupies the pivotal position is v(i) = pi/n!.
Underlying this measure of value are two crucial sociological assumptions: (1) Since the notion of pivoting is defined with respect to minimal winning coalitions, one infers that membership of a coalition in excess of the minimally winning size is irrelevant. This is a version of the size principle to be discussed below. (2) The expectation about imputations is quite different from imputations prescribed in the Von Neumann-Morgenstern solution. In a solution, the division of winnings among equally weighted members is, regardless of their position in the coalition, equal. But in the sociological theory underlying Shapley’s value, there is a time dimension to membership in a coalition so that the player in the pivotal position can expect to receive more than others. Over time, these advantages are expected to average out; but still the single imputation in the specific play is not an imputation in the solution. The difference is, of course, the assumption of the existence of a time dimension (and perhaps of a differentiation of roles in the coalition-formation process).
Which players should become partners
The main contributions here are those by Gamson (1961a; 1961b) and Riker (1962). Underlying these contributions is the size principle, which is the assertion that, with perfect and complete information, players should prefer minimally winning coalitions to larger winning ones. Using this principle, it is possible to show that if players have unequal weights then some possible coalitions are preferable to others, and indeed in some distributions of weights one possible coalition is uniquely preferable to others. Similarly, some players have unique advantages over others in the sense that the advantaged players can expect to be included in any preferable coalition. Since these two kinds of advantage are a function of the kind of ways in which the several weights can add up to k, which is the minimally winning size, it is not possible to specify in general these advantages for players and coalitions. Tables specifying these advantages for relevant variations in partitions of the total weight in the set of players are set forth in Gamson and in Riker for n = 3, 4, or 5. The most interesting result of such specifications is that usually the least weighty players ought to combine with each other, which leads to the somewhat paradoxical assertion that the weightiest player is usually the weakest in terms of combinatorial advantages. Note, however, that this conclusion follows from an argument in which perfect and complete information is assumed. Weakening the assumptions about information also weakens the force of this conclusion. As information is rendered less perfect and less complete, players may be expected to attempt to increase the size of winning coalitions above the minimum in order to guarantee victory.
Verification
Much more energy has been expended on the elaboration of the theory of coalitions than on the verification of it. The paucity of attempts at verification is explicable in terms of a theoretical difficulty : The whole theory is normative in the sense that it specifies what rational players should do to obtain the best possible payoff. It does not specify what real players will in fact do. To render the normative theory into a descriptive one, it must be assumed (1) that some players (that is, at least the winners) are rational, in the sense that they prefer to win rather than lose, and (2) that some political situations are sufficiently analogous to abstract games so that the theory of games is also a theory of political coalitions. Many social scientists are suspicious of both assumptions, and, given their reluctance to adopt them, the theory remains unverifiable just as all normative theories are unverifiable as truth-functional sentences. Nevertheless, some social scientists have been willing to assume that institutions that favor winners encourage the existence of rational political men and that political situations in which participants perceive their problem as one of winning are quite analogous to abstract games. Hence they have regarded the theory of coalitions as descriptive and have sought to verify it, hoping thereby to verify as well the assumption about rationality.
Verification and the “solution.”
Several experiments, most of which are well summarized by Rapoport and Orwant (1962) and Riker and Niemi (1964), have been conducted to test some features of the notion of a solution. In general, the results of these have not verified the Von Neumann-Morgenstern theory, except when special precautions have been taken in the experimental design to encourage subjects to behave highly competitively. Usually the imputations arrived at have been of the sort (1/n, 1/n, …, 1/n), instead of the prescribed a1, a2, …, ak, ?1 ?2,…, ?n-k, where ai = ai and ai> 0 > ?i. The latter imputation has been achieved with fair consistency only when, for example, stooges have been inserted among sophisticated subjects to insist upon imputations of the prescribed kind. Several explanations for the failure have been advanced, to wit, that the experimental design (1) failed to protect the outcome from the influence of variables in subjects’ personalities (e.g., their attitudes toward gambling); (2) failed to provide stakes large enough to induce rational behavior; and (3) permitted subjects to perceive the experimenter as another player in an (n + l)-player game, in which the imputation (1/n, 1/n, …, 1/n,–n) may be in the solution (provided the game is regarded as discriminatory). Further experiments with improved designs are necessary before any conclusion can be drawn on the truth or falsity of solution theory.
