A Theory of Coalitions in the Triad

Theodore Caplow

American Sociological Review, Vol. 21, No. 4. (Aug., 1956), pp. 489-493.

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A THEORY OF COALITIONS IN THE TRIAD THEODORECAPLOW University of Minnesota

STUDY of the triad is one of the most in- teresting and satisfactory areas of cur-rent research activity in sociology. The relevant theory is unusually simple and straightforward. I t lends itself to empirical verification more readily than most other models of interaction. The theory of the triad promises to be applicable to situations of different scale, although this advantage has not yet been fully exploited. I t seems likely that generalizations developed in the three-person group will be applicable to situations where the interacting units are organized groups, even such very large groups as political parties and states. In his original discussion, Simmel deliberately com- pared intimate human situations such as the contention of two men for one woman with large scale situations like the European balance of power. For many years Simmel's powerful but

unsystematic analysis was admired and neg- lected. The development of research evidence on the triad has come about very recently. The last two years have seen the publication of a series of brilliant experiments by Mills, Strodtbeck, Torrance, Bales and Borgatta. These experiments have demonstrated the reality of the phenomenon which Simmel stressed-the tendency of the triad to become a coalition of two against one. They have also demonstrated that small differences in "power," "activity," and other characteris- tics of the members of the triad have con- siderable influence upon the formation and persistence of coalitions.

The purpose of this paper is to examine the model of the triad whose members are not identical in power and to call attention to a neglected feature of this model, namely, that the formation of given coalitions depends upon the initial distribution of power in the triad and, other things being equal, may be ~redicted to some extent when the initial

unsymmetric distribution, i.e., those in which different coalitions receive different results. They remark, "it seems that what a player can get in a definite coalition depends not only on what the rules of the game provide for that eventuality, but also on the other (competing) possibility of coalition for him- self and for his partner. Since the rules of the game are absolute and inviolable, this means that under certain conditions conzpen-sations must be paid among coalition part- ners; i.e., that a player must have to pay a well defined price to a prospective coalition partner. The amount of the compensations will depend on what other alternatives are open to each of the players." l Concealed in this and subsequent discussions is the as-sumption of equality of power among the three players, even when they have different possibilities of gain or loss. (See especially their figure 51). This assumption does not fit many triads of sociological interest in which the typical gain consists of domination over other triad members, and not in an ex- ternal reward to be obtained by a given coalition. This is the case in three-sibling rivalries, in three-cornered political contests, and in many other concrete situations. We shall consider six types. The following

asumptions will apply: Assumption 1. Members of a triad may

differ in strength. A stronger member can control a weaker member, and will seek to do so. Assumption 2. Each member of the triad

seeks control over the others. Control over two others is preferred to control over one other. Control over one other is preferred to control over none. Assumption 3. Strength is additive. The

strength of a coalition is equal to the sum of the strengths of its two members. Assumption 4. The formation of coalitions

takes place in an existing triadic situation,

distribution of power is known. l John von Neumann and Oskar Morgenstern, In their discussion of the zero-sum three- Theory o f Games and Econolnic Behavior, Prince-

ton: Princeton University Press, 1947, p. 227. See person game, van Neumann and MOrgell- also Chapter 5, especially 22.1.1 to 22.3.4 and stern consider at some length the case nf Chapter 6, 29.2.1 and 29.2.2.

490 AMERICAN SOCIOLOGICAL REVIEW so that there is a pre-coalition condition in every triad. Any attempt by a stronger member to coerce a weaker member into joining a non-advantageous coalition will pro- voke the formation of an advantageous coalition to oppose the coercion. The question of how well these assump-

tions are met by the triads we encounter in laboratory situations must be left for later consideration.

Type 1: In this simplest case, all three members are of equal strength. This is the classic, but probably not the most common- type of triad. The coalitions AB, BC and CA are equally likely, and each member strives to enter a coalition within which he is equal to his ally and stronger (by virtue of the coalition) than the isolate.

Type 2: One member is stronger than the other two, but not much stronger. Again, all three members seek a coalition, since to be isolated is unequivocally disadvantageous. However, the three possible coalitions are no longer of equal advantage. Consider the position of B. If he forms a coalition with A, he will (by virtue of the coalition) be stronger than C, but within the coalition he will be weaker than A. If,on the other hand, he forms a coalition with C, he will be equal to C within the coalition and stronger than A by virtue of the coalition. The position of C is identical with that of B, so that other things being equal, the coalition BC will be formed and the individually strongest mem- ber of the triad will ordinarily turn out to be the weakest after the formation of the most probable coalition.

Type 3: Here again, two members of the triad are equal in strength, but this time the third member is weaker. I t will readily be seen that A may strengthen his position by forming a coalition with either B or C, and will be welcomed as an ally by either B or C. On the other hand, if B joins C, he does not improve his pre-coalition position of equality with C and superiority to A. His only motive to enter a coalition with C is to block the coalition AC. However, C's position is identi- cal with B and he, too, will prefer A to B as an ally. Thus there are two probable coalitions, AB and AC.

