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Coalition formation in characteristic function games: Competitive tests of three theories
JOURNAL
OF
Coalition
EXPERIMENTAL
SOCIAL
Formation Competitive
PSYCHOLOGY
16, 61-76
in Characteristic Tests of Three
(i980)
Function Theories
Games:
CHARLES E. MILLER Northern
Illinois
University
Received August 16, 1977 Two experiments tested the bargaining, minimum resource, and minimum power theories of coalition formation in situations involving different payoffs for some of the winning coalitions (characteristic function games). In the first experiment, a triadic resource distribution was employed and payoffs for the coalitions were specified in such a way that each of the three theories predicted the formation of a different coalition. The coalition predicted by minimum power theory formed the most frequently, and the mean divisions of the payoffs among coalition members were also closest to the predictions of minimum power theory. Ilowever, the most frequent coalition was not only the one predicted by minimum power theory, it was also the one having the largest payoff per member. Therefore. a second experiment was conducted, which empIoyed a tetradic resource distribution and specified the coalition payoffs in such a way that (a) each of the theories predicted the formation of a different coalition and (b) none of the predicted coalitions was the one with the largest payoff per member. Although the mean payoff divisions in the coalitions in this experiment were closest to those predicted by bargaining theory, the coalition that formed most frequently was not one of those predicted by any of the theories. Rather, it was the one having the largest payoff per member. None of the three theories is able to account adequately for the results of both experiments. The difficulties that the theories have in dealing with coalition formation in situations in which there are different payoffs for winning are discussed.
The formation of coalitions is an important and pervasive aspect of social interactions. A coalition may be defined as the joint use of resources to determine the outcome of a decision in a mixed-motive situation involving more than two individuals (Gamson, 1964). For experimental mu-poses, resources are ordinarily represented as weights controlled by the individuals, such that some specified quota of these weights is needed in order to win a payoff. When no individual has enough weight to This research was supported by a grant from the Council of Academic Deans at Northern Illinois University. Requests for reprints should be sent to Charles E. Miller, Department of Psychology, Northern Illinois University, DeKalb, IL 60115. 61
~22-~~31/g0/01~6~-~6$02.50/0 Copyright @ 1980 by Academic Press, Inc. Ail right, of reproduction in any form reserved.
62
CHARLES
E. MILLER
win the payoff by himself, it is necessary for a coalition to form. The members of the winning coalition can divide the payoff among themselves in whatever way they want. Theories of coalition formation are faced with two fundamental questions: (a) Which one of the possible winning coalitions will form? And (b) how will the members of a winning coalition divide the payoff? Although a number of coalition theories have been proposed, only those theories that undertake to answer both of these questions are considered in the present paper. This restriction excludes several solutions from the theory of n-person games, such as the kernel (Davis & Maschler, 1965) and the Aumann and Maschler (1964) bargaining set, because they do not generally predict which coalition will form. It also excludes Caplow’s (1956) theory of coalitions in triads because the theory does not predict how coalition members will divide the payoff for winning. Most experimental studies of coalition formation have been concerned with situations in which the payoffs are the same for every winning coalition. Yet it seems likely that in the “real world,” the payoffs that different coalitions can obtain are very often not constant. For example, several small manufacturing firms may form a coalition to obtain a contract from a large buyer, but the size of the contract may depend upon which firms enter into the coalition. The present paper considers only those theories that can be applied to situations in which there are different payoffs for some of the winning coalitions. This restriction excludes Komorita’s (1974) weighted probability theory. There are three theories which are not excluded by either of the above criteria: minimum resource theory (Gamson, 1961a, 1961b), minimum power theory (Gamson, 1961a, 1964; Shapley, 1953; Shapley & Shubik, 1954), and bargaining theory (Komorita & Chertkoff, 1973). This paper presents two experiments that test these theories against one another in situations where they make competing predictions. Before turning to the particulars of the experiments, however, the assumptions of each of the theories need to be taken up. All three of the theories are alike in assuming that each individual is motivated to maximize his own payoff, but the theories differ in their assumptions about how an individual decides which of the coalitions will allow him to do this. Minimum resource theory assumes that the payoff for a coalition will be divided according to a norm of equity, or parity (Gamson, 1961a), which prescribes that the payoff be divided in proportion to the resources that the members contribute to the coalition. In situations where the payoff is the same no matter which coalition wins, an individual gets the most by maximizing his share (proportion) of the payoff for the coalition. Since the individual’s resources are the same regardless of which coalition he joins, the smaller the combined resources of the coalition, the larger will be his share. The theory therefore predicts that
COALITION
FORMATION
63
the coalition which will form is the one having the minimum amount of combined resources needed to win. It is this coalition, out of all the possible ones, which maximizes the payoffs to its members. Minimum resource theory also makes predictions for situations 111 which some of the winning coalitions receive different payoffs (Gamson, 196la, p. 376). In these situations the theory does not necessarily predict the formation of the winning coalition with the minimum amount of resources, Here, an individual’s payoff from joining a coalition is equal to the proportion of resources he contributes to the coalition, multiplied by the coalition payoff. A small share of the payoff in a coalition in which the payofT is large may be better than a large share of the payoff in a coalition in which the payoff is small. Minimum power theory assumes that the members of a coalition will divide the payoff in proportion to their Shnpley values (Shapley, 19533, rather tban in proportion to their resources (see Gamson, 1961a, p* 382). Individuals’ Shapley values are considered an indication of their “‘power” in a situation. (When the payoffs in a situation are equal for all e winning coalitions, the individuals’ proportional Shapley values are called their pivotal power; Shapley & Shubik. 1954.) A person’s Shapley value is found by examining those possible winning coalitions of which he is a member and determining what would happen if he were to withdraw from them. An individual has a large Shapley value if, by withdrawing, he can change numerous possible winning coalitions with high payoffs (values) into losing coalitions, or into winning coalitions with much lower payoffs. The formula for calculating Person i’s Shapley value is: (s - l)! (n - s)! si=c
n,
~4s) - es - ($1,
SEN
where n is the total number of persons in the coalition situation, s is the number of persons in a given coalition (subset), V(S) is the payoff (value) for a given coalition, v(S - (i}) is the payoff for the coalition if Person i withdraws from the coalition, and the summation ranges over a!1 the coalitions (subsets) S of which Person i can be a member ( pp. 108-109). Bargaining theory assumes that the division of tbe payo coalition will be a compromise between two conflicting norms. member who is strong in resources will argue for a division of the pa according to the equity norm, while a member who is weak in resources will advocate that the payoff be divided according to a norm of equality. It is assumed that the coalition members will adopt a “split&e-difference” principle (Schelling, 1960) regarding these two norms, with each member’s share of the payoff lying halfway between what it would be if the payoff were divided in proportion to the resources of the members and
64
CHARLES
E. MILLER
what it would be if the payoff were divided equally among the members. An individual’s payoff in a coalition is the product of the total payoff for the coalition and the individual’s share of the payoff. A coalition is predicted to form if it maximizes the payoffs to all its members when the payoff is divided in this manner. l For most situations of interest, there will be one such coalition. Attempts to test the minimum resource, minimum power, and bargaining theories against each other (e.g., Komorita & Moore, 1976; Michener, Fleishman, & Vaske, 1976; Murnighan, Komorita, & Szwajkowski, 1977) have been limited exclusively to situations in which all of the winning coalitions receive the same payoff. There are at least two ways in which this limitation is somewhat unfortunate: First, as already mentioned, in many real-world situations there are likely to be differences in the payoffs which various coalitions can acquire. Second, it turns out that, when the payoff for winning is constant, the theories generally make competing predictions with regard to the question of how the payoff will be divided but nof with regard to the question of which coalition will form. For example, much of the early research on coalition formation involved triads (e.g., Kelley & Arrowood, 1960; Vinacke & Arkoff, 1957), and as long as every winning coalition receives an equal payoff, there are no resource distributions in triads for which each of the theories predicts the formation of a different coalition. Recent coalition research has dealt increasingly with tetrads, but even in tetrads there are no resource distributions for which the formation predictions of each theory differ from those of the other two (see Michener et al., 1976). It appears not generally to have been recognized that if the limitation of constant payoffs is dropped, it then becomes possible to find situations, both in triads and in tetrads, for which the minimum resource, minimum power, and bargaining theories make competing predictions not only with regard to the division of payoffs, but also with regard to the formation of coalitions. Such situations were employed in the two experiments that are reported here. EXPERIMENT
1
Method Resources. The first experiment involved triads in which the resource distribution was D(.5:4-3-2), where the first number denotes the quota of 1 This prediction applies only to initial instances of coalition formation. Bargaining theory also provides asymptotic predictions for repeated instances of coalition formation, but both of the experiments in this paper were intended to test only the predictions for initial instances. However, it is possible to argue that the asymptotic predictions of the theory apply to initial instances of coalition formation, to the extent that the bargaining which leads to initial coalitions involves a number of rounds of negotiations. As a check against this possibility, asymptotic predictions were derived for the situations used in both of the present experiments. On the whole, these predictions proved to be less accurate than the predictions for initial instances of coalition formation, and they are not discussed here.