As for the notion of Y -stability, the one attempt to use it in analyzing a real situation (which attempt amounts indirectly to an attempt at verification) has produced interesting and intuitively plausible, but far from definitive, results. Luce and Rogow (1956) analyzed a simplified version of the three houses of the national legislature in the United States (i.e., the president, in his legislative capacity, the Senate, and the House of Representatives) in which there were assumed to exist two parties each with some members who always voted with the majority of the party (die-hards) and some who might on occasion vote with a majority of the other party (defectors). They examined the legislative branch according to variations in numbers of party members and numbers of die-hards and defectors in each party and produced two conclusions that do not otherwise seem immediately obvious: (1) a president is weak when either party has a more than two-thirds majority in Congress and (2) a president is strong when there is no party that can obtain a two-thirds majority even with the help of defectors from the other party. These nonobvious, but intuitively satisfying, conclusions suggest that much can be learned by further application of Y -stability theory to other situations.
Verification and use of value
It is hard to imagine how the notion of value might be verified, for it is a method to evaluate rather than predict or prescribe. Riker tried to determine by statistical analyses of roll calls in an assembly whether or not legislators seek to improve their chances of occupying the pivotal position; but his data and procedures were too crude to lead to any sure conclusions (1959). The notion of value has, however, been used to evaluate several real constitutions, treating them as games to be played: the Congress in the United States; the Security Council of the United Nations (Shapley & Shubik 1954); the Electoral College in the United States (Mann & Shapley I960; 1962); and the federal relationship in the United States, Canada, and Australia (Riker & Schaps 1957).
Verification of prescriptions on partners
Vinacke and Arkoff (1957) and Gamson (1961b) have attempted to verify Caplow’s predictions in experimental situations. Their results, with some reservations, tend to verify his specifications of preferable coalitions. Riker has offered historical evidence for the acceptance of the size principle by politicians in national and international political situations (1962, chapters 3, 7). Barth, assuming the size principle, has shown that Afghan chieftains conduct their intertribal diplomacy in terms of it (1959). Although far from satisfactorily verified, this feature of the theory of coalitions is closer to verification than any other.
Further developments
Considering the previous development of the theory of coalitions, it seems likely that future expansion of the theory can be expected only if additional sociological assumptions can be incorporated in the mathematical theory. The original theory of solutions to n-person games contained a bare minimum of assumptions about behavior, viz., (1) rational motives, i.e., participants’ desire to win rather than to lose; (2) sets and subsets or the notion of coalitions itself; and (3) super additivity. Out of these minimal assumptions, Von Neumann and Morgenstern were able to arrive at the theory of solutions, which, however, seemed inadequate because it specified only the division of winnings and not the choice of partners. Luce’s step toward the discussion of partners (i.e., ? -stability) involved the introduction of an additional sociological assumption, namely, the existence of a standard of behavior, ? , that admitted some kinds of bargaining about membership in coalitions but not others. Caplow’s discussion of partners and Riker’s elaboration of the size principle required the introduction of both the notion of differentials in weights (which is substantially equivalent to the introduction of the notion of “power”) and the notion of a majority. Since these elaborations of the theory of coalitions depended upon the addition of sociological premises, it can be expected that future elaborations will depend on future additions.
Dynamics of growth
One promising additional sociological assumption is the notion that coalitions go through a process of growth. Let the situation at the beginning of any decision-making or allocative process be such that participants are partitioned into n single-member subsets. Define coalitions as winning or losing subsets when a winning subset exists, and define protocoalitions as subsets when no winning coalition exists. Then the process of decision making is the transformation of some protocoalition into a winning coalition. If a winning coalition consists of k members, where 1/2n ? k ? n, then the process of decision making or allocation is the development of a protocoalition from 1 member to 2 members, from 2 to 3 members,… from (k – 2) members to (k – 1) members, and from (k – 1) to k members. There are several crucial stages in this process, especially the movement from one-member to two-member protocoalitions and the movement from (k– 1) to k members (which latter may be called the “end-play”). Riker has set forth some of the strategic considerations of the end-play, but these have as yet to be developed in a general statement (1962, chapter 6). His conclusions can be summarized, however, as the assertion that the strategy of end-play is to find, for a given structure of protocoalitions, that coalition that most nearly approaches the minimally winning size. Doubtless many other strategic considerations enter into the growth process, although these are as yet unspecified.