Type 4: I n this case, the strength of A exceeds the combined strength of B and C. B and C have no motive to enter a coalition with each other. Once formed, the coalition

would still be weaker than A and they would still be equal within it. A, on the other hand, has no motive to form a coalition with B or C, since he is already stronger than each of them and is not threatened by their coalition. No coalition will be formed, unless either B or C can find some extraneous means of in- ducing A to join them.

Type 5: In this case, no two members of the triad are equal in strength but the com- bined strength of any two members exceeds that of the third. This resembles Type 3 in that the weakest member of the triad has a definite advantage, being sure to be included in whatever coalition is formed. A seeks to join both B and C, and C seeks to join both A and B, but B has no incentive to enter a coalition with A and A has a very strong incentive to enter a coalition with C. Whether the differential strength of A and B will make them differentially attractive to C lies out- side the scope of our present assumptions.

Type 6: This is like the previous type in that the three members of the triad are un- equal, but here A is stronger than B and C combined and has no motive to form a coali- tion. As in Type 4, true coalition is impossi- ble. However, while in Type 4 both of the weaker members seek to join the stronger member, only C can improve his position by finding some extraneous means of inducing A to join him. The most surprising conclusion which

emerges from this analysis is that the nature of the triadic situation often favors the weak over the strong. Thus in Type 2, by virtue of the fact that A is stronger than B or C, he will ordinarily become the isolate and therefore he is functionally the weakest. In Type 3, the formation of either of the two probable coalitions will benefit A. Though he is the weakest member of the triad and will continue to be weaker than his partner i~ either coalition, he may nevertheless be described as the most secure of the three. The same is again true of Type 5, where both A and B but not C are in danger of being isolated. The practical implications are con- siderable.

AVAILABLE EVIDENCE

Unfortunately, the data of the several ex- cellent laboratory studies of interaction proc- esses in the triad are not so arranged as to

A THEORY O F COALITIONS IN THE TR IAD 491

permit the direct test of the hypotheses suggested b y the foregoing analysis. Never- theless, there is a certain amount of frag-mentary evidence which suggests tha t this may b e a useful model with which t o ap-proach the s tudy of the coalition process.

Th e first application consists of a n example given b y Simmel :

I t was the general custom of the Incas to divide a newly conquered tribe in two ap-proximately equal halves and to place a supervisor over each of them, but to give these

two supervisors slightly different ranks. This was indeed the most suitable means for pro- voking rivalry between the two heads, which prevented any united actions against the ruler on the part of the subjected territory. By con- trast, both identical ranks and greatly dif- ferent ranks would have made identification much easier. If the two heads had had the same rank, an equal distribution of leader-ship in case of action would have been more likely than any other arrangement; and, since there would have been need for subordination, peers would have most probably submitted to

492 AMERICAN SOCIOI REVIEW

such a technical necessity. If the two heads had had very different ranks, the leadership of one would have found no opposition. The slight difference in rank least of all allows an organic and satisfactory arrangement in the unification feared, since the one would doubt- less have claimed unconditional prerogative because of his superiority, which, on the other hand, was not significant enough to suggest the same claim to the other."

When these two alternative arrangements are translated into the language of our model, we see that the first alternative would correspond to Type 2.3 I n place of the original explanation, we may say that the slightly unequal ranks are a device for converting Type 2 to Type 5. I t a t once appears that, whereas in Type 2 only the "rebellious" coalition BC is probable, in Type 5 the "loyal" coalition AC is equally probable under our assumptions. Whether it would be equally probable in the real life situation depends upon other factors, such as the distances among A, B and C, but regardless of these other factors, some of the elements of the basic situation are likely to make themselves felt as the real situation unfolds. For example, i t is clear that in a situation of Type 5 , C can extract concessions from B in return for entering the coalition BC, despite the fact that B is stronger. I n Mills' first study of power relations in

three-person groups, attention is focused upon the relationship between the most ac- tive member and the next most active mem- ber, as measured by the number of acts scored according to Bales' method of inter- action process analysis. If the two most active members show a mutually supportive pattern, then their relationship is charac-terized as solidary. Otherwise it is described as conflicting, dominant, or contending. Rlills concludes that '(medians of support rate exchanged between three members in a series of 48 problem solving sessions con-

"urt H. Wolff (translator and editor), The Sociology o f Georg Simmel, Glencoe: The Free Press, 1950, pp. 165-166.

"ecause the coalition is feared in the future situation where successful rebellion would be pos- sible, we must assume A

A FACTOR AXALYSIS OF SIXTEEN PRIMITIVE SOCIETIES 493

nificant a t the .01 level of confidence)." This is pretty much what we would expect from the disadvantageous position of B with respect to potential coalitions in Type 5 . But it more-of a straw in the wind than real evidence.