COALITION
FORMATION
65
resources required to win, and the subsequent numbers denote the resources of the individuals. This resource distribution was selected as a particularly interesting one to study because there has been more research conducted on it, using constant payoffs for winning, than on any other triadic distribution. Payoffs. With the D(5:4-3-2) distribution of resources, any two of the triad members can get together and form a winning coalition. When the payoffs are the same for all the winning coalitions, minimum resource theory and bargaining theory predict that the coalition between 3 and 2, denoted C(3-2), is the most likely to form, while minimum power theory predicts that all of the two-person coalitions are equally likely. When there are differences in the payoffs for the winning coalitions, however, it is possible for each of the theories to predict the formation of a different coalition, depending upon the exact values of the payoffs. In the present experiment, subjects played a coalition game in which the payoffs for the coalitions were specified in such a way that the three theories would make competing predictions. Coalition games in which different winning coalitions receive different payoffs are commonly described in terms of their characteristic functions. The characteristic function of a game specifies a value, or payoff, for each possible coalition, including one-person coalitions. The characteristic function for the game used in the present experiment was: v(4) = v(3) = u(2) = 0; ~(4-3) = 240; ~(4-2) = 230; ~(3-2) = 200; ~(4-3-2) = 240, where v(4-2), for example, indicates the payoff to C(4-2). The upper part of Table 1 shows, for these payoffs, the predictions of the theories regarding which coalitions will form and how payoffs in the coalitions will be divided. As can be seen, a different coalition is predicted by each theory. Subjects. Subjects were 81 male students enrolled in an introductory psychology course. Participation in the experiment allowed the subjects to earn extra credit toward their course grades. Procedure. Subjects were scheduled to take part in the experiment in three-person groups, three or four groups at a time. The experiment was presented to subjects as a study of coalition formation in legislative bodies (cf. Gamson, 1961b). Each subject was asked to play the role of a Eegislator in a coalition game. Before the game began, each subject drew, at random, the letter A, B, or C. The letter drawn determined the number of votes that the subject controlled for the game. A conrrolled 40 votes; B controlled 30 votes; and C controlled 20 votes. (That is, the votes were in the ratio 4:3:2.)
66
CHARLES
E. MILLER
TABLE PREDICTED Theory
AND
ACTUAL
C(4-3)
COALITIONS
C(4-2)
1 AND
PAYOFF
C(3-2)
DIVISIONS
Predicted coalitions
Predicted division of payoffs Minimum resource Minimum power Bargaining
137-103 131-109 129-111
153-77 129-101
134-96
120-80 103-97 110-90
C(3-2) C(4-3) C(4-2)
Actual division of payoffs 128-112
125-10.5
100-100
Actual coalition frequencies 15
11
1
Note. C(4-3-2), not shown here, did not occur. The value of C(4-3-2) was 240 points. The values of C(4-3), C(4-2), and C(3-2) were 240, 230, and 210 points, respectively. The Shapley values for 4, 3, and 2 were 91.7, 76.7, and 71.7, respectively.
The object of the game was for some of the legislators to form a coalition that controlled a majority of the 90 votes. Any legislators who formed such a coalition won a payoff, but the amount of the payoff depended upon which coalition formed, as explained above. A coalition could be formed only if the subjects making up the coalition agreed on how to divide the payoff for the coalition. The game consisted of a number of trials on which the subjects made offers to one another by means of written offer slips. On these offer slips, subjects indicated with whom they wished to form a coalition and how they proposed to divide. the payoff for the coalition. Subjects were permitted to send an offer for only one coalition on any given trial. After all of the subjects had completed their offer slips, the experimenter collected and examined the slips and distributed them to the appropriate legislators. When a subject received an offer, he had to accept or reject it by writing “Accept” or “Reject” on the offer slip. If a subject received more than one offer, he could accept only one of them. After the subjects had accepted or rejected the offers that they received, the offer slips were collected and the winning coalition was announced, if one had formed. A coalition was considered to have formed if one and only one coalition was accepted. For example, if A sent an offer to B and B accepted it, and B sent an offer to C and C accepted it, neither the A-B nor the B-C coalition was considered to have formed, and subjects went on to the next trial. Trials continued until some coalition formed, thus ending the game. Each subject was told to try to win as many points for himself as he possibly could and to do this without regard to how much or how little any
COALITION
FORMATION
67
of the other subjects might win. In each experimental session, there were three monetary prizes at stake. The subject who won the most points in position A was awarded a prize of $3.00, as were the subjects who won the most points in positions B and C. The subjects were not informed in advance about the amount of the prizes, nor were they informed of the basis on which the prizes would be awarded. Each subject was simply told that if he wanted to win one of the monetary prizes, he should attempt to win as many points as possible. To assist the subjects in understanding the procedure of the experiment, several examples were provided of how the offer slips might be filled out. Any questions that the subjects had concerning the procedure were also answered. Finally, once the experiment began, no oral communication was permitted.
The lower part of Table 1 shows the frequencies of the various csaiitions and the mean divisions of the payoffs in the coalitions. The difference in frequencies among the coalitions was significant, with C(4-3) being the most frequent coalition, as predicted by minimum power theory,
x2(2) = 11.56,p < .Ol.” Table 2 shows the proportion (frequency) of initial offers made by subjects in each position to form each of the coalitions. As might be expected, there appears to be a strong correspondence between the proportion of offers made to form coalitions and the frequency with which the coalitions actually formed (cf. Komorita & Moore, 1976). The data on initial coalition offers were analyzed by means of x2~ For purposes of calculating the expected frequencies, offers to form C(4-3-2) were disregarded, and it was assumed that, by chance, subjects in each position would direct an equal number of offers toward each of the two-person coalitions between which they had a choice. Player 4 made significantly more initial offers to form C(4-3) than to form C(4-2), x211$ = 8.33, p < .O%. Player 3 made significantly more initial offers to form C(4-3) than to form C(3-2), x2(1) = 10.70, p < .01. And 2 made significantly more initial offers to form C(4-2) than to form C(?-2). x’(i) =
3.85,~ < .05. ? The x2 test indicates that not all coalitions were equaily frequent. but does not necessarily indicate that C(4-3) was more frequent than C(4-2) or C(3-2). In an attempt to address this problem, in lieu of a suitable one-sample x2 analog to tests for simple effects, the frequency of C(4-3) was tested separately against the frequency of C(4-2) and the frequency of C(3-2). The overall (Ylevel for both tests was fixed at .05, under the assumption that the tests were independent. This was done because of the difficulty of fixing an appropriate 01 level for multiple tests on portions of the same data set. This procedure at ieast provides a somewhat more stringent 01level, even though the tests were not independent. The tests showed that there was no significant difference in frequency between C(4-3) and C(4-2), but C(4-3) was significantly more frequent than C(3-2), x2(1) = 12.25, p < .05.
68
CHARLES
E. MILLER
TABLE PROPORTION
OF INITIAL FORM
OFFERS EACH
2 MADE
BY EACH
POSITION
TO
COALITION
Offers made to form Offers made by 4 3 2 M
C(4-3)
C(4-2)
C(3-2)
C(4-3-2)
.78 (21) .81 (22) .80
.22 (6) .67 (18) .44
.19 (5) .30 (8) .24
.oo (0) .oo (0) .04 (1) .Ol
Note. Numbers in parentheses are the actual frequencies of initial coalition offers.