Roles
Defining roles as positions specified by the rules of any particular process of coalition formation, Von Neumann and Morgenstern assumed that while roles of players may differ with respect to advantages in the rules, still the roles are identical with respect to the kind of behavior in bargaining. Furthermore, they assumed that players could rotate indiscriminately among roles. There is no reason to suppose that either assumption is appropriate for the description of the natural world. Indeed, it is likely that in reality there exists a variety of roles in the coalition-formation process: for example, leader, the role of initiating that bargaining by which protocoalitions are enlarged; follower, the role of moving, at the instigation of a leader, from a single-member protocoalition to a larger one; pivot, already defined as that follower who becomes the kth member of a minimally winning coalition; reliable follower, the role of irrevocably accepting membership in a multimember protocoalition; defecting follower, the role of accepting and subsequently rejecting membership in a multimember protocoalition; wallflower, a member of a single-member coalition who is not sought out for the role of follower; etc. Doubtless the theory of coalition formation may be rendered more appropriate for the study of nature by mathematizing the essential features of behavior in each of these roles. For example, with respect to the role of leader, it is likely, as Riker has argued, that his payments are different in kind from the payments to followers, that the leader (who quite possibly starts out as a wallflower) is willing to forgo material reward for the sake of obtaining the psychic reward of leadership and that the leader may, for the sake of retaining his leadership, pay out more material rewards than his prospective coalition can win. Assuming these possibilities do in fact prevail, then the notion of a solution must be modified. Solution theory requires a symmetric kind of imputation (although it allows for unequal rewards for roles with unequal advantages in the rules). But if differentials in behavior may affect imputations, then the notion of “equal rewards for equal advantages in the rules” must be abandoned for a notion of unequal rewards according to whether the participant plays the role of leader or follower, with the leader accepting the lesser material reward. (The sociological, but not the mathematical, consequences of accepting the notion of these differentials in roles is set forth in Riker 1962.)
In the beginning of this article, it was suggested that a theory of coalitions amounted to a theory of politics. The subsequent considerations, however, suggest that a theory of coalitions adequate to serve as a theory of politics has not yet been developed. The hope for the next generation is that it will be.
William H. Riker
[Also relevant are the entriesDecision making;Game theory;Simulation.]
BIBLIOGRAPHY
Barth, Fredrik 1959 Segmentary Opposition and the Theory of Games: A Study of Pathan Organization. Journal of the Royal Anthropological Institute of Great Britain and Ireland 89:5–21.
Buchanan, James M.; and Tullock, Gordon 1962 The Calculus of Consent: Logical Foundations of Constitutional Democracy. Ann Arbor: Univ. of Michigan Press.
Caplow, Theodore 1956 A Theory of Coalitions in the Triad. American Sociological Review 21:489–493.
Gamson, William A. 196la A Theory of Coalition Formation. American Sociological Review 26:373–382.
Gamson, William A. 1961b An Experimental Test of a Theory of Coalition Formation. American Sociological Review 26:565–573.
Luce, R. Duncan 1954 A Definition of Stability for n-Person Games. Annals of Mathematics 59:357–366.
Luce, R. Duncan; and Raiffa, Howard 1957 Games and Decisions: Introduction and Critical Survey. A Study of the Behavioral Models Project, Bureau of Applied Social Research, Columbia University. New York: Wiley. ? First issued in 1954 as A Survey of the Theory of Games, Columbia University, Bureau of Applied Social Research, Technical Report No. 5.
Luce, R. Duncan; and Rogow, Arnold A. 1956 A Game Theoretic Analysis of Congressional Power Distributions for a Stable Two-party System. Behavioral Science 1:83–95.
Mann, Irwin; and Shapley, L. S. 1960 Values of Large Games, IV: Evaluating the Electoral College by Montecarlo Techniques. U.S. Air Force Project, RAND, Memorandum RM-2651. Santa Monica, Calif.: RAND Corp.
