Theories of coalition formation and the effects of reference groups

J O U R N A L OF E X P E R I M E N T A L SOCIAL PSYCHOLOGY 13, 1 6 6 - 1 8 1

(1977)

Theories of Coalition Formation and the Effects of Reference Groups J. K E I T H M U R N I G H A N , S. S. KOMORITA, AND E U G E N E SZWAJKOWSKI

University of lllinois Received April 22, 1976 The predictions of four theories of coalition behavior were compared to the results obtained from thr~e coalition games conducted under two reference group conditions. Participants were given a set to use a reference group composed either of the other players in their experimental session or of players in other groups in the same position as themselves ("similar others"). While the different games had an impact on the accuracy of the theoretical predictions, the data as a whole tended to support Bargaining theory (Komorita & Chertkoff, 1973) and the Weighted Probability model (Komorita, 1974) over Minimum Resource theory (Gamson, 1961) and Minimum Power theory (Shapley & Shubik, 1954). The results also indicated that a reference group of "similar others" led to more accurate theoretical predictions and to higher payoffs for the powerful player in each of the games, even though his demands were higher in these conditions. The use of four-person coalition games in coalition research was also discussed.

The study of coalition behavior has employed several research procedures. Gamson's (1961) "full-fledged coalition situation" specifies the general conditions that most studies have implemented. This situation consists of more than two players who are attempting to maximize their share of the rewards, with no player having dictatorial or veto power, and with no condition in which all the players can jointly maximize their payoffs. The players' task is to form coalitions that determine the division of the rewards to the coalition members. Recent research no longer separates the formation and division components. Instead, coalitions are formed through the use of written offers or proposals for the particular reward division among the proposed coalition members. Upon acceptance of such an offer by each of the proposed members, a coalition is said to have formed. In most cases, the agreement is announced and further agreements are then attempted. Thus, most studies consider a number of trials on which coalitions are formed. The first author would like to gratefully acknowledge the support he received from the Center for Advanced Study at the University of Illinois during this project. This study was also supported in part by a research grant from the National Science Foundation (SOC74-13399) to the second author. Requests for reprints should be sent to J. Keith Murnighan, Organizational Behavior Group, Department of Business Administration, University of Illinois, Urbana, IL 61801. 166 Copyright 9 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.

ISSN 0022-1031

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In almost all coalition research, regardless of the presence or absence of monetary rewards, the instructions have included a statement exhorting the players to attempt to maximize their rewards. However, even with such exhortations, the instructions may not have been sufficient to instill a completely competitive spirit in the players. Several studies (e.g., Kalisch, Milnor, Nash, & Nering, 1954; Stryker & Psathas, 1960; Trost, 1965; Vinacke, 1959) have noted a tendency for some of the participants to exhibit noncompetitive behavior. The present study, therefore, was designed (in part) to investigate the possibility that the manipulation of an individual's reference group (Kelley, 1952) determines an individual's comparison level (Thibaut & Kelley, 1959), and that this in turn may determine whether he adopts a competitive or noncompetitive orientation. An individual who holds a powerful position and whose reference group is the other people in his immediate situation will be able to maintain his "superiority" (Laing & Morrison, 1974) even though he may allow the weaker members of his group to obtain rewards that are greater than he thinks they deserve. Such behavior can be interpreted as noncompetitive. In other situations, however, an individual in a powerful position may compare himself to individuals in other groups who also hold powerful positions. In this case, the powerful person will be motivated to extract as much as he can from the payoffs, thus increasing his power relative to the leaders of other groups. A political example of this type of individual might be a prime minister, a dictator, or a president. If his reference group consists of other prime ministers, dictators, or presidents, he may attempt to increase his power relative to these other individuals at the expense of the people in his country. Presidents of industrial firms, labor unions, and chairmen of financial institutions and university departments may experience these same motivations (Marris, 1963). This behavior, then, can be interpreted as competitive. The present study was also designed to compare the accuracy of the predictions of four descriptive theories of coalition behavior, all of which assume that the players are motivated to maximize their rewards. Given this assumption, this study attempted to answer the question, "Does a reference group that induces noncompetitive motives in the players render the predictions of the theories less accurate than a reference group that induces competitive motives?" Theories o f Coalition Behavior Although many theories of coalition formation have been proposed, only those theories that predict both which coalitions are likely to form and the reward division in these coalitions were considered in this