With regard to the division of the payoffs in the various coalitions, it appears from Table 1 that the predictions made by minimum power theory and bargaining theory were more accurate than the predictions made by minimum resource theory. The predictions of the minimum power and bargaining theories were closer to equal divisions of the payoffs than were the predictions of minimum resource theory, although the actual payoff divisions were more equal yet. To assess the accuracy of each of the theories, 95% confidence limits were calculated for 4’s mean payoffs of 128 (SD = 5.6) in C(4-3) and 125 (SD = 11 .O) in C(4-2). The predictions of minimum power theory fell within these confidence limits for both of the coalitions. The bargaining theory predictions fell within the confidence limits for C(4-3) but not for C(4-2). The predictions of minimum resource theory fell outside of the confidence limits for both of the coalitions. The relative accuracy of the theories is also indicated by discrepancy scores, d, which were calculated for all of the triads (cf. Rapoport & Kahan, 1976). For each triad, this score is the sum of the absolute differences between each member’s observed payoff and his predicted payoff:
a’ = 1% - PAI + lo, - ~~1+ lo, - ~~1. The smaller the discrepancy score, the more accurate the theory. The mean discrepancy scores for minimum power theory, bargaining theory, and minimum resource theory were 13.1, 15.5, and 35.1, respectively. Discussion While the present results seem to lend support to minimum power theory, it ‘is possible to suggest an alternative interpretation, at least with respect to the formation of coalitions. Notice that the coalition which formed most frequently was the one with the largest payoff per member, the coalition which formed next most frequently was the one with the next
COALITION
FORMATION
69
largest payoff per member, and so on. This suggests that when there are differences in payoffs among the winning coalitions, there may be a tendency for coalitions with larger payoffs per member to form more frequently than coalitions with smaller payoffs per member. According to this interpretation, the present results do not really offer support for any of the three theories. Rather, the apparent support for minimum power theory is only an incidental consequence of the fact that the coalition predicted by minimum power theory also happened, because of the way in which the resources and payoffs were specified, to be the coalition having the largest payoff per member. To investigate this possibility and to provide a further test of the theories against one another, a second experiment was conducted. EXPERIMENT
2
Method For this experiment it was necessary to devise a situation in which the resources of the individuals and the payoffs for the various coalitions were such that (a) each of the coalition theories would predict the formation of a different coalition and (b) none of these predicted coahtions would be the one with the largest payoff per member. Resources. The resource distribution employed was D(l0:8-5-4-2). There are four minimal winning coalitions for this distribution: C(8-59, C&-4), C(8-2), and C(5-4-2).3 Payoffs. Subjects played a coalition game for which the characteristic function was: u(8) = v(5) = v(4) = v(2) = ~(5-4) = ~(5-2) = ~(4-2) = 0; v(8-5) = 250; ~(8-4) = 240; ~(8-2) = 210; u(5-4-2) = 200; $3-5-4) = ~(8-5-2) = ~(8-4-2) = ~(8-5-4-2) = 250.
The upper part of Table 3 shows the predictions of the three theories regarding which of the coalitions will form and how the payoffs in the coalitions will be divided in this game. As can be seen, each theory predicts the formation of a different coalition, and none of the theories predicts the formation of C&5), which is the coalition having the largest payoff per member. 3 A minimal winning coalition is a winning coalition such that, if any member is deleted from the coalition, the coalition will no longer be a winning one. Any of the three theories may predict the formation of a nonminimal winning coalition for certain situations involving different payoffs for different winning coalitions. However, the payoffs in the present experiment were deliberately specified so that the coalitions predicted by the theories, along with the coalition having the largest payoff per member, were all minimal winning coalitions.
CHARLES
70
E. MILLER
TABLE PREDICTED
Theory
AND ACTUAL
C(S-5)
3
COALITIONS
C(S-4)
C(8-2)
AND PAYOFF
C(5-4-2)
DIVISIONS
Predicted coalition
Predicted division of payoffs Minimum resource Minimum power Bargaining
1.54-96 191-59 139-111
160-80 185-55 140-100
168-42 166-44 137-73
91-73-36 71-69-60 79-70-51
C(8-2) C(5-4-2) C(S-4)
Actual division of payoffs 137-l 13 143-97
145-65
64-68-69
Actual coalition frequencies 31
16
9
2
Note. Only one of the nonminimal winning coalitions, not shown here, occurred. The value of all nonminimal winning coalitions was 250 points. The values of C(S-5), C(S-4), C(S-2), and C(5-4-2) were 250, 240, 210, and 200 points, respectively. The Shapley values for 8, 5, 4, and 2 were 133.3, 41.7, 40.0, and 35.0, respectively.
Subjects. Subjects were 236 male students enrolled in an introductory psychology course. Participation in the experiment allowed the subjects to earn extra credit for the course. Procedure. Either two or three groups composed of four subjects each took part in every experimental session. The procedure for the experiment was basically the same as for Experiment 1, with subjects playing the role of legislators in a coalition game. In this game, A controlled 80 votes; B controlled 50 votes; C controlled 40 votes; and D controlled 20 votes (i.e., votes were in the ratio 8:5:4:2). The amount of the payoff to be won in the game depended upon which coalition formed, as described above. As in the first experiment, subjects exchanged written offer slips with one another, indicating with whom they wanted to form a coalition and how they wanted to divide the payoff for the coalition. Subjects were permitted to send an offer for only one coalition on any given exchange of offers. (However, if B, say, wanted to offer the B-C-D coalition, he could send offers to both C and D on a single exchange, since this still constituted an offer for just one coalition.) When a subject received more than one offer, he could accept only one of them, unless the offers were based on forming the same coalition. For example, if B received an offer from C to form the B-C-D coalition and also an offer from D to form the B-C-D coalition, he could accept both of them if both offers involved exactly the same division of the payoff. A coalition was considered to have formed if all the members of the coalition to whom an offer was made accepted the coalition, and if no other coalition was accepted by all
COALITION
FORMATION
71
the members to whom it was offered. Exchanges ofoffers continued until some coalition formed. There were four monetary prizes at stake in the entire experiment. The subject who won the most points throughout the experiment in position A was awarded a prize of $5.00. The subjects who won the most points throughout the experiment in positions B, C, and D were each awarded like prizes. Subjects were not told ahead of time of the amount of the prizes or the basis on which the prizes would be awarded. They were only informed that if they wished to win a monetary prize they should try to win as many points as possible. The procedure for the experiment was explained further through the use of examples, and any questions that the subjects had were answered. Once an experimental session got under way, no oral communication was allowed. Results The lower part of Table 3 shows the frequencies of the various coalitions and the divisions of the payoffs in the coalitions. The difference in the frequencies of the minimal winning coalitions was significant, x2(3) = 31.79, p < .OOI .* The coalition that formed most frequently was not any of the ones predicted by the theories, but was instead the coalition with the largest payoff per member. In fact, as in the fn-st experiment, there was a strong positive relationship between the payoffs per member in the coalitions and the frequencies with which the coalitions occurred. Table 4 shows the proportion of initial offers made by each position to form the various coalitions. A comparison of Tables 3 and 4 reveals that the strong correspondence, found in Experiment 1, between the proportions of initial offers to form coalitions and the frequencies with which the coalitions actually formed was present in Experiment 2 as well. The data on initial offers to form coalitions were analyzed, position by position, using x2. For purposes of calculating expected frequencies, offers to form nonminimal winning coalitions were disregarded. There was a significant difference in the frequencies of the coalitions initially offered by 8, with C(8-5) being offered the most frequently, x2(2) = 45.19, p < .OOl. This finding is important because the payoffs predicted by the bargaining theory for 8 in C(8-5) and C(E-4) differ only slightly (see Table 3). It might have been argued that, as predicted by the theory, 8 was pretty much indifferent between the two coalitions, but C(8--5) formed more frequently because 5 made more offers to form C(S-5) than 4. made 4 The frequency of (38-5) was tested separately against the frequency ofC(%4), and C(S-4-2), with the overall ty level for the tests fixed at .O5 (see footnote difference in frequency between C(8-5) and (3-4) approached significance, $(I) p i .06; and C(8-5) was significantly more frequent than C(8-2), x2(1) = 12.10,p < C&-4-2), x2(1) = 25.48, p < .05.
C&.2), 2). The = 4.79, .05. arid
72
CHARLES
E. MILLER
TABLE PROPORTION
OF INITIAL FORM
4
OFFERS
MADE
VARIOUS
BY EACH
POSITION
TO
COALITIONS
Offers made to form Offers made by 8 5 4 2 M
C(8-5)
C(8-4)
C(8-2)
.75 (44) .92 (54) .83
.12 (7) .78 (46) .45
.14 (8) .47 (28) .31
C(S-4-2) .03 (2) .03 (2) .05 (3) .04
Other .oo .05 .19 .47 .I8
(0) (3) (11) (28)
Note. Numbers in parentheses are the actual frequencies of initial coalition offers.
to form C(8-4). Clearly, however, bargaining theory cannot account for 8’s overwhelming preference for C(8-5).5 Turning to the initial offers from the remaining positions, 5 made significantly more initial offers to form C(8-5) than to form C(.5-4-2), x2(1) = 48.29, p < .OOl; 4 made significantly more initial offers to form C(8-4) than to form C(5-4-2), x2(1) = 40.33, p < .OOl; and 2 made significantly more initial offers to form C(8-2) than to form C(.5-4-2), x2(1) = 16.03, p < .OOl. With regard to the division of payoffs in the coalitions, the data are obviously more in accord with the predictions of bargaining theory than with the predictions of minimum resource theory or minimum power theory (see Table 3). The predictions made by the bargaining theory were generally closer to equal divisions of the payoffs than were the predictions made by the minimum resource or minimum power theories. As in Experiment 1, the payoff data were analyzed by calculating 95% confidence limits-in this case, for 8’s mean payoffs of 137 (SD = 9.4) in C(8-5), 143 (SD = 9.6) in C(8-4), and 145 (SD = 24.0) in C(8-2). This analysis revealed that the predictions of bargaining theory were within the confidence limits for all three coalitions, while the predictions of minimum resource theory and minimum power theory were outside the confidence limits for all three coalitions. Also as in Experiment 1, discrepancy scores were calculated for each of the theories. The mean discrepancy scores were 18.6 for bargaining theory, 39.1 for minimum resource theory, and 90.4 for minimum power theory. An additional aspect of some interest regarding the payoff divisions in the coalitions is that 8’s mean payoff was largest in C(8-2), next largest in C(8-4), and smallest in C(8-5), and yet the frequency of occurrence of the 5 The frequency of 8’s initial offers to form C(8-5) differed significantly from the frequency of 8’s initial offers to form C(8-4), ~71) = 26.84,~ i .05, and C(8-2), x*(l) = 24.92, p < .05.