Mann, Irwin; and Shapley, L. S. 1962 Values of Large Games, VI: Evaluating the Electoral College Exactly. Rand, Memorandum RM-3158-PR. Santa Monica, Calif.: RAND Corp.
Rapoport, Anatol 1960 Fights, Games, and Debates. Ann Arbor: Univ. of Michigan Press.
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Riker, William H. 1959 A Test of the Adequacy of the Power Index. Behavioral Science 4:120–131.
Riker, William H. 1962 The Theory of Political Coalitions. New Haven: Yale Univ. Press.
Riker, William H.; and Niemi, Donald 1962 The Stability of Coalitions on Roll Calls in the House of Representatives. American Political Science Review 56: 58–65.
Riker, William H.; and Niemi, Richard G. 1964 Anonymity and Rationality in the Essential Three-person Game. Human Relations 17:131–141.
Riker, William H.; and Schaps, Ronald 1957 Disharmony in Federal Government. Behavioral Science 2: 276–290.
Shapley, L. S. 1953 A Value for n-Person Games. Volume 2, pages 307–317 in H. W. Kuhn and A. W. Tucker (editors), Contributions to the Theory of Games. Princeton Univ. Press.
Shapley, L. S.; and Shubik, Martin 1954 A Method for Evaluating the Distribution of Power in a Committee System. American Political Science Review 48: 787–792.
Shubik, Martin (editor) 1964 Game Theory and Related Approaches to Social Behavior: Selections. New York: Wiley.
Vinacke, W. E.; and Arkoff, Abe 1957 An Experimental Study of Coalitions in the Triad. American Sociological Review 22:406–414.
Von neumann, John; and Morgenstern, Oskar (1944) 1964 Theory of Games and Economic Behavior. 3d ed. New York: Wiley.
II COALITION FORMATION
Few areas exhibit less external uniformity than the alliances of men. “Politics makes strange bed-fellows,” we say to express our bewilderment at some new coalition which belies our expectations from past knowledge of the participants. But this is not the only kind of surprise that the study of coalition formation has in store. “Strength is weakness” and “Playing to win is playing to lose” are only two examples of the lessons to be drawn from this area of human behavior. The fact that these lessons are not yet proverbial indicates only that the term “coalition” is in need of clarification, since the phenomena to which it should be applied are already sufficiently familiar.
As with many other terms, it is possible to use “coalition” in a manner that robs it of much of its meaning. Thus, it is sometimes used to mean no more than the kind of joint activity displayed by, say, two young children who hide in the same place when it is time for them to go to bed. A common extension of this usage is to denote the mutuality of affective support that often accompanies such activity; for instance, if the children agreed to tell their mother that they were fish and did not need to sleep, this little fantasy might be said to express a temporary coalition against the mother.
It is necessary to guard against such uses of “coalition” because the immense influence of Georg Simmel in this area has made them all too frequent. Simmel ([1902–1917] 1950, pp. 135–136) argued that in any closely knit group of three persons— a “triad,” as he called it—situations were bound to arise in which two of the three would regard the third as an intruder. But Simmel’s interest in triadic situations extended mainly to those in which there is no coalition of two against one, even in the sense of shared affective support. The situations to which he gave most attention are those in which two members of the triad are in conflict with each other and the third either mediates between them or acts as tertius gaudens— that is, as one who draws comfort and advantage from the conflict. Nevertheless, a number of social scientists have drawn on Simmers formulation to support the proposition that any triad tends to split into a coalition of two and a third who is excluded.
The old proverb that when two is company, three is a crowd, undoubtedly has some truth in it and is worth investigating. But the truth of this proverb, or its scientific equivalent, is less at issue here than its relevance for the topic of coalition formation. If every clique within a larger group is to be called a coalition, then any partitioning of a group into subparts on whatever basis could be viewed as a “coalition structure,” and the study of coalitions would be coextensive with the study of cliques or even with the entire field of sociometry. In order to avoid this kind of confusion, coalitions should be defined far more narrowly.
Defining a coalition
In this article, “coalition” will be used to mean the joint use of resources to determine the outcome of a decision, where a resource is some weight such that some critical quantity of it in the control of two or more parties to the decision is both necessary and sufficient to determine its outcome. Participants will be said to be using their resources jointly only if they coordinate their deployment of resources with respect to some decision. That is what is meant by saying that they have formed a coalition.