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study. This restriction excludes some important theories, such as Caplow's (1956) and several normative theories of n-person cooperative games (cf. Luce and Raiffa, 1957; Rapoport, 1970). Four theories meet the above criterion: (1) Gamson's (1961) Minimum Resource theory; (2) an extension of Shapley and Shubik's index of"pivotal power," suggested by Gamson (1964), hereafter called Minimum Power theory; (3) Komorita and Chertkoff's (1973) Bargaining theory; and (4) Komorita's (1974) Weighted Probability model. To compare and evaluate the four theories, groups of four persons participated in one of three coalition situations (games). In each game, the resources of the participants were manipulated by assigning a certain number of votes to each person. These votes determine the combinations of players that could form a winning (majority) coalition. The three games were: (1) 9(8-3-3-3); (2) 9(8-7-1-1); and (3) 15(8-7-7-7), where the first number denotes the number of votes needed to form a winning (majority) coalition, and subsequent numbers denote the number of votes (resources) at the disposal of each player. By identifying the players by letters (A, B, C, and D), in descending order of resources, one can see that the same four minimal-winning coalitions I exist in each of the three games: AB, AC, AD, and BCD. In the following description of the four theories, the 9(8-7-1-1) game will be used to illustrate their different predictions. Minimum Resource theory. Gamson's (1961) theory is based on the assumption of a "parity norm" which specifies that rewards be divided in direct proportion to the resources of the coalition members. Thus, in the 9(8-7-1-1) game, if the 8-1 coalition should form, the parity norm specifies that the prize should be divided 8/9 for A and 1/9 for C or D, hereafter denoted 89-I1 (percentage division of the prize). Assuming that individuals are motivated to maximize their share of the reward, the theory predicts the formation of the coalition that minimizes joint resources and is just large enough to win ("the cheapest winning"). Since the 8-1 and 7-1-1 coalitions are just large enough to win (each exactly meets the quota of 9 votes), these coalitions should be the most likely to form. Minimum Power theory. Minimum Power theory is based on an index of pivotal power proposed by Shapley and Shubik (1954). The pivotal power of a player is determined by dividing the number of times a person's resources (votes) are "pivotal" (in the sense that a losing coalition is converted into a winning one by adding this person to A minimal-winningcoalition is a winning coalition such that the deletion of any one player converts it into a losing coalition. All four theories to be compared in this study predict that nonminimal-winningcoalitions will not occur. Also, in the present study the "grand coalition" of all four players was not permitted.

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the coalition) by the total number of permutations of the players. For the four persons in the 9(8-7-1-1) game, 24 permutations result in winning coalitions. Player A is pivotal in 12 cases (he has pivotal power of 1/2), and players B, C, and D are each pivotal in four cases (pivotal power of 1/6 each). If players are motivated to maximize their rewards, and if they believe that rewards should be divided in direct proportion to pivotal power (Gamson, 1964), the theory predicts that players will attempt to form a coalition that minimizes the coalition's total pivotal power. (This approach is very similar to Minimum Resource theory; the focus, however, is on pivotal power rather than resources.) Thus, the BCD coalition, which results in a total pivotal power of 1/2, is preferred to any of the twoperson coalitions, which total 2/3 pivotal power. In addition, the theory predicts that the reward division for the BCD coalition should be 33-33-33 (and 75-25 for the 8-7 and 8-1 coalitions). Bargaining theory. The basic assumption underlying Bargaining theory (Komorita and Chertkoff, 1973) is that, in a given coalition, those members who are "strong" in resources (above average) will expect and demand a share of the rewards based on the parity norm, while those who are " w e a k " (below average) will demand equality. For an iterated game over trials, the theory makes differential predictions on the initial trial and at the asymptotic level. For the initial trial, the theory predicts that the rewards allocated will be the average of those prescribed by the parity and equality norms. At the asymptote, the theory predicts that, for a given coalition, rewards will be divided in direct proportion to each member's maximum expectation in alternative coalitions. For example, if players A and C are negotiating the division of rewards in the 9(8-7-1-1) game, A's maximum expectation in alternative coalitions will be 89 (based on parity in the AD coalition) while C's maximum expectation will.be 33 (based on equality in the BCD coalition). Converting the ratio of these expectations to a base of 100% yields asymptotic predictions of 73-27 for the AC coalition. Similarly, the theory also predicts a reward division of 53-47 in the AB coalition, of 73-27 in the AD coalition, and of 33-33-33 in the BCD coalition. The theory postulates that the most likely coalition to form is the one that minimizes coalition members' temptation to defect. This temptation is represented by the quantity X(O,j - Ei~), where O~j denotes the predicted reward of individual i in coalition j; Eij denotes his maximum expectation in alternative coalitions; and the summation is over the members of coalition j. In the 9(8-7-1-1) games, the temptation to defect is minimized in the 8-1 coalitions. In the 8-7 coalition (for a 53-47 division), A will be tempted to form 8-1, while in the BCD coalition (for a 33-33-33 split), all three players will be tempted to form a coalition with A. Hence the theory predicts that the 8-1 coalitions are most likely to occur. The Weighted Probability model. The basic assumption underlying the