COALITION
FORMATION
73
coalitions ran in the opposite direction. On the face of it, this finding would appear to call into question the validity of the assumption, central to all of the theories, that each individual will attempt to join whichever coalition maximizes his payoffs. Within the present experiment, however, the only direct evidence available regarding this assumption is whether there was some tendency on the part of subjects with the weight g--or any of the other weights-to reject the offer of a relatively large payoff and to accept instead the offer of a relatively small payoff. Therefore, all instances in which an individual received more than one offer at the same time and accepted one of them were examined, in order to see in how many of these instances the individual accepted an offer that was smaller than one of the offers he rejected. There were 70 instances in which 8 received more than one offer and accepted one of them, and in only 9 of these instances (13%) was the accepted offer smaller than one of the rejected offers. There were an additional 39 instances in which one of the other individuals (i.e., 5, 4, or 2) received more than one offer and accepted one of them, and in only 5 of these instances (13%) was the accepted offer smaller than one of the rejected offers. Thus, as far as the acceptance and rejection of offers of coalition are concerned, the in viduals do appear, for the most part, to have tried to maximize their payoffs. GENERAL
DlSCUSSlON
None of the three theories was able to predict accurately, for both of the experiments, which coalition would form most frequently or how the payoffs would be divided. With regard to the formation of coalitions, minimum power theory did accurately predict the most frequent coalition in Experiment 1. The results of Experiment 2, however, in which the theory’s prediction of the most frequent coalition was inaccurate, suggest that the accurate prediction in the first experiment was probably due to the fact that the predicted coalition was the one which got the largest payoff per member. With regard to the division of payoffs, which of the theories was the most accurate appears to have depended, at least to some extent, on which theory made predictions that were closest to an equal division. The predictions of minimum power theory in Experiment 1 and bargaining theory in Experiment 2 were closest to an equal payo sion, and these theories were the most accurate in the respective ments. Tt seems possible that the theories cannot deal adequately with situations such as those studied in the present experiments because the theories tend to attribute too much influence to the distribution of resources and not enough influence to the payoffs for the coalitions. Prior research on the theories, by focusing mainly on situations involvi constant payoffs for winning (so-called “simple” games) has neglect
74
CHARLES
E. MILLER
the possible effects of coalition payoffs and has limited itself to conditions under which the effects of resources are likely to be most pronounced. When there are no differences in payoffs among the winning coalitions, the distribution of resources provides individuals with almost their only basis for deciding which of the coalitions will maximize their payoffs. Numerous studies have shown that under such conditions, resources will affect both the formation of coalitions and the division of payoffs in the coalitions. On the other hand, when there are differences in payoffs among the winning coalitions, but no differences in resources among the individuals, one would expect individuals to be sensitive to the payoff differences. The results of the few previous studies that have examined coalition behavior in characteristic function games indicate that this is in fact the case (e.g., Horowitz & Rapoport, 1974; Kahan & Rapoport, 1974; Rapoport & Kahan, 1976), with coalition payoffs affecting both the coalitions that form and the way that the payoffs for the coalitions are divided. Unfortunately, there is little empirical evidence regarding the important question of what is likely to happen when, as in the present experiments, there are differences in both the resources of the individuals and the payoffs for the winning coalitions (but see Gamson, 1961b; Wolf & Shubik, 1977). Situations of this sort are considerably more complicated than those in which either the payoffs or the resources are the same. Such situations may be accompanied by considerable uncertainty about what constitutes appropriate behavior, either in an individually rational or a socially normative sense. Recent work by Komorita, Chertkoff, and their colleagues (Chertkoff & Esser, 1977; Komorita & Brinberg, 1977) suggests that the weight which individuals give to different norms may vary from situation to situation. It may be that when there are differences in the payoffs for coalitions, the individuals will tend to focus on these differences rather than on any differences in resources which might exist. To the extent that individuals ignore resources, the norm of equity will carry little weight relative to the norm of equality. One fairly easy rule of thumb for an individual to adopt is to try to join the coalition with the largest payoff per member. The greater the weight given to the norm of equality and, hence, the more nearly equal the individuals expect the payoff division to be, the more likely it is that the coalition with the largest payoff per member will be preferred, since it is this coalition which will be expected to maximize the payoffs to all its members. In the two experiments reported here, relatively small differences in coalition payoffs led to relatively large differences in coalition frequencies. The present research suggests a change in focus for the study of coalitions. It clearly points up the importance of studying situations in which both coalition payoffs and individual resources vary. It shows that the minimum power, minimum resource, and bargaining theories cannot ac-
COALITION
FORMATION
75
count adequately for coalition behavior in such situations. It also demonstrates the influence of coalition payoffs on the formation of coalitions and calls into question the heavy emphasis on resources in prior research. Most real-world situations probably involve differences in coalition payoffs as well as individual resources (see Axelrod, 1970), and more attention ought to be paid to such situations. REFERENCES Aumann, R. J., & Maschler, M. The bargaining set for cooperative games. In M. Dresher, L. S. Shapley, & A. W. Tucker (Eds.), Advances in game theory. Annals of Mathematics Study 52. Princeton, N. J.: Princeton Univ. Press, 1964. Pp. 443-476. Axelrod, R. Conjlict of interest: A theory of divergent goals with applications to politics. Chicago: Markham, 1970. Caplow, T. A theory of coalitions in the triad. American Sociological Review, 1956, 21% 271-280.
Chertkoff, J. M., & Esser, J. K. A test of three theories of coalition formation when agreements can be short-term or long-term. Journal of Personality attd Social Psychulogy, 1977, 35, 237-249. Davis, M., & Maschler, M. The kernel of a cooperative game. Navnl Reseurck Logistics Quarterly, 1965, 12, 223-259. Gamson, W. A. A theory of coalition formation, American Sociologicai Review, 1961, 26, 373-382. (a) Gamson, W. A. An experimental test of a theory of coalition formation. American Sociological Review, 1961, 26, 585-573. (b) Gamson, W. A. Experimental studies of coalition formation. In L. Berkowitz (Ed.), Advances in experimental social psychology. New York: Academic Press, 1964. Vol. 1. Horowitz. A. D.. & Rapoport, A. Test of the kernel and two bargaining set models in fourand five-person games. In A. Rapoport (Ed.), Game theory as a theory of conj?ict resolution. Dordrecht: Reidel, 1974. Pp. 161-192. Kahan, 3. P., & Rapoport, A. Test of the bargaining set and kernel models in three-person games. In A. Rapoport (Ed.), Grime theory as n theory ofconjict resolution. Dordrecht: Reidel, 1974. Pp. 119-160. 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. Psycho/ogica/ Review. 1974, 81, 242-256.
Komorita,
S. S., & Brinberg,
Sociometry,
D. The effects of equity norms in coalition formation.
1977, 40, 351-361.
Komorita, S. S., & Chertkoff, J. M. A bargaining theory of coalition formation. Psychological Review, 1973, 80, 149-162. Komorita, S. S., & Moore, D. Theories and processes of coalition formation. Journal qf Personality and Social Psychology, 1976, 33, 371-381. Michener, H. A., Fleishman, J. A., & Vaske, J. J. A test of the bargaining theory of coalition formation in four-person groups. Joarna( of Persona&y and Social Psychology, 1976, 34, 1114-1126.
Murnighan, 3. K., Komorita, S. S., & Szwajkowski, E. Theories of coalition formation and the effects of reference groups. Journal of Experimental Social Psychology, 1977, 13, 166-381.
Rapoport, A. N-person game theory: Concepts and applications. of Michigan Press, 1970.
Ann Arbor, Mich.: Univ.
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Rapoport, A., & Kahan, J. P. When three is not always two against one: Coalitions in experimental three-person cooperative games. Journal ofExperimental Social Psychology, 1976, 12, 253-273. Schelling, T. C. The strategy of conflict. Cambridge, Mass.: Harvard Univ. Press, 1960. Shapley, L. S. A value for N-person games. Annals of Mathematics Studies, 19.53, 28, 307-3 17. Shapley, L. S., & Shubik, M. Method for evaluating the distribution of power in a committee system. American Political Science Review, 19.54, 48, 787-792. Vinacke, W. E., & Arkoff, A. An experimental study of coalitions in the triad. American Sociological Review, 1957, 22, 406-414. Wolf, C., & Shubik, M. Beliefs about coalition formation in multiple resource three-person situations. Behavioral Science, 1977, 22, 99-106.