Some studies which have been cast in terms of coalition formation do not seem to meet the above criteria. For example, in Mills (1954) subjects were asked to decide (not necessarily on a group basis) what verdict they would have cast had they been present at the trial of Herman Melville’s hero Billy Budd, and various indices of support between each pair of participants were calculated. But if such an experiment deals with coalition formation, it must be possible to say what kinds of resources are being deployed. To include the Mills study in our definition, we might consider “arguments” as the resource, so that persons who argued in favor of the same verdict would be coordinating their use of resources, and this would count as a coalition. But the fact that two people who argue for the same verdict have come to like or respect each other would still be only incidental to the coalition situation.
Not all situations in which resources are used to affect the outcome of a decision involve coalition formation. This can be made clearer for situations involving several parties by first examining the simpler two-person situation. Schelling (1958) classifies two-person games of strategy into “pure coordination” games, “pure conflict” (or “zerosum”) games, and “mixed-motive” games. One good example of a pure coordination game is the relation between two bridge partners, since they have the same interests and their only problem is one of coordinating their strategy. In contrast, the relation between two teams in bridge is one of pure conflict, since their interests are diametrically opposed. The classic example of a mixed-motive game is the situation known in game theory as the “prisoner’s dilemma,” in which neither of two prisoners accused of complicity in the same crime can be condemned if neither confesses; if only one prisoner confesses, the other will be condemned and the one who confessed set free; and if both confess, both will be condemned (Rapoport & Chammah 1965; Rapoport & Orwant 1962). Thus the players are partly in conflict, because one prisoner can be set free only at the expense of the other, and partly in harmony, because both prisoners stand to gain if they agree not to confess, though this joint strategy will bring each of them less than the maximum he might have gained independently. The situation poses a dilemma because the prisoners are assumed to be held incommunicado, and each therefore has to make his decision in ignorance of what the other will do.
Schelling’s classification can be adapted to cover situations involving more than two parties. Thus pure coordination games would be those in which there exists a solution that brings the greatest possible return to all parties at once, and there is no reason to exclude any participant, since in pursuing his own interests, each party is aiding the others to achieve theirs. The problem would then be one of achieving this goal as efficiently as possible by coordinating the use of all available resources. But to call this a coalition situation would be to dilute the meaning of the term, since any group in which there was no conflict of interest and which was pursuing some common goal could then be called a coalition. Much the same considerations apply to pure conflict situations; the issue of whether to form a coalition does not arise, since no player has anything to gain by forming one— indeed, the joint use of resources may be forbidden by the rules of the game. It should be noted that some n-person, zero-sum games are not pure conflict games, since they may allow some of the players to pool their resources and so gain at the expense of the other players. A pure conflict game, however, is one from which coalitions are excluded, either by definition or because there is no incentive for them.
It therefore seems necessary to conclude that coalitions can take place only within the context of mixed-motive, n-person games, in which both conflict and common interest are simultaneously present, and must govern the courses of action chosen. This is because in such games there is no outcome that brings the greatest possible return to all players at once, while there is always the possibility that at least two of the players may do better if they pool their resources.
The term “coalition” can now be defined with greater precision. It is the joint use of resources to determine the outcome of a decision in a mixedmotive situation involving more than two units. Many studies of three-person groups that have been cast in coalition terms have not studied coalition formation under the above definition, though this does not mean that they lack relevance for the study of other aspects of group behavior.
Theories of coalition formation should predict who will combine with whom and how they will split the proceeds. Social scientists are not directly concerned with how people should behave in coalition situations, though obviously they can study normative prescriptions by treating them as statements about expected behavior. Each of the different theories of coalition formation that will be described depends on a different set of basic assumptions. In order to show how each type of theory can be applied, the following example of a coalition situation will be used. Suppose that successive polling at a political convention has reduced the list of candidates to three: candidate A, who controls 48 per cent of the votes; candidate B, who controls 30 per cent; and candidate C, who controls 22 per cent. Each of the three has absolute control over the votes of his supporters. The rules state that whoever obtains a simple majority is nominated.