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Weighted Probability model (Komorita, 1974) is that, because of the logistic problem of communicating offers and counteroffers, large coalitions are more difficult to form than small ones. As the number of potential coalition members increases, the severity of the problem of achieving both reciprocity and unanimous agreement on the terms of the offer also increases. The number of potential defectors from the coalition also increases with its size; hence, a large coalition is not only more difficult to form, but may be more difficult to maintain. These hypotheses are supported by the inverse relationship between group size and the cohesiveness of a group reported by Cartwright and Zander (1968, p. 102). In contrast to Bargaining theory, the Weighted Probability model assumes that an individual's share of the prize should be a function of the number of alternative winning coalitions available to him/her, rather than the quality of these alternatives. In the 9(8-7-1-1) game, if A and B are considering an AB coalition, A has two other alternatives (AC and AD) while B has only one other alternative (BCD). The model, therefore, predicts that rewards in the AB coalition should be divided in a two-to-one ratio (i.e., 67-33). The same prediction is made for the other two-way coalitions. The model also predicts a 33-33-33 reward division in the 7-1-1 coalition because all these players have the same number of alternatives. Finally, unlike the other theories, the Weighted Probability model makes an exact probability prediction for each of the four minimal-winning coalitions. In a manner similar to the derivation of the reward division predictions, Player A should be included twice as often as the other players. Thus, the predicted probability is 2/7 for each of the two-way coalitions and 1/7 for the BCD coalition. Table 1 shows the predictions of the four theories for the three games TABLE 1 COALITIONS AND REWARD DIVISION (IN PARENTHESES) PREDICTED BY FOUR THEORIES FOR THREE GAMES

Games Theory

9 (8-3-3-3)

9 (8-7-1-1)

15 (8-7-7-7)

Minimum power

3-3-3 (33-33-33)

7-1-1 (33-33-33)

7-7-7 (33-33-33)

Minimum resource

3-3-3 (33-33-33)

7-1-1 (77-11-11) or 8-1 (89-11)

8-7 (53-47)

Bargaining

8-3 (69-31)

8-1 (73-27)

8-7 (61-38)

Weighted probability

8-3 (67-33)

8-1 (67-33) or 8-7 (67-33)

8-7 (67-33)

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used in this study. The theories can be differentiated as follows: (1) Minimum Power theory, unlike the other three theories, uniformly predicts that the BCD coalition will form; (2) Minimum Resource theory can be differentiated from the remaining two theories on the basis of coalition frequencies in the first two games, 9(8-3-3-3) and 9(8-7-1-1), and on the basis of the predicted reward division in the 15(8-7-7-7) game; and (3) the last two theories make very similar predictions but can be differentiated on the basis of the frequency of the 8-7 coalition in the 9(8-7-1-1) game and on the basis of reward division in the 15(8-7-7-7) game. The present research also extends previous research on four-person groups in coalition situations. Willis (1962) studied tetrads in two different games. Unfortunately, a strict comparison between Willis' results and those of the present study is difficult to make due to the use of different procedures. (Formation of a majority coalition in Willis' study did not guarantee that the coalition would obtain the payoff.) Of greater relevance to the present research is a study by Shears (1967). Shears studied two four-person games. One, a 5(4-2-2-1) game, has the same minimal winning coalitions (AB, AC, AD, and BCD) as those in the three games considered in the present study. (The second game studied by Shears gave one player veto power, and therefore was not a fullfledged coalition situation.) Although each of the four theories considered here makes predictions for this game, it is difficult to evaluate them since Shears did not distinguish between the 4-2 and the 4-1 coalitions in presenting her data. Nevertheless, the data do indicate that the two-person coalitions formed significantly more often than the three-person coalitions (counter to Minimum Power theory's predictions), and that the payoffs averaged approximately 65-35 in these coalitions. The present study extends Shears' research to three additional four-person games and allows not only for a test of the effects of different reference groups on the outcomes but also for a more pointed evaluation of the recent theories.

METHOD Subjects. The subjects were 168 male undergraduates, predominantly juniors, enrolled in an introductory course in organizational behavior. They participated in groups of four persons. Subjects received course credit and small monetary payoffs (dependent on their performance) for participating. Design. Two between-subjects variables were manipulated: games and reference groups. A within-groups variable, trials, was also included in an attempt to depict the bargaining process. Twenty-eight subjects (seven groups of four players) participated in each game/reference group combination. The three games used in the study were those described in Table 1. Players' reference groups were manipulated by telling them that they would receive a monetary reward for participating, and that the amount of money they received would be determined after the experiment by comparing the points they accumulated during the game to either (a) the points accumulated by other participants in the same experimental

