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F1111? .1...1111......1311191u2o 1...... n. .4911- N 1. . 1 ..1. «1.31.3...111111113141500114 7.21.1.1 1:121... 04h“.fo1~ouh “a . 1-. 0 1 1r: I III. .II- LIBRAR; 'l I f IHlllllllllllllHllllUlllNllllllllllllllllllllllllllllllllll . p..-a;3;;::a:;ffan 31293 10390 4953 ”h V ‘w can’t-mi. l ABSTRACT AN EXAMINATION OF THE POWER INVERSION EFFECT IN THREE-PERSON MIXED MOTIVE GAMES by Steven G. Cole One of the phenomenon predicted by the theories which have attempted to describe behavior in a three person mixed—motive situa- tion is the power inversion effect. The present paper examined the power inversion effect as it relates to both the truel, i.e., a three person duel, and the triadic coalition formation situation. Due to the paucity of studies dealing with the truel, there is a lack of data to support the power inversion effect in that situa— tion. On the other hand, the coalition formation situation has been studied extensively over the last decade and the power inversion effect has been reported by virtually every study. The focus of the coalition research has been on the determini- stic coalition formation situation —- those situations in which some unit or combination of units will win with probability one. The probabilistic coalition situation —- those situations in which no unit or combination of units will win with probability one -— has been virtually ignored. This study extended coalition formation theory to explain individual behavior in the probabilistic situation. To test the proposed theory, the present study utilized a truel game paradigm. Four conditions were examined: (1) A deterministic truel which allowed coalitions, (2) a deterministic truel which did not allow coalitions, (3) a probabilistic truel which allowed coali- tions, and (4) a probabilistic truel which did not allow coalitions. The main hypothesis was that the power inversion effect would be replaced by the strength is strength effect in the probabilistic coalition formation situation. It was also predicted that power in- version effect would be observed in the truel situations in which no coalition was allowed to form and in the deterministic coalition formation situation. The results supported the main hypothesis. However, the support was weakened by the fact that the power inver- sion effect was not observed in the deterministic coalition formation situation. The power inversion effect was observed in the propensity to attack the stronger of the other two players in the truel situations and in the coalition formation situations in which coalitions did not form. However, it was not observed in the relative chance for survival. A theory based on the relative disparity of strengths was proposed which predicts that the power inversion effect, with reSpect to distri- bution of attacks, will occur in the truel situation as long as the strengths of the participants are not equal. When the relative chance of survival is considered, the power inversion effect will be a function of the relative strength of the strongest player. A . , - I: . ~-/ )1/ 1‘" . The31s Committee: Approved:_L ,t;?L¢7 g3 J//&;’:f7‘7 ,/T ,/ James L. Philli 8 Chairman 1; 1 A ‘w -- p ’ Datez‘thguta;Qi 175Lg//6 Q) I 7' 7* Lawrence A. Messe v Donald M. Johnson AN EXAMINATION OF THE POWER INVERSION EFFECT IN THREE—PERSON MIXED MOTIVE GAMES by Steven G. Cole A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF ARTS Department of Psychology 1968 ACKNOWLEDGEMENTS The author wishes to express his sincere appreciation to Dr. James L. Phillips for his encouragement and criticism during the preparation of this thesis. Appreciation is also expressed to the members of my committee Dr. Lawrence Messe and Dr. Donald Johnson. The author also wishes to thank his wife Betty, for her assistance and love. ii ACKNOWLEDGEMENTS . LIST OF TABLES . . LIST OF FIGURES LIST OF APPENDICES INTRODUCTION . . . PROBLEM . . . . . METHOD . . . . . . RESULTS . . . . . DISCUSSION . . SUMMARY AND CONCLUSION REFERENCES . . . APPENDICES . . TABLE OF CONTENTS iii Page‘ ii iv vi 20 25 29 45 6O 64 67 TABLE LIST OF TABLES Caplow's eight triadic types and the coalitions which are predicted by the four major theories. . . . . Who to whom matrix for offers in the Pc and Dc conditions 0 O O O O O O O O O O O O C O O O O O O O O 0 Frequency with which attacks are directed toward the stronger and weaker players for each condition on eaCh trial. 