OF
Coalition
EXPERIMENTAL
SOCIAL
Formation Competitive
PSYCHOLOGY
16, 61-76
in Characteristic Tests of Three
(i980)
Function Theories
Games:
CHARLES E. MILLER Northern
Illinois
University
Received August 16, 1977 Two experiments tested the bargaining, minimum resource, and minimum power theories of coalition formation in situations involving different payoffs for some of the winning coalitions (characteristic function games). In the first experiment, a triadic resource distribution was employed and payoffs for the coalitions were specified in such a way that each of the three theories predicted the formation of a different coalition. The coalition predicted by minimum power theory formed the most frequently, and the mean divisions of the payoffs among coalition members were also closest to the predictions of minimum power theory. Ilowever, the most frequent coalition was not only the one predicted by minimum power theory, it was also the one having the largest payoff per member. Therefore. a second experiment was conducted, which empIoyed a tetradic resource distribution and specified the coalition payoffs in such a way that (a) each of the theories predicted the formation of a different coalition and (b) none of the predicted coalitions was the one with the largest payoff per member. Although the mean payoff divisions in the coalitions in this experiment were closest to those predicted by bargaining theory, the coalition that formed most frequently was not one of those predicted by any of the theories. Rather, it was the one having the largest payoff per member. None of the three theories is able to account adequately for the results of both experiments. The difficulties that the theories have in dealing with coalition formation in situations in which there are different payoffs for winning are discussed.
The formation of coalitions is an important and pervasive aspect of social interactions. A coalition may be defined as the joint use of resources to determine the outcome of a decision in a mixed-motive situation involving more than two individuals (Gamson, 1964). For experimental mu-poses, resources are ordinarily represented as weights controlled by the individuals, such that some specified quota of these weights is needed in order to win a payoff. When no individual has enough weight to This research was supported by a grant from the Council of Academic Deans at Northern Illinois University. Requests for reprints should be sent to Charles E. Miller, Department of Psychology, Northern Illinois University, DeKalb, IL 60115. 61
~22-~~31/g0/01~6~-~6$02.50/0 Copyright @ 1980 by Academic Press, Inc. Ail right, of reproduction in any form reserved.
62
CHARLES
E. MILLER
win the payoff by himself, it is necessary for a coalition to form. The members of the winning coalition can divide the payoff among themselves in whatever way they want. Theories of coalition formation are faced with two fundamental questions: (a) Which one of the possible winning coalitions will form? And (b) how will the members of a winning coalition divide the payoff? Although a number of coalition theories have been proposed, only those theories that undertake to answer both of these questions are considered in the present paper. This restriction excludes several solutions from the theory of n-person games, such as the kernel (Davis & Maschler, 1965) and the Aumann and Maschler (1964) bargaining set, because they do not generally predict which coalition will form. It also excludes Caplow’s (1956) theory of coalitions in triads because the theory does not predict how coalition members will divide the payoff for winning. Most experimental studies of coalition formation have been concerned with situations in which the payoffs are the same for every winning coalition. Yet it seems likely that in the “real world,” the payoffs that different coalitions can obtain are very often not constant. For example, several small manufacturing firms may form a coalition to obtain a contract from a large buyer, but the size of the contract may depend upon which firms enter into the coalition. The present paper considers only those theories that can be applied to situations in which there are different payoffs for some of the winning coalitions. This restriction excludes Komorita’s (1974) weighted probability theory. There are three theories which are not excluded by either of the above criteria: minimum resource theory (Gamson, 1961a, 1961b), minimum power theory (Gamson, 1961a, 1964; Shapley, 1953; Shapley & Shubik, 1954), and bargaining theory (Komorita & Chertkoff, 1973). This paper presents two experiments that test these theories against one another in situations where they make competing predictions. Before turning to the particulars of the experiments, however, the assumptions of each of the theories need to be taken up. All three of the theories are alike in assuming that each individual is motivated to maximize his own payoff, but the theories differ in their assumptions about how an individual decides which of the coalitions will allow him to do this. Minimum resource theory assumes that the payoff for a coalition will be divided according to a norm of equity, or parity (Gamson, 1961a), which prescribes that the payoff be divided in proportion to the resources that the members contribute to the coalition. In situations where the payoff is the same no matter which coalition wins, an individual gets the most by maximizing his share (proportion) of the payoff for the coalition. Since the individual’s resources are the same regardless of which coalition he joins, the smaller the combined resources of the coalition, the larger will be his share. The theory therefore predicts that
COALITION
FORMATION
63
the coalition which will form is the one having the minimum amount of combined resources needed to win. It is this coalition, out of all the possible ones, which maximizes the payoffs to its members. Minimum resource theory also makes predictions for situations 111 which some of the winning coalitions receive different payoffs (Gamson, 196la, p. 376). In these situations the theory does not necessarily predict the formation of the winning coalition with the minimum amount of resources, Here, an individual’s payoff from joining a coalition is equal to the proportion of resources he contributes to the coalition, multiplied by the coalition payoff. A small share of the payoff in a coalition in which the payofT is large may be better than a large share of the payoff in a coalition in which the payoff is small. Minimum power theory assumes that the members of a coalition will divide the payoff in proportion to their Shnpley values (Shapley, 19533, rather tban in proportion to their resources (see Gamson, 1961a, p* 382). Individuals’ Shapley values are considered an indication of their “‘power” in a situation. (When the payoffs in a situation are equal for all e winning coalitions, the individuals’ proportional Shapley values are called their pivotal power; Shapley & Shubik. 1954.) A person’s Shapley value is found by examining those possible winning coalitions of which he is a member and determining what would happen if he were to withdraw from them. An individual has a large Shapley value if, by withdrawing, he can change numerous possible winning coalitions with high payoffs (values) into losing coalitions, or into winning coalitions with much lower payoffs. The formula for calculating Person i’s Shapley value is: (s - l)! (n - s)! si=c
n,
~4s) - es - ($1,
SEN
where n is the total number of persons in the coalition situation, s is the number of persons in a given coalition (subset), V(S) is the payoff (value) for a given coalition, v(S - (i}) is the payoff for the coalition if Person i withdraws from the coalition, and the summation ranges over a!1 the coalitions (subsets) S of which Person i can be a member ( pp. 108-109). Bargaining theory assumes that the division of tbe payo coalition will be a compromise between two conflicting norms. member who is strong in resources will argue for a division of the pa according to the equity norm, while a member who is weak in resources will advocate that the payoff be divided according to a norm of equality. It is assumed that the coalition members will adopt a “split&e-difference” principle (Schelling, 1960) regarding these two norms, with each member’s share of the payoff lying halfway between what it would be if the payoff were divided in proportion to the resources of the members and
64
CHARLES
E. MILLER
what it would be if the payoff were divided equally among the members. An individual’s payoff in a coalition is the product of the total payoff for the coalition and the individual’s share of the payoff. A coalition is predicted to form if it maximizes the payoffs to all its members when the payoff is divided in this manner. l For most situations of interest, there will be one such coalition. Attempts to test the minimum resource, minimum power, and bargaining theories against each other (e.g., Komorita & Moore, 1976; Michener, Fleishman, & Vaske, 1976; Murnighan, Komorita, & Szwajkowski, 1977) have been limited exclusively to situations in which all of the winning coalitions receive the same payoff. There are at least two ways in which this limitation is somewhat unfortunate: First, as already mentioned, in many real-world situations there are likely to be differences in the payoffs which various coalitions can acquire. Second, it turns out that, when the payoff for winning is constant, the theories generally make competing predictions with regard to the question of how the payoff will be divided but nof with regard to the question of which coalition will form. For example, much of the early research on coalition formation involved triads (e.g., Kelley & Arrowood, 1960; Vinacke & Arkoff, 1957), and as long as every winning coalition receives an equal payoff, there are no resource distributions in triads for which each of the theories predicts the formation of a different coalition. Recent coalition research has dealt increasingly with tetrads, but even in tetrads there are no resource distributions for which the formation predictions of each theory differ from those of the other two (see Michener et al., 1976). It appears not generally to have been recognized that if the limitation of constant payoffs is dropped, it then becomes possible to find situations, both in triads and in tetrads, for which the minimum resource, minimum power, and bargaining theories make competing predictions not only with regard to the division of payoffs, but also with regard to the formation of coalitions. Such situations were employed in the two experiments that are reported here. EXPERIMENT
1
Method Resources. The first experiment involved triads in which the resource distribution was D(.5:4-3-2), where the first number denotes the quota of 1 This prediction applies only to initial instances of coalition formation. Bargaining theory also provides asymptotic predictions for repeated instances of coalition formation, but both of the experiments in this paper were intended to test only the predictions for initial instances. However, it is possible to argue that the asymptotic predictions of the theory apply to initial instances of coalition formation, to the extent that the bargaining which leads to initial coalitions involves a number of rounds of negotiations. As a check against this possibility, asymptotic predictions were derived for the situations used in both of the present experiments. On the whole, these predictions proved to be less accurate than the predictions for initial instances of coalition formation, and they are not discussed here.