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session or (b) the points accumulated by other players who had been in the same bargaining position in previous studies. Each four-person group (seven under each combination of game and reference group conditions) completed 12 trials. A trial consisted of the formation of a winning coalition. (Participants were not told how many trials would be completed.) The dependent variables in the study were the frequency of occurrence of the different coalitions, the payoffs the players received as a member of the winning coalition, and the demands the players made on each trial. In addition, deviations between predicted and observed payoffs for each theory were used to determine the accuracy of the theories. P r o c e d u r e . The four participants in each group were seated around a set of opaque partitions that shielded them from view of each other and the experimenter. The participants were given written as well as tape-recorded instructions. Several examples of coalition formation in everyday life were initially given; subjects were then asked to assume they were legislative leaders attempting to form a majority coalition. They were informed that there would be many trials of the game (unspecified), and on each trial, the winning coalition would be allowed to divide a prize of 100 points among its members. They were instructed to do as well as they could (i.e., maximize points) and that the points they accumulated would be converted to money at the end of the experiment. The instructions diverged at this point for the two reference group conditions. Half of the groups in each game condition received the following instructions, designed to elicit a Present Group reference: " . . . The more points you win, the more money you will receive. A conversion scale will be used to determine how much money you receive, but it will not be based on one cent for one point." The remaining subjects, in the Similar Others reference condition, were told that their payoffs depended on their performance relative to the performance of players in their position in other groups: 9 . . The more points you win, the more money you will receive. However, the conversion scale to determine how much money you receive will not be the same for each person; that is, the number of points you win will be compared to the performance of other subjects in the s a m e p o s i t i o n as y o u . Thus, the conversion from points to money will depend on your position in the game. Substitution of this single passage was the only difference in the two sets of instructions. After the initial orientation to the task, subjects were instructed that they would make offers on each trial by means of written "offer slips." These offer slips required each subject to "address" his offers (indicating to whom they wished to send their offers) and also required a proposal regarding the division of rewards for the coalition members. For example, if Person X wished to form the X-Y-Z coalition, he addressed offers to both Persons Y and Z and specified a division of the rewards (e.g., 50-25-25 for Persons X, Y, and Z, respectively). For two-person coalitions, a player was required to send a single offer slip. For three-person coalitions, a player was required to send two offer slips. In the latter case, players were instructed that the two offers must be identical with regard to the proposed division of rewards; for example, they could not send an offer to one person to form one coalition and a second, different offer to another person to form another coalition. This procedure allowed both two- and three-person coalition to form in a single step. Although three-person coalitions are more difficult to form, the difficulty is not inherent in the procedure. After the players had completed their offers, the experimenter collected, examined, and distributed them to the proper persons. After receiving an offer, each person could accept or reject it by checking a space marked " A c c e p t " or "Reject" at the bottom of the offer slips9 If a person received more than one offer, the person was allowed to accept only one of them, unless two offers proposed the same coalition. Hence, each person could only accept offers to form a single coalition on each trial. Furthermore, in determining a winning coalition, any player's proposal, if

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accepted, was considered to have priority over any offer he might accept, thus committing him to his own offer. After the players had either accepted or rejected their offer(s), the experimenter collected the offer slips and announced the winning coafition, if one had formed. Subjects were informed that a coalition would be declared the winning one if all the proposed coalition partners accepted the offer. If no coalition formed because at least one person rejected each of the proposed coafitions, the experimenter announced that a coalition had not formed, and the procedure was repeated until a coalition formed. 2 After each winning coalition, the experimenter notified the coalition members in writing of the number of points they had obtained for that trial. In order to familiarize the players with the procedure, a practice trial was conducted before the start of the session. Immediately after the practice trial, the players were randomly reassigned to a position designated by letter (A, B, C, and D). No verbal communication was permitted thereafter; hence, the players could not identify each other once the experiment had begun.

RESULTS F o r t h e six e x p e r i m e n t a l c o n d i t i o n s , T a b l e 2 s h o w s t h e m e a n p r o p o r tion o f t i m e s t h a t e a c h o f t h e f o u r m i n i m a l - w i n n i n g c o a l i t i o n s o c c u r r e d , a n d a l s o t h e m e a n r e w a r d d i v i s i o n in t h e s e c o a l i t i o n s . N o n m i n i m a l winning coalitions (ABC, ABD, and ACD) were not predicted by any of the theories and occured only 4% of the time; hence, they were excluded from the analyses. T o d e t e r m i n e if t h e i n d e p e n d e n t v a r i a b l e s a f f e c t e d t h e f r e q u e n c i e s o f forming the four coalitions, a multivariate analysis of variance was cond u c t e d u s i n g g a m e s , r e f e r e n c e g r o u p s , a n d t h r e e b l o c k s o f f o u r trials as i n d e p e n d e n t v a r i a b l e s , a n d t h e f r e q u e n c i e s o f t h e f o u r m i n i m a l w i n n i n g c o a l i t i o n s in e a c h trial b l o c k as t h e d e p e n d e n t v a r i a b l e s . E a c h o f t h e c o a l i t i o n f r e q u e n c i e s w a s t r a n s f o r m e d to a r c s i n v a l u e s p r i o r to t h e a n a l y s i s . N e i t h e r t h e m u l t i v a r i a t e n o r t h e u n i v a r i a t e F - r a t i o w a s significant f o r a n y o f t h e m a i n e f f e c t s o r i n t e r a c t i o n s (p > . 10). S i n c e t h e i n d e p e n d e n t v a r i a b l e s h a d n o significant e f f e c t on t h e freq u e n c i e s o f f o r m i n g t h e s e c o a l i t i o n s , t h e d a t a in t h e s u b s e q u e n t a n a l y s i s were pooled over the two independent variables. The frequencies of the f o u r c o a l i t i o n s (for e a c h g r o u p in e a c h c o n d i t i o n ) w e r e c o n v e r t e d t o ranks and Friedman's test (analysis of variance by ranks) was used to test for d i f f e r e n c e s in c o a l i t i o n f r e q u e n c i e s . T h e r e s u l t s o f this t e s t i n d i c a t e d t h a t t h e r e w e r e significant d i f f e r e n c e s in t h e f r e q u e n c i e s o f t h e f o u r c o a l i t i o n s : X2(3) = 9.31, p < .03. T o d e t e r m i n e if this r e s u l t c o u l d b e a t t r i b u t e d to t h e l o w f r e q u e n c y o f the BCD coalition, the Friedman test was again used, but with the excluz In some situations, two or three coalitions could form on the same trial. For instance, if A sent an offer to C, D sent an offer to A, and both offers were accepted, AC would be declared the winning coalition because A was committed to his offer (invalidating his acceptance). If three coalitions formed in this manner, with each being invalidated by another, the players were informed of the situation and the trial was rerun.