Predicting coalitions
The first type of theory to be dealt with is minimum resource theory, which was first developed by Gamson (1961a) and Riker (1962), though some features of the theory were anticipated by Caplow (1956; 1959). The central hypothesis of the theory is that a coalition will form in which the total resources are as small as possible while still being sufficient. Thus, to use the convention example, since the coalition BC, with 52 per cent of the vote, is the smallest of three possible winning coalitions, it is clearly the one that will be formed. In this situation, strength, as defined by the amount of resources possessed before any coalition is formed, is really weakness, since the strongest candidate, with control over 48 per cent of the votes, is excluded from the winning coalition.
The cheapest coalition principle can be applied to groups of any size. It does not necessarily imply defeat for the strongest players; for example, if there had been a four-man game in which candidates A and B each controlled 24 per cent of the votes, with candidate C controlling 22 per cent and candidate D 30 per cent, candidate D would have been both the strongest player and a member of the cheapest winning coalition. Thus it is not true to say that initial strength never pays off in a coalition situation; strength is weakness only under certain conditions—conditions that can be clearly specified by minimum resource theory.
But what are the grounds on which the cheapest coalition prediction rests? One answer is that in many coalition situations the players believe that no one should gain proportionately more or less from a coalition than the amount of resources he is able to contribute to it. A convenient name for this belief is the parity norm. It should be noted that it is a belief about what the players feel in general that they deserve, not an empirical estimate of the distribution of power in any particular situation.
The parity norm is merely the ancient principle of distributive justice applied to coalitions. “A man in an exchange relation with another will expect that… the net rewards, or profits, of each man be proportional to his investments” (Homans 1961, p. 75). This norm discourages the formation of coalitions that are more powerful than necessary, because if a player gains from a coalition an amount proportional to the resources he brings to it, it is obviously to his advantage to join the coalition in which his resources will represent the largest possible share of the total resources pooled.
A second theory, the theory of minimum power, is based on an interpretation of L. S. Shapley’s method for evaluating the worth of an n-person game for any player (Shapley 1953). This method is based on the number of times a player is the “pivotal” member who turns an insufficient coalition into a winning one. The method is, of course, part of mathematical game theory, and the descriptive application of it proposed here is in no sense intended as a contribution to that normative theory. One may interpret Shapley’s measure of pivotal power as a measure of a player’s initial bargaining power. This type of power is clearly distinct from the amount of a player’s initial resources. In simple terms, a player’s pivotal power is the proportion of times his resources can change a losing coalition into a winning one. It is expressed by the index P/N, where N is the total number of permutations among the players and P is the number of permutations in which his resources are pivotal (Shapley & Shubik 1954). Thus, in the three-man convention situation, since there are 3! or six permutations, each candidate will be pivotal twice. The relative power of the candidates is there-fore equal, even though their initial resources are unequal.
One way of giving descriptive relevance to the concept of pivotal power is to assume that all players in a coalition situation will demand a share of the payoff proportional to their pivotal power. By deductions analogous to those in minimum resource theory, it follows that the winning coalition will be the smallest one possible in terms of the total pivotal power of its members. Thus, in the convention example, since each possible twoman coalition is the same “size,” each is equally likely to occur. It also follows that, since partners have the same power, the rewards of coalition will be divided in equal shares.
A third theory of coalition formation known as anti-competitive theory is an outgrowth of a series of experiments by Vinacke (1959; Bond & Vinacke 1961), many of which involved female subjects. The basic assumption of anti-competitive theory is that players in the coalition situation do not want to compete with each other; on the contrary, they are concerned mainly with preserving social relationships within the group. It follows that coalitions in such a group will form along the lines of least resistance. This behavior is quite unlike that predicted by minimum resource theory, which assumes that players are trying to get as much as they feel is owed to them, or by minimum power theory, which assumes that players are trying to get as much they can. The “line of least resistance” in anti-competitive theory can be defined as the relation that exists between players who, if they combined, would have no problems about how to divide their gains. This would seem to be especially true of players who have equal amounts of initial resources, since both the parity norm and the principle of pivotal power would prescribe that, if they combined, they would share their gains equally.
Of course, if either the parity norm or the pivotal power principle were followed to the exclusion of any other principle, the players would not need to bargain even if their resources or power were unequal. But in actual coalition situations the possibility that one principle may be invoked against another can never be ruled out, and it therefore seems likely that, where resources are unequal, bargaining of some kind will take place.