COALITION
FORMATION
65
resources required to win, and the subsequent numbers denote the resources of the individuals. This resource distribution was selected as a particularly interesting one to study because there has been more research conducted on it, using constant payoffs for winning, than on any other triadic distribution. Payoffs. With the D(5:4-3-2) distribution of resources, any two of the triad members can get together and form a winning coalition. When the payoffs are the same for all the winning coalitions, minimum resource theory and bargaining theory predict that the coalition between 3 and 2, denoted C(3-2), is the most likely to form, while minimum power theory predicts that all of the two-person coalitions are equally likely. When there are differences in the payoffs for the winning coalitions, however, it is possible for each of the theories to predict the formation of a different coalition, depending upon the exact values of the payoffs. In the present experiment, subjects played a coalition game in which the payoffs for the coalitions were specified in such a way that the three theories would make competing predictions. Coalition games in which different winning coalitions receive different payoffs are commonly described in terms of their characteristic functions. The characteristic function of a game specifies a value, or payoff, for each possible coalition, including one-person coalitions. The characteristic function for the game used in the present experiment was: v(4) = v(3) = u(2) = 0; ~(4-3) = 240; ~(4-2) = 230; ~(3-2) = 200; ~(4-3-2) = 240, where v(4-2), for example, indicates the payoff to C(4-2). The upper part of Table 1 shows, for these payoffs, the predictions of the theories regarding which coalitions will form and how payoffs in the coalitions will be divided. As can be seen, a different coalition is predicted by each theory. Subjects. Subjects were 81 male students enrolled in an introductory psychology course. Participation in the experiment allowed the subjects to earn extra credit toward their course grades. Procedure. Subjects were scheduled to take part in the experiment in three-person groups, three or four groups at a time. The experiment was presented to subjects as a study of coalition formation in legislative bodies (cf. Gamson, 1961b). Each subject was asked to play the role of a Eegislator in a coalition game. Before the game began, each subject drew, at random, the letter A, B, or C. The letter drawn determined the number of votes that the subject controlled for the game. A conrrolled 40 votes; B controlled 30 votes; and C controlled 20 votes. (That is, the votes were in the ratio 4:3:2.)
66
CHARLES
E. MILLER
TABLE PREDICTED Theory
AND
ACTUAL
C(4-3)
COALITIONS
C(4-2)
1 AND
PAYOFF
C(3-2)
DIVISIONS
Predicted coalitions
Predicted division of payoffs Minimum resource Minimum power Bargaining
137-103 131-109 129-111
153-77 129-101
134-96
120-80 103-97 110-90
C(3-2) C(4-3) C(4-2)
Actual division of payoffs 128-112
125-10.5
100-100
Actual coalition frequencies 15
11
1
Note. C(4-3-2), not shown here, did not occur. The value of C(4-3-2) was 240 points. The values of C(4-3), C(4-2), and C(3-2) were 240, 230, and 210 points, respectively. The Shapley values for 4, 3, and 2 were 91.7, 76.7, and 71.7, respectively.
The object of the game was for some of the legislators to form a coalition that controlled a majority of the 90 votes. Any legislators who formed such a coalition won a payoff, but the amount of the payoff depended upon which coalition formed, as explained above. A coalition could be formed only if the subjects making up the coalition agreed on how to divide the payoff for the coalition. The game consisted of a number of trials on which the subjects made offers to one another by means of written offer slips. On these offer slips, subjects indicated with whom they wished to form a coalition and how they proposed to divide. the payoff for the coalition. Subjects were permitted to send an offer for only one coalition on any given trial. After all of the subjects had completed their offer slips, the experimenter collected and examined the slips and distributed them to the appropriate legislators. When a subject received an offer, he had to accept or reject it by writing “Accept” or “Reject” on the offer slip. If a subject received more than one offer, he could accept only one of them. After the subjects had accepted or rejected the offers that they received, the offer slips were collected and the winning coalition was announced, if one had formed. A coalition was considered to have formed if one and only one coalition was accepted. For example, if A sent an offer to B and B accepted it, and B sent an offer to C and C accepted it, neither the A-B nor the B-C coalition was considered to have formed, and subjects went on to the next trial. Trials continued until some coalition formed, thus ending the game. Each subject was told to try to win as many points for himself as he possibly could and to do this without regard to how much or how little any
COALITION
FORMATION
67
of the other subjects might win. In each experimental session, there were three monetary prizes at stake. The subject who won the most points in position A was awarded a prize of $3.00, as were the subjects who won the most points in positions B and C. The subjects were not informed in advance about the amount of the prizes, nor were they informed of the basis on which the prizes would be awarded. Each subject was simply told that if he wanted to win one of the monetary prizes, he should attempt to win as many points as possible. To assist the subjects in understanding the procedure of the experiment, several examples were provided of how the offer slips might be filled out. Any questions that the subjects had concerning the procedure were also answered. Finally, once the experiment began, no oral communication was permitted.
The lower part of Table 1 shows the frequencies of the various csaiitions and the mean divisions of the payoffs in the coalitions. The difference in frequencies among the coalitions was significant, with C(4-3) being the most frequent coalition, as predicted by minimum power theory,
x2(2) = 11.56,p < .Ol.” Table 2 shows the proportion (frequency) of initial offers made by subjects in each position to form each of the coalitions. As might be expected, there appears to be a strong correspondence between the proportion of offers made to form coalitions and the frequency with which the coalitions actually formed (cf. Komorita & Moore, 1976). The data on initial coalition offers were analyzed by means of x2~ For purposes of calculating the expected frequencies, offers to form C(4-3-2) were disregarded, and it was assumed that, by chance, subjects in each position would direct an equal number of offers toward each of the two-person coalitions between which they had a choice. Player 4 made significantly more initial offers to form C(4-3) than to form C(4-2), x211$ = 8.33, p < .O%. Player 3 made significantly more initial offers to form C(4-3) than to form C(3-2), x2(1) = 10.70, p < .01. And 2 made significantly more initial offers to form C(4-2) than to form C(?-2). x’(i) =
3.85,~ < .05. ? The x2 test indicates that not all coalitions were equaily frequent. but does not necessarily indicate that C(4-3) was more frequent than C(4-2) or C(3-2). In an attempt to address this problem, in lieu of a suitable one-sample x2 analog to tests for simple effects, the frequency of C(4-3) was tested separately against the frequency of C(4-2) and the frequency of C(3-2). The overall (Ylevel for both tests was fixed at .05, under the assumption that the tests were independent. This was done because of the difficulty of fixing an appropriate 01 level for multiple tests on portions of the same data set. This procedure at ieast provides a somewhat more stringent 01level, even though the tests were not independent. The tests showed that there was no significant difference in frequency between C(4-3) and C(4-2), but C(4-3) was significantly more frequent than C(3-2), x2(1) = 12.25, p < .05.
68
CHARLES
E. MILLER
TABLE PROPORTION
OF INITIAL FORM
OFFERS EACH
2 MADE
BY EACH
POSITION
TO
COALITION
Offers made to form Offers made by 4 3 2 M
C(4-3)
C(4-2)
C(3-2)
C(4-3-2)
.78 (21) .81 (22) .80
.22 (6) .67 (18) .44
.19 (5) .30 (8) .24
.oo (0) .oo (0) .04 (1) .Ol
Note. Numbers in parentheses are the actual frequencies of initial coalition offers.