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MEAN PROPORTION OF MINIMAL-WINNING COALITIONS AND MEAN REWARD DIVISION (IN PARENTHESES) IN THESE COALITIONS (POOLED OVER 12 TRIALS AYD 7 GROUPS IN EACH CONDITION)a

Games (8-3-3-3)

(8-7-1-1)

(8-7-7-7)

Combined

Reference group conditions

Minimal-winning coalitions AB

AC

AD

BCD

.20 (80-20)

.25 (69-31)

.38 (73-27)

.12 (34-34-32)

.25 (59-41)

.29 (62-38)

.20 (63-37)

.19 (34-33-33)

Similar others Present group

.21 (57-43)

.19 (65-35)

.38 (66-34)

.20 (36-32-31)

.29 (52-48)

.19 (82-18)

.36 (58-42)

.14 (39-31-30)

Similar others Present group

.29 (63-37)

.29 (64-36)

.25 (64-36)

.16 (34-33-34)

.27 (56-44)

.24 (57-43)

.33 (55-45)

.21 (33-35-33)

.25 (60-40)

.24 (66-34)

.31 (63-37)

.17 (35-33-32)

Similar others Present group

a Proportions across each row do not sum to unity because nonminimal-winning coalitions, which occurred in approximately 4% of the games, were excluded.

sion of the BCD coalition. This test is analogous to a planned comparison in the analysis of variance in that Minimum Power theory predicts that this coalition should be most frequent in all three games, while the Weighted Probability model predicts that the two-person coalitions should be most frequent. Based on the frequency ranks of the three twoperson coalitions, the Friedman test was not significant at the 5% level, X2(2) = 2.37. This second analysis indicates that the frequency of the BCD coalition made a substantial contribution to the previous significant result, and suggests that the BCD coalition occurred less frequently than the two-person coalitions (see Table 2). These results strongly support the predictions of the Weighted Probability model. Indeed, the model's predictions, of .28 probabilities for the two-person coalitions and a . 14 probability for the BCD coalition, are very accurate. However, the results are completely counter to the predictions of Minimum Power theory. They are also inconsistent with the predictions of Minimum Resource theory, which predicts the BCD coalition in both the 9(8-3-3-3) game and the 9(8-7-1-1) game. Bargaining theory, on the other hand, is partially supported because it predicts the formation of a two-person coalition in each game. However, it

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TABLE 3 MEAN OUTCOME OF STRONGEST PLAYER (A) WHEN INCLUDED IN THE WINNING COALITIONa Games Reference group Similar others Present group Mean b

9 (8-3-3-3)

9 (8-7-1-1)

15 (8-7-7-7)

Mean b

72.7 (70) 61.1 (62) 66.9

62.1 (66) 61.5 (71) 61.8

62.9 (70) 55.9 (65) 59.4

65.9 59.5

Outcomes in AXX coalitions are not included. b Means in each case are unweighted.

does not predict the formation of the AB coalition in the 9(8-7-1-1) game. Reward-Division Data In predicting the division of rewards, the most sensitive tests are derived from the mean outcome of the "strong" person (A) in each coalition. The most frequent coalitions were the two-person coalitions, and all of these coalitions included Player A. 3 Hence, a 3 (games) x 2 (reference groups) • 3 (trial blocks) analysis of variance was conducted on A's average outcome from winning coalitions within each trial block. The results showed significant main effects for game, F(2,36)= 3.73, p < .05, and for reference groups, F(1,36) = 7.85, p < .01. The effects for trial blocks was marginally significant, F(2,72) = 2.58, p < . 10, indicating increasing outcomes over blocks. The interaction between games and reference groups was not significant, F(2,36) -- 1.92, p < .20, even though the means across the six conditions shown in Table 3 might lead one to expect significance. Player A's outcomes are higher in the Similar Others condition than in the Present Group condition, as hypothesized. With regard to the main effect of games, post hoc tests (Newman-Keuls) indicated that A's outcomes in the 9(8-3-3-3) game were significantly greater than in the 15(8-7-7-7) game and were marginally greater, p < .10, than in the 9(8-7-1-1) game. This result is probably due in large part to A's high payoffs in the Similar Others, 9(8-3-3-3) condition. To evaluate the relative accuracy of the four theories, 1% confidence intervals were determined for Player A's mean outcome. For Minimum Resource theory, the predictions of 89 for the 8-1 coalition in the a Although all the theories do not predict that A will be included in the winning coalitions, almost all the predictions for the BCD coalition are the same, i.e., 33-33-33 reward divisions. Minimum Resource theory, however, predicts a 77-I 1-11 reward division for BCD in the 9(8-7-1-1) game.