,4 more direct test is found in Mills' second published study. In this experiment, two mem- bers of each task triad were collaborators, acting out predetermined instructions. The third member was a naive subiect. The triads were arranged in such a way that each naive subject was of definitely higher or lower status than his fellow members. In other words, each triad corresponded either to Type 2 or Type 3. The first major hypoth- esis was "that the coalition would be more apt to persist when the isolate is of higher status relative to others in the group than when he is of lower status. [This involves the assumption that] the controlling re-sources of the high status member are less

6 E . Paul Torrance, "Some Consequences of Power Differences on Decision Making in Permanent and Temporary Three-Man Groups," in A. Paul Hare, Edgar F. Borgatta, Robert F. Bales (edi-tors), Small Goups: Studies i n Social Interaction, New York: Alfred K. Knopf, 1955, pp. 488-489.

contingent upon actual support from others, and, consequently, that both the value and the loss of the ally is less for him than for the low status member. With superior re-sources following rejection, he is expected to stand against the coalition, while the isolate of fewer resources is expected to submit." The findings, although not statis- tically significant, were directly contrary to the hypothesis. I t was found that "the low status persons is less apt to conform to the majority opinion than the high status isolate" and that "the low status isolate offers more resistance to the coalition than the high status member."

To the extent that the conditions of Type 2 were reproduced with the high-status sub- jects and of Type 3 with the low-status sub- jects in these experiments, this finding is consistent with the model. We would except less resistance to the "inevitable" coalition of BC in Type 2 than to the improbable and therefore unstable coalition of BC in Type 3.

Theodore M. Mills, "The Coalition Pattern in Three Person Groups," American Sociological Re-view, 19 (December, 1954), p. 663.

8 Ibid., pp. 664-665.

A FACTOR ANALYSIS OF SIXTEEN PRIMITIVE SOCIETIES *

KARLF. SCHUESSLERAND HAROLDDRIVER

Indiana University

THIS paper discusses the utility of factor analysis in cultural sociology. A factor analysis of sixteen primitive societies is here re~orted not so much for its socio- logical interest but rather to project into the open certain methodological issues which are likely to arise in this type of study. In a sense the study is a continuation of efforts to arrive a t a more refined under-standing of these groups and their inter-relations by correlational procedures.

In 1939 Kroeber constructed a classifica-tion of tribes by arranging correlation in-dexes (Yule's Q) in a matrix so that tribes

* Expanded version of a paper read at the Ameri- can Sociological Society meetings, September, 1955. This research was supported by a grant-in-aid from the Indiana University Graduate School.

with relatively high coefficients inter se were alongside one an0ther.l Recently in an effort to validate cluster analysis and to explore its power Clements applied this technique to the same coefficients which Kroeber had previously clustered by inspec t i~n .~ re-His sults agreed closely with those obtained by Kroeber, and the agreement was viewed as a validation of cluster analysis. The question of whether factor analysis would yield in-formation about intergroup connections

Alfred L. Kroeber, "Local Ethnographic and Methodological Inferences," in Harold E. Driver, Culture Element Distributions X : Northwest Cali-fornia, University of. California Anthropological Records, 1 (1939), pp. 425-429.

2 Forrest E. Clements, "The Use of Cluster Analysis with Anthropological Data," Alnerican

You have printed the following article:A Theory of Coalitions in the TriadTheodore CaplowAmerican Sociological Review, Vol. 21, No. 4. (Aug., 1956), pp. 489-493.Stable URL:http://links.jstor.org/sici?sici=0003-1224%28195608%2921%3A4%3C489%3AATOCIT%3E2.0.CO%3B2-R

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[Footnotes]4 Power Relations in Three-Person GroupsTheodore M. MillsAmerican Sociological Review, Vol. 18, No. 4. (Aug., 1953), pp. 351-357.Stable URL:http://links.jstor.org/sici?sici=0003-1224%28195308%2918%3A4%3C351%3APRITG%3E2.0.CO%3B2-95 The Family as a Three-Person GroupFred L. StrodtbeckAmerican Sociological Review, Vol. 19, No. 1. (Feb., 1954), pp. 23-29.Stable URL:http://links.jstor.org/sici?sici=0003-1224%28195402%2919%3A1%3C23%3ATFAATG%3E2.0.CO%3B2-Z7 The Coalition Pattern in Three Person GroupsTheodore M. MillsAmerican Sociological Review, Vol. 19, No. 6. (Dec., 1954), pp. 657-667.Stable URL:http://links.jstor.org/sici?sici=0003-1224%28195412%2919%3A6%3C657%3ATCPITP%3E2.0.CO%3B2-T

8 The Coalition Pattern in Three Person GroupsTheodore M. MillsAmerican Sociological Review, Vol. 19, No. 6. (Dec., 1954), pp. 657-667.Stable URL:http://links.jstor.org/sici?sici=0003-1224%28195412%2919%3A6%3C657%3ATCPITP%3E2.0.CO%3B2-T

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