With regard to the division of the payoffs in the various coalitions, it appears from Table 1 that the predictions made by minimum power theory and bargaining theory were more accurate than the predictions made by minimum resource theory. The predictions of the minimum power and bargaining theories were closer to equal divisions of the payoffs than were the predictions of minimum resource theory, although the actual payoff divisions were more equal yet. To assess the accuracy of each of the theories, 95% confidence limits were calculated for 4’s mean payoffs of 128 (SD = 5.6) in C(4-3) and 125 (SD = 11 .O) in C(4-2). The predictions of minimum power theory fell within these confidence limits for both of the coalitions. The bargaining theory predictions fell within the confidence limits for C(4-3) but not for C(4-2). The predictions of minimum resource theory fell outside of the confidence limits for both of the coalitions. The relative accuracy of the theories is also indicated by discrepancy scores, d, which were calculated for all of the triads (cf. Rapoport & Kahan, 1976). For each triad, this score is the sum of the absolute differences between each member’s observed payoff and his predicted payoff:
a’ = 1% - PAI + lo, - ~~1+ lo, - ~~1. The smaller the discrepancy score, the more accurate the theory. The mean discrepancy scores for minimum power theory, bargaining theory, and minimum resource theory were 13.1, 15.5, and 35.1, respectively. Discussion While the present results seem to lend support to minimum power theory, it ‘is possible to suggest an alternative interpretation, at least with respect to the formation of coalitions. Notice that the coalition which formed most frequently was the one with the largest payoff per member, the coalition which formed next most frequently was the one with the next
COALITION
FORMATION
69
largest payoff per member, and so on. This suggests that when there are differences in payoffs among the winning coalitions, there may be a tendency for coalitions with larger payoffs per member to form more frequently than coalitions with smaller payoffs per member. According to this interpretation, the present results do not really offer support for any of the three theories. Rather, the apparent support for minimum power theory is only an incidental consequence of the fact that the coalition predicted by minimum power theory also happened, because of the way in which the resources and payoffs were specified, to be the coalition having the largest payoff per member. To investigate this possibility and to provide a further test of the theories against one another, a second experiment was conducted. EXPERIMENT
2
Method For this experiment it was necessary to devise a situation in which the resources of the individuals and the payoffs for the various coalitions were such that (a) each of the coalition theories would predict the formation of a different coalition and (b) none of these predicted coahtions would be the one with the largest payoff per member. Resources. The resource distribution employed was D(l0:8-5-4-2). There are four minimal winning coalitions for this distribution: C(8-59, C&-4), C(8-2), and C(5-4-2).3 Payoffs. Subjects played a coalition game for which the characteristic function was: u(8) = v(5) = v(4) = v(2) = ~(5-4) = ~(5-2) = ~(4-2) = 0; v(8-5) = 250; ~(8-4) = 240; ~(8-2) = 210; u(5-4-2) = 200; $3-5-4) = ~(8-5-2) = ~(8-4-2) = ~(8-5-4-2) = 250.
The upper part of Table 3 shows the predictions of the three theories regarding which of the coalitions will form and how the payoffs in the coalitions will be divided in this game. As can be seen, each theory predicts the formation of a different coalition, and none of the theories predicts the formation of C&5), which is the coalition having the largest payoff per member. 3 A minimal winning coalition is a winning coalition such that, if any member is deleted from the coalition, the coalition will no longer be a winning one. Any of the three theories may predict the formation of a nonminimal winning coalition for certain situations involving different payoffs for different winning coalitions. However, the payoffs in the present experiment were deliberately specified so that the coalitions predicted by the theories, along with the coalition having the largest payoff per member, were all minimal winning coalitions.
CHARLES
70
E. MILLER
TABLE PREDICTED
Theory
AND ACTUAL
C(S-5)
3
COALITIONS
C(S-4)
C(8-2)
AND PAYOFF
C(5-4-2)
DIVISIONS
Predicted coalition
Predicted division of payoffs Minimum resource Minimum power Bargaining
1.54-96 191-59 139-111
160-80 185-55 140-100
168-42 166-44 137-73
91-73-36 71-69-60 79-70-51
C(8-2) C(5-4-2) C(S-4)
Actual division of payoffs 137-l 13 143-97
145-65
64-68-69
Actual coalition frequencies 31
16
9
2
Note. Only one of the nonminimal winning coalitions, not shown here, occurred. The value of all nonminimal winning coalitions was 250 points. The values of C(S-5), C(S-4), C(S-2), and C(5-4-2) were 250, 240, 210, and 200 points, respectively. The Shapley values for 8, 5, 4, and 2 were 133.3, 41.7, 40.0, and 35.0, respectively.
Subjects. Subjects were 236 male students enrolled in an introductory psychology course. Participation in the experiment allowed the subjects to earn extra credit for the course. Procedure. Either two or three groups composed of four subjects each took part in every experimental session. The procedure for the experiment was basically the same as for Experiment 1, with subjects playing the role of legislators in a coalition game. In this game, A controlled 80 votes; B controlled 50 votes; C controlled 40 votes; and D controlled 20 votes (i.e., votes were in the ratio 8:5:4:2). The amount of the payoff to be won in the game depended upon which coalition formed, as described above. As in the first experiment, subjects exchanged written offer slips with one another, indicating with whom they wanted to form a coalition and how they wanted to divide the payoff for the coalition. Subjects were permitted to send an offer for only one coalition on any given exchange of offers. (However, if B, say, wanted to offer the B-C-D coalition, he could send offers to both C and D on a single exchange, since this still constituted an offer for just one coalition.) When a subject received more than one offer, he could accept only one of them, unless the offers were based on forming the same coalition. For example, if B received an offer from C to form the B-C-D coalition and also an offer from D to form the B-C-D coalition, he could accept both of them if both offers involved exactly the same division of the payoff. A coalition was considered to have formed if all the members of the coalition to whom an offer was made accepted the coalition, and if no other coalition was accepted by all
COALITION
FORMATION
71
the members to whom it was offered. Exchanges ofoffers continued until some coalition formed. There were four monetary prizes at stake in the entire experiment. The subject who won the most points throughout the experiment in position A was awarded a prize of $5.00. The subjects who won the most points throughout the experiment in positions B, C, and D were each awarded like prizes. Subjects were not told ahead of time of the amount of the prizes or the basis on which the prizes would be awarded. They were only informed that if they wished to win a monetary prize they should try to win as many points as possible. The procedure for the experiment was explained further through the use of examples, and any questions that the subjects had were answered. Once an experimental session got under way, no oral communication was allowed. Results The lower part of Table 3 shows the frequencies of the various coalitions and the divisions of the payoffs in the coalitions. The difference in the frequencies of the minimal winning coalitions was significant, x2(3) = 31.79, p < .OOI .* The coalition that formed most frequently was not any of the ones predicted by the theories, but was instead the coalition with the largest payoff per member. In fact, as in the fn-st experiment, there was a strong positive relationship between the payoffs per member in the coalitions and the frequencies with which the coalitions occurred. Table 4 shows the proportion of initial offers made by each position to form the various coalitions. A comparison of Tables 3 and 4 reveals that the strong correspondence, found in Experiment 1, between the proportions of initial offers to form coalitions and the frequencies with which the coalitions actually formed was present in Experiment 2 as well. The data on initial offers to form coalitions were analyzed, position by position, using x2. For purposes of calculating expected frequencies, offers to form nonminimal winning coalitions were disregarded. There was a significant difference in the frequencies of the coalitions initially offered by 8, with C(8-5) being offered the most frequently, x2(2) = 45.19, p < .OOl. This finding is important because the payoffs predicted by the bargaining theory for 8 in C(8-5) and C(E-4) differ only slightly (see Table 3). It might have been argued that, as predicted by the theory, 8 was pretty much indifferent between the two coalitions, but C(8--5) formed more frequently because 5 made more offers to form C(S-5) than 4. made 4 The frequency of (38-5) was tested separately against the frequency ofC(%4), and C(S-4-2), with the overall ty level for the tests fixed at .O5 (see footnote difference in frequency between C(8-5) and (3-4) approached significance, $(I) p i .06; and C(8-5) was significantly more frequent than C(8-2), x2(1) = 12.10,p < C&-4-2), x2(1) = 25.48, p < .05.
C&.2), 2). The = 4.79, .05. arid
72
CHARLES
E. MILLER
TABLE PROPORTION
OF INITIAL FORM
4
OFFERS
MADE
VARIOUS
BY EACH
POSITION
TO
COALITIONS
Offers made to form Offers made by 8 5 4 2 M
C(8-5)
C(8-4)
C(8-2)
.75 (44) .92 (54) .83
.12 (7) .78 (46) .45
.14 (8) .47 (28) .31
C(S-4-2) .03 (2) .03 (2) .05 (3) .04
Other .oo .05 .19 .47 .I8
(0) (3) (11) (28)
Note. Numbers in parentheses are the actual frequencies of initial coalition offers.
to form C(8-4). Clearly, however, bargaining theory cannot account for 8’s overwhelming preference for C(8-5).5 Turning to the initial offers from the remaining positions, 5 made significantly more initial offers to form C(8-5) than to form C(.5-4-2), x2(1) = 48.29, p < .OOl; 4 made significantly more initial offers to form C(8-4) than to form C(5-4-2), x2(1) = 40.33, p < .OOl; and 2 made significantly more initial offers to form C(8-2) than to form C(.5-4-2), x2(1) = 16.03, p < .OOl. With regard to the division of payoffs in the coalitions, the data are obviously more in accord with the predictions of bargaining theory than with the predictions of minimum resource theory or minimum power theory (see Table 3). The predictions made by the bargaining theory were generally closer to equal divisions of the payoffs than were the predictions made by the minimum resource or minimum power theories. As in Experiment 1, the payoff data were analyzed by calculating 95% confidence limits-in this case, for 8’s mean payoffs of 137 (SD = 9.4) in C(8-5), 143 (SD = 9.6) in C(8-4), and 145 (SD = 24.0) in C(8-2). This analysis revealed that the predictions of bargaining theory were within the confidence limits for all three coalitions, while the predictions of minimum resource theory and minimum power theory were outside the confidence limits for all three coalitions. Also as in Experiment 1, discrepancy scores were calculated for each of the theories. The mean discrepancy scores were 18.6 for bargaining theory, 39.1 for minimum resource theory, and 90.4 for minimum power theory. An additional aspect of some interest regarding the payoff divisions in the coalitions is that 8’s mean payoff was largest in C(8-2), next largest in C(8-4), and smallest in C(8-5), and yet the frequency of occurrence of the 5 The frequency of 8’s initial offers to form C(8-5) differed significantly from the frequency of 8’s initial offers to form C(8-4), ~71) = 26.84,~ i .05, and C(8-2), x*(l) = 24.92, p < .05.