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9(8-7-1-1) game, and of 53 for the 8-7 coalitions in the 15(8-7-7-7) game, were outside the 1% confidence band. Since Minimum Resource theory received little support in predicting coalition frequency, these results suggest that the theory is quite inadequate for the games used in this study. Similarly, Minimum Power theory is also inadequate. Though it predicts that the BCD coalition should occur in all three games, it predicts a 75-25 split in the two-person coalitions. This prediction is outside the 1% confidence band in all three games. The Weighted Probability model predicts that Player A's outcomes should be 66.7 in all three games. This prediction is, of course, inconsistent with the significant main effect of games reported earlier. Also, this prediction is outside the 1% confidence band for the 9(8-7-1-1) and 9(8-7-7-7) games. Bargaining theory seems to be the most accurate in predicting reward division. Its predictions are within the 1% confidence band for the 9(8-3-3-3) game and the 9(8-7-7-7) game. However, its prediction of 73 for 8-1 coalitions in the 9(8-7-1-1) game was outside the confidence band. An additional set of analyses was conducted to determine the effects of reference groups on the accuracy of these theoretical predictions. Since a reference group of Similar Others presumably leads to competitive motives, and since all the theories assume competitive motives, the theories should be more accurate in the Similar Others condition. Deviation scores between Player A's payoffs and each of the theory's predictions were the dependent variables in four separate 3 (game) by 2 (reference groups) by 3 (trial blocks) analyses of variance, one for each theory. Because Minimum Power theory and the Weighted Probability model make constant reward division predictions for the three games, the results for these two theories are identical to the results of the analysis of Player A's outcomes. The main effect for reference groups was significant for each of the theories, indicating that in each case the discrepancies between predicted and observed payoffs for the strong player were significantly smaller (p < .05) in the Similar Others condition, as predicted. The analysis also revealed a main effect of games for Minimum Resource theory, F(2,36)= 28.01, p < .001. This effect was due in large part to the large discrepancy between the predictions and the data in the 9(8-7-1-1) game. In summary, for the three games used in this study, the results for reward division indicate that all four theories have weaknesses in predicting the mean outcome of Player A. Minimum Power theory is clearly inadequate, and Minimum Resource theory also appears quite deficient. The Weighted Probability model's predictions for coalition frequencies are excellent, but its predictions for reward division are somewhat less accurate. Bargaining theory seems to be the most accurate predictor of

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reward divisions. However, it was deficient in predicting the reward division for the 8-1 coalition in the 9(8-7-1-1) game. In addition, none of the theories was particularly accurate in the "present group" conditions.

Analysis of Offers (Demands) An analysis of the players' demands was conducted to examine the bargaining processes mediating the outcomes of coalition formation. The players' position was included as one of the factors (A vs. B vs. C and D combined) in a 3 x 2 x 3 x 3 (games by reference groups by players by trial blocks) analysis of variance. The main effect of players' position was significant, F(2,72) = 147.95, as was its interaction with games, F(4,72) = 5.87, with reference groups, F(2,73) = 12.28, and with trial blocks, F(4,144) = 9.29, in all cases,p < .001. The demands of Player A in all conditions were significantly higher than the other players. However, the player by game interaction, depicted in Table 4, indicates that A demanded relatively less in the 9(8-7-7-7) game, when he was faced by weaker players who controlled seven votes. Correspondingly, the demands made by B, C, and D were less when they had either one or three votes (32.3, 33.3, and 31.4) than when they had seven votes, (40.8, 38.7, and 38.9). Post hoc tests of the players by reference group interaction indicated that the demands of Players B, C, and D did not differ significantly in the two reference group conditions; Player A, however, demanded significantly more in the Similar Others condition (X = 66.7) than in the Present Group condition (X = 54.8). Post hoc tests of the players by trial block interaction indicated that Player A's demands increased significantly from the first trial block (51.6) to the second (61.3), but not significantly from the second to the third (63.6). Player B's demands dropped significantly from the

TABLE 4 MEAN DEMANDS OF FOUR PLAYERS IN THREE GAMES (POOLED OVER REFERENCE GROUPS AND TRIALS) Games Players

9 (8-3-3-3)

9 (8-7-1-1)

15 (8-7-7-7)

A B C and D

63.9a 32.3d 33.9ca

62.2a 40.8c 31.4o

56.2b 38.7~a 38.9ca

N o t e . Values with a common subscript denote that they do not differ significantly, based on the Newman-Keuls test.