Players who are interested in minimizing the disruptive effects of bargaining are likely to avoid the hard and skillful bargainer. It follows that, where the anti-competitive norm exists, playing to win is playing to lose, since the more openly a player seeks to get as much as he can, the less likely it is that he will find a partner who will help him to get it. This can lead to the ironic situation that those players who ultimately profit most from a coalition situation are those who have made least effort to do so.
In the convention example it is not clear what the outcome would be under the assumptions of anti-competitive theory, especially since initial resources are unequally divided between the candidates and the possibility of bargaining, as already explained, cannot be ruled out. If candidate A builds up his lead in voting strength by his known skill in bargaining, he may find himself without support from the other, initially weaker candidates if both the parity norm and the anti-competitive norm are at work in the situation. A player who looks for allies from a position of strength may find not only that playing to win is playing to lose but also that, in coalition situations, strength is weakness.
Coalitions and social organization
Coalition theory is relevant for understanding social organization in two ways: (1) in the rise of new forms of social organization and (2) in the operation of existing social systems.
The rise of social organization
Coalitions may be viewed as a nascent form of social organization. They may, of course, simply break up into their constituent parts, or they may become stable and institutionalized. In the latter case, the process of coalition formation is a social organizational process; it is an important mechanism for the creation of new social organizational forms.
In the primary group and ad hoc laboratory groups which have so frequently served as the object of empirical studies of coalition formation, individuals form coalitions and their coalitions are unlikely to be given any formal or permanent status. When, in more complex forms of social organization, the coordination of resources does become highly stable and enduring, are we still justified in calling them coalitions? The answer seems to depend on (1) the degree of institutionalization of the coalition and (2) the maintenance of the original boundaries of the units which form the coalition. For example, consider a situation in which a number of originally autonomous units join in an association that draws on the resources of the units to compete for the achievement of their collective goals. The new association might become so successful that its maintenance becomes more important for the members than the achievement of any advantage relative to each other. Finally, the autonomous nature of the original units may disappear completely.
The formation of a labor union, for example, does not seem too different from such a situation. Individuals who are seeking better working conditions, material benefits, and sometimes more general social changes are induced to join an association. This union comes to symbolize the collective interest of the members to such a degree that they are willing to forgo some short-run interest to themselves—for example, the kind of interest that would be served by “scabbing” or strikebreaking. Or they will be willing to risk personal injury and arrest to administer sanctions to those who do participate in such activities. Such forms of social organization are no longer nascent; they have arrived.
Coalitions in existing social systems
Coalition formation is important through its connection with cleavages and integration in a social system and through its impact on the content of decision. The constituent parts of a coalition do not always lose their separate identity. The Democratic party has a stable existence, but the diverse set of interests which constitute it have maintained their own boundaries. One faction of a political party may continue to act as an autonomous unit and to form coalitions on certain issues with other parties (Luce & Rogow 1956). Even when the original units dis-appear, coalitions may be quite relevant. New is-sues may create a new set of autonomous factions, perhaps quite different from the original units which formed the association. In either case, coalition formation is intimately connected with the structure of cleavages within a social system; such a structure provides the units from which coalitions are formed.
Since coalitions form around decisions or sets of decisions, the maintenance of a coalition will frequently require the tacit neutrality of the coalition on issues which divide the members. Lerner and Aron (1957) argue that the defeat of the European Defense Community (EDC) by the French National Assembly was particularly understandable in light of the coalition demands of the Mendes-France government. The MRP, chief supporters of EDC, were in opposition because of other issues (Indochina), while the Gaullists, who were part of the governing coalition, opposed EDC. Vigorous sup-port on this issue by the Mendes-France government would have made salient a set of cleavages that crosscut the governing coalition.
A shifting coalition structure tends to reduce the severity of cleavage, since today’s enemy may be needed tomorrow as an ally. On the other hand, a shifting coalition structure also reduces the degree of consensus required for action to occur; it need be based only on the temporary conjunction of interests. Coalition formation, then, involves one process by which cleavage and consensus are balanced in social organization.
William A. Gamson
[See alsoCohesion, social; Cooperation; Game theory; Groups; Parties, political; Sociometry;and the biography ofSimmel.]
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