COALITION
FORMATION
73
coalitions ran in the opposite direction. On the face of it, this finding would appear to call into question the validity of the assumption, central to all of the theories, that each individual will attempt to join whichever coalition maximizes his payoffs. Within the present experiment, however, the only direct evidence available regarding this assumption is whether there was some tendency on the part of subjects with the weight g--or any of the other weights-to reject the offer of a relatively large payoff and to accept instead the offer of a relatively small payoff. Therefore, all instances in which an individual received more than one offer at the same time and accepted one of them were examined, in order to see in how many of these instances the individual accepted an offer that was smaller than one of the offers he rejected. There were 70 instances in which 8 received more than one offer and accepted one of them, and in only 9 of these instances (13%) was the accepted offer smaller than one of the rejected offers. There were an additional 39 instances in which one of the other individuals (i.e., 5, 4, or 2) received more than one offer and accepted one of them, and in only 5 of these instances (13%) was the accepted offer smaller than one of the rejected offers. Thus, as far as the acceptance and rejection of offers of coalition are concerned, the in viduals do appear, for the most part, to have tried to maximize their payoffs. GENERAL
DlSCUSSlON
None of the three theories was able to predict accurately, for both of the experiments, which coalition would form most frequently or how the payoffs would be divided. With regard to the formation of coalitions, minimum power theory did accurately predict the most frequent coalition in Experiment 1. The results of Experiment 2, however, in which the theory’s prediction of the most frequent coalition was inaccurate, suggest that the accurate prediction in the first experiment was probably due to the fact that the predicted coalition was the one which got the largest payoff per member. With regard to the division of payoffs, which of the theories was the most accurate appears to have depended, at least to some extent, on which theory made predictions that were closest to an equal division. The predictions of minimum power theory in Experiment 1 and bargaining theory in Experiment 2 were closest to an equal payo sion, and these theories were the most accurate in the respective ments. Tt seems possible that the theories cannot deal adequately with situations such as those studied in the present experiments because the theories tend to attribute too much influence to the distribution of resources and not enough influence to the payoffs for the coalitions. Prior research on the theories, by focusing mainly on situations involvi constant payoffs for winning (so-called “simple” games) has neglect
74
CHARLES
E. MILLER
the possible effects of coalition payoffs and has limited itself to conditions under which the effects of resources are likely to be most pronounced. When there are no differences in payoffs among the winning coalitions, the distribution of resources provides individuals with almost their only basis for deciding which of the coalitions will maximize their payoffs. Numerous studies have shown that under such conditions, resources will affect both the formation of coalitions and the division of payoffs in the coalitions. On the other hand, when there are differences in payoffs among the winning coalitions, but no differences in resources among the individuals, one would expect individuals to be sensitive to the payoff differences. The results of the few previous studies that have examined coalition behavior in characteristic function games indicate that this is in fact the case (e.g., Horowitz & Rapoport, 1974; Kahan & Rapoport, 1974; Rapoport & Kahan, 1976), with coalition payoffs affecting both the coalitions that form and the way that the payoffs for the coalitions are divided. Unfortunately, there is little empirical evidence regarding the important question of what is likely to happen when, as in the present experiments, there are differences in both the resources of the individuals and the payoffs for the winning coalitions (but see Gamson, 1961b; Wolf & Shubik, 1977). Situations of this sort are considerably more complicated than those in which either the payoffs or the resources are the same. Such situations may be accompanied by considerable uncertainty about what constitutes appropriate behavior, either in an individually rational or a socially normative sense. Recent work by Komorita, Chertkoff, and their colleagues (Chertkoff & Esser, 1977; Komorita & Brinberg, 1977) suggests that the weight which individuals give to different norms may vary from situation to situation. It may be that when there are differences in the payoffs for coalitions, the individuals will tend to focus on these differences rather than on any differences in resources which might exist. To the extent that individuals ignore resources, the norm of equity will carry little weight relative to the norm of equality. One fairly easy rule of thumb for an individual to adopt is to try to join the coalition with the largest payoff per member. The greater the weight given to the norm of equality and, hence, the more nearly equal the individuals expect the payoff division to be, the more likely it is that the coalition with the largest payoff per member will be preferred, since it is this coalition which will be expected to maximize the payoffs to all its members. In the two experiments reported here, relatively small differences in coalition payoffs led to relatively large differences in coalition frequencies. The present research suggests a change in focus for the study of coalitions. It clearly points up the importance of studying situations in which both coalition payoffs and individual resources vary. It shows that the minimum power, minimum resource, and bargaining theories cannot ac-
COALITION
FORMATION
75
count adequately for coalition behavior in such situations. It also demonstrates the influence of coalition payoffs on the formation of coalitions and calls into question the heavy emphasis on resources in prior research. Most real-world situations probably involve differences in coalition payoffs as well as individual resources (see Axelrod, 1970), and more attention ought to be paid to such situations. REFERENCES Aumann, R. J., & Maschler, M. The bargaining set for cooperative games. In M. Dresher, L. S. Shapley, & A. W. Tucker (Eds.), Advances in game theory. Annals of Mathematics Study 52. Princeton, N. J.: Princeton Univ. Press, 1964. Pp. 443-476. Axelrod, R. Conjlict of interest: A theory of divergent goals with applications to politics. Chicago: Markham, 1970. Caplow, T. A theory of coalitions in the triad. American Sociological Review, 1956, 21% 271-280.
Chertkoff, J. M., & Esser, J. K. A test of three theories of coalition formation when agreements can be short-term or long-term. Journal of Personality attd Social Psychulogy, 1977, 35, 237-249. Davis, M., & Maschler, M. The kernel of a cooperative game. Navnl Reseurck Logistics Quarterly, 1965, 12, 223-259. Gamson, W. A. A theory of coalition formation, American Sociologicai Review, 1961, 26, 373-382. (a) Gamson, W. A. An experimental test of a theory of coalition formation. American Sociological Review, 1961, 26, 585-573. (b) Gamson, W. A. Experimental studies of coalition formation. In L. Berkowitz (Ed.), Advances in experimental social psychology. New York: Academic Press, 1964. Vol. 1. Horowitz. A. D.. & Rapoport, A. Test of the kernel and two bargaining set models in fourand five-person games. In A. Rapoport (Ed.), Game theory as a theory of conj?ict resolution. Dordrecht: Reidel, 1974. Pp. 161-192. Kahan, 3. P., & Rapoport, A. Test of the bargaining set and kernel models in three-person games. In A. Rapoport (Ed.), Grime theory as n theory ofconjict resolution. Dordrecht: Reidel, 1974. Pp. 119-160. 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. Psycho/ogica/ Review. 1974, 81, 242-256.
Komorita,
S. S., & Brinberg,
Sociometry,
D. The effects of equity norms in coalition formation.
1977, 40, 351-361.
Komorita, S. S., & Chertkoff, J. M. A bargaining theory of coalition formation. Psychological Review, 1973, 80, 149-162. Komorita, S. S., & Moore, D. Theories and processes of coalition formation. Journal qf Personality and Social Psychology, 1976, 33, 371-381. Michener, H. A., Fleishman, J. A., & Vaske, J. J. A test of the bargaining theory of coalition formation in four-person groups. Joarna( of Persona&y and Social Psychology, 1976, 34, 1114-1126.
Murnighan, 3. K., Komorita, S. S., & Szwajkowski, E. Theories of coalition formation and the effects of reference groups. Journal of Experimental Social Psychology, 1977, 13, 166-381.
Rapoport, A. N-person game theory: Concepts and applications. of Michigan Press, 1970.
Ann Arbor, Mich.: Univ.
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Rapoport, A., & Kahan, J. P. When three is not always two against one: Coalitions in experimental three-person cooperative games. Journal ofExperimental Social Psychology, 1976, 12, 253-273. Schelling, T. C. The strategy of conflict. Cambridge, Mass.: Harvard Univ. Press, 1960. Shapley, L. S. A value for N-person games. Annals of Mathematics Studies, 19.53, 28, 307-3 17. Shapley, L. S., & Shubik, M. Method for evaluating the distribution of power in a committee system. American Political Science Review, 19.54, 48, 787-792. Vinacke, W. E., & Arkoff, A. An experimental study of coalitions in the triad. American Sociological Review, 1957, 22, 406-414. Wolf, C., & Shubik, M. Beliefs about coalition formation in multiple resource three-person situations. Behavioral Science, 1977, 22, 99-106.