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first trial block (41.2) to the second (35.5), but not to the third (35.0). This change was probably due to a large drop in B's demands in the 9(8-7-1-1) game (from 47.0 to 36.6). Players C and D showed nonsignificant drops in their demands over the three trial blocks (from 35.9 to 34.9 to 33.5), Thus, while A's demands were increasing over trials, the weaker players' demands were decreasing over trials. These results further support Siegel and Fouraker's (1960) Level of Aspiration model. While most of the research testing this model has been in the two-person bargaining literature (e.g., Yukl, 1974a, b), the present study extends its support to n-person games.

DISCUSSION The comparison of the four theories indicates that the predictions of Bargaining theory and the Weighted Probability model were more accurate than the predictions of the other two theories. In all three games, Minimum Power theory could not account for the preponderance of two-person coalitions (or their reward divisions), while Minimum Resource theory could not account for either the observed coalition frequencies or the rewards received by the strong player. This lack of support indicates that the basic assumption of the two theories--that the players' expected rewards are directly proportional to their resources or pivotal p o w e r - - m a y be invalid. This assumption shows a strong resemblance to Adams' equity theory (1965); the failure of proportionality suggests that the results of bargaining in coalition situations may not be amenable to the predictions of a "simple" (single variable) equity-like principle. The two more complicated theories considered here, Bargaining theory and the Weighted Probability model, fared considerably better. The Weighted Probability model's focus on the size of different minimalwinning coalitions may have been a key determinant in a player's choice of coalitions. The support for Bargaining theory, especially in predicting the reward division (with one noted exception), suggests that a player may indeed use the quality of his alternatives as a threat during negotiations. Although the Bargaining theory and the Weighted Probability model were clearly superior to Minimum Resource theory and Minimum Power theory, their support was not unequivocal. The Weighted Probability model predicted coalition frequencies extremely accurately but its predictions of reward divisions were relatively inaccurate. Thus, these results suggest that the model's assumption that rewards will be divided in proportion to a player's predicted probability of inclusion in the winning coalition may be invalid. The present data indicate that payoffs are determined, in part, by the magnitude of a player's resources; consequently, in order to increase the accuracy of its reward division predictions, the Weighted Probability

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model may have to incorporate assumptions pertaining to the resource distribution of the players. The Bargaining theory's predictions were consistently supported with two exceptions. As predicted, two-person coalitions were more frequent than the BCD coalition; Player A's payoffs increased over trials; and reward division predictions were correctly predicted in two of the three games. The two prediction errors were in the 9(8-7-1-1) game: the theory cannot account for the large frequency of the AB (8-7) coalition, and it failed to accurately predict Player A's payoff in this game. There are two plausible explanations for these discrepancies. First, the bargaining in this game may not have reached its asymptotic level by the 12th trial. While the other games all distributed the same number of votes to Players B, C, and D, this game gave the weak players different amounts of resources. This may have led to greater instability within the overall bargaining process, as suggested by the large drop in Player B's demands in the early trials of this game. Alternatively, the theory may have failed here because of the large disparity in resources between the players in this game. In particular, the opportunity for A to form either of two 8-1 coalitions leads the theory to make quite extreme predictions (due to its use of the parity norm as one of the determinants of bargaining threats). In deriving its 73-27 prediction, the theory assumes that Hayer A will use his alternative maximum of 89 as a threat during negotiation. Such a threat may have been too extreme to be credible. While more moderately profitable alternatives such as 73 in the 9(8-3-3-3) game and 53 in the 15(8-7-7-7) game may be within the bounds of "reasonability," more excessive demands could exceed these bounds. Future research may be able to specify where the criterion for credible threats lies, indicating the specific areas where the parity norm mig~ be modified. Among the most intriguing results o r t h i s study were the significant effects of reference groups. The data Showed that the strong player demanded more when his reference group was other similar players, and that he succeeded in reaping better outcomes and even more frequent inclusion in the winning coalition in this condition. It is understandable that he would attempt to increase his payoffs by increasing his demands in this situation, but one would expect that these increased demands would have resulted in less frequent inclusion in winning coalitions. One explanation of this finding focuses on the subtle interaction between a player's reference group and his power. For the strong player, the Present Group condition led him to compare himself with weaker players, while the Similar Others condition led him to compare himself with other powerful players. The results indicated that when a powerful player's reference group was other powerful players, he was more demanding and may have intimidated an opponent who held a weak position.