0 O O O O O O O O O O O O O O O O O O O O 0 Games won by players A, B, and C when not in a coalition O O C O O O I O O O O O O O O O O O O O O 0 Frequency with which the desired share of the payoff (desired payoff), the share of the payoff the other player is expected to desire (expected payoff), and the final division of the payoff (final payoff) approximated the parity split or was an equal split in the Pc and Dc conditions. . . . . Frequency with which A was the preferred player or one of the other players was the preferred player in each condition before and after the game was played . . Summary of the analysis of variance on the preferred playing position before and after the game was played . Means and Standard deviations for the pre and post game preferences of player position . . . . . . . . . . Correlations between the dependent variables and the individual difference measurements in the Dc, D, Pc, and P conditions. . . . . . . . . . . . . . . . . . . . iv PAGE 31 33 36 38 40 42 42 44 FIGURE LIST OF FIGURES Page The predicted probability that the stronger of the other two members of a truel will be attacked based on the disparity of relative strengths . . . . . . . . . . . . . . . . . . 53 The predicted probability that the strongest member of a truel will win based on the relative disparity of strengths . . . . . . . 53 APPENDIX LIST OF APPENDICES Page Experimental Materials. . . . . . . . . . . 67 Factor Analysis of Experimenter Evaluation Data . . . . . . . . . . . . . . 88 A Review of Theories and Research Relevant to Coalition Formation in the Triad . . . . 91 vi Introduction One of the prominent phenomena predicted by the theories which have attempted to describe behavior in a three person mixed—motive situation1 is that the person who is the ”strongest" with respect to initial resources is often functionally the weakest. This phenomenon, referred to as the "strength is weakness" effect, has been predicted by theories which deal with coalition formation (Caplow, 1956; Gamson, 1961a) as well as theories which deal with the truel, i.e., a three person duel (Shubik, 1954). However, the literature has virtually ignored the fact that in many cases the person who is the ”weakest" with respect to initial resources is functionally the strong- est, i.e., the ”weakness is strength” effect. It is sensible to think of these two effects as sub-phenomena of a more general ”power inversion" effect. The present paper examined the power inversion effect as it relates to both the truel and the triadic coalition formation situation. Due to the paucity of studies dealing with the truel, there is a lack of data to support the power inversion effect in that situation. On the other hand, coalition formation has been studied extensively over the last decade and the power inversion effect has been reported by almost every study. Therefore the major portion of the present paper concentrated on coalition research. The mixed-motive situation is defined as a situation in WhiCh (1) there is no outcome which will maximize the payoff to all of the participants, and (2) at least two of the participants can increase their payoff by cooperative activity. For this study a coalition is the joint use of resources to determine the outcome of a decision in a mixed-motive situation involving more than two units." (Gamson, 1964, p. 85) Coalition theory and research, which is included within the boundary condition of this definition, has focused on the deterministic coalition formation situation —- those situations in which some unit or combination of units will win with probability one. The probabilistic coalition situation —- those situations in which no unit or combination of units will win with probability one -- has been virtually ignored. The present paper extends coalition formation theory to explain individual behavior in the probabilistic situation. Since the present study proposed to examine the power inver— sion effect in the truel as well as in the coalition situation, the experimental game paradigm was based on the truel. Although it was kept as similar as possible, the basic truel was manipulated so that the following four conditions were examined: (1) a deterministic truel which allowed coalitions, (2) a deterministic truel which did not allow coalitions, (3) a probabilistic truel which allowed coalitions, and (4) a probabilistic truel which did not allow coalitions. Iruel Theory and Research The truel was first discussed by Shubik (1954) and is based on the concept of a three person duel. It is well known that in a duel, two individuals attempt to eliminate each other and the winner is that person who survives. The truel is a similar situation which differs only in that three individuals participate. In the pure truel no eXplicit coalitions may form. This does not, however, exclude the formation of implicit coalitions. In fact, in the truel sit- uation it is often the case that two of the participants may increase their chance of survival by eliminating the third participant