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For the weak player, however, the effects of reference groups were not so salient. Indeed, the weak players evidenced no significant change in demands when their reference groups differed. One possible explanation for this result is that the lack of power inherent in a weak position may be such a strong determinant of a player's responses that different reference groups may be relatively ineffectual. As regards the impact of reference groups on the theoretical assumption of competitive motivations, the present study indicates that a reference group of similar others resulted in significantly more accurate predictions for each of the theories. Thus, given the relative stability of the weak players' behavior in the two reference group conditions, the strong player's low demands (and outcomes) in the Present Group conditions can be labeled noncompetitive behavior. 4 This indicates that in testing each of the models, care must be taken to insure that the players are competitively motivated. The consideration of reference groups may be very important in this effort. The data collected here were much richer than the data reported in earlier coalition studies (cf., Stryker, 1972) based on three-person groups. In the present study, one player not only had more apparent power than the other players (i.e., he had more resources than anyone else), he also had more " r e a l " power than anyone else. These findings, therefore, have greater relevance for the study of power than previous research. In addition, the present study increased the salience of the power dimension by manipulating the players' reference groups. Although the generalizability of the present findings to other situations (i.e., industrial firms, university departments, etc.) can be questioned, the use of four players in a coalition situation in order to vary both real and apparent power leads to a more realistic representation of "real world" power struggles. Studying only three players restricts full-fledged coalition situations to instances where the only power differences are apparent and not real (Kelley and Arrowood, 1960). In the present study, "strength" did not become "weakness" as the three-person studies have concluded. With four or more players, the use of power becomes crucial. The weak players can usurp the strong player's power, but he can also retain it if his strategies are well-founded. With this in mind, studies of coalition behavior may do well to expand their horizons from the study of triads to groups of larger sizes. 4 The noncompetitive behavior observed in the present study may be analogous to the anticompetitive behavior noted by Gamson (1964). While anticompetitiveness has been associated with the bargaining behavior of females, this research suggests future investigations which might explore the possibility that females used the present group as their reference (resulting in anti- or noncompetitive behavior) and that males in the s a m e study used similar others as their reference (resulting in competitive behavior).

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REFERENCES Adams, J. S. Inequity in social exchange. In L. Berkowitz (Ed.), Advances in Experimental Social Psychology. Vol. 2. New York: Academic Press, 1965. pp. 265-299. Caplow, T. A theory of coalitions in the triad. American Sociological Review, 1956, 21, 271-280. Cartwright, D., & Zander, A. Group Dynamics. New York: Harper and Row, 1968. Gamson, W. A theory of coalition formation. American Sociological Review, 1961, 26, 373-382. Gamson, W. Experimental studies of coalition formation. In L. Berkowitz (Ed.), Advances in Experimental Social Psychology. Vol. 1. New York: Academic Press, 1964. Kalisch, G., Milnor, J. W., Nash, J., & Nering E. D. Some experimental n-person games. In Thrall, R. M., Coombs, C. H., & Davis, R. L. (Eds.) Decision Processes. New York: John Wiley, 1954. Kelley, H. H. Two functions of reference groups. In G. E. Swanson, T. M. Newcomb, and E. L. Hartley (Eds.) Readings in Social Psychology. New York: Holt, Rinehart, and Winston, 1952. Kelley, H. H., & Arrowood, A. J. Coalitions in the triad: Critique and experiment. Sociometry, 1960, 23, 231-244. Komorita, S. S. A weighted probability model of coalition behavior. Psychological Review, 1974, 81, 242-256. Komorita, S. S., & Chertkoff, J. M. A bargaining theory of coalition formation. Psychological Review, 1973, 80, 149-162. Laing, J. D., & Morrison, R. J. Sequential games of status. Behavioral Science, 1974, 19, 177-196. Marris, R. A model of "managerial enterprise." Quarterly Journal of Economics, 1963, 77, 185-209. Luce, R. D., & Raiffa, H. Games and Decisions. New York: John Wiley, 1957. Rapoport, A. N-Person Game Theory. Ann Arbor: University of Michigan Press, 1970. Shapley, L. & Shubik, M. A method for evaluating the distribution of power in a committee system. The American Political Science Review, 1954, 48, 787-792. Shears, L. M. Patterns of coalition formation in two games played by male tetrads. Behavioral Science, 1967, 12, 130-137. Siegel, S., & Fouraker, L. E. Bargaining and Group Decision Making. New York: McGrawHill, 1960. Stryker, S. Coalition formation. In C. G. McClintock (Ed.) Experimental Social Psychology. New York: Macmillan Press, 1972. Stryker, S., & Psathas, G. Research on coalitions in the triad: Findings, problems, and strategy. Sociometry, 1960, 23, 217-230. Thibaut, J., & Kelley, H. H. The Social Psychology of Small Groups. New York: John Wiley, 1959. Trost, J. Coalitions in triads. Acta Psychologica, 1965, 8, 226-243. Vinacke, W. E. Sex roles in a three-person game. Sociometry, 1959, 22, 343-360. Willis, R. H. Coalitions in the tetrad. Sociometry, 1962, 25, 358-376. Yukl, G. Effects of situational variables and opponent concessions on a bargainer's perception, aspirations, and concessions. Journal of Personality and Social Psychology, 1974, 29, 227-236. (a) Yukl, G. Effects of the opponent's initial offer, concession magnitude, and concession frequency on bargaining behavior. Journal of Personality and Social Psychology, 1974, 30, 323-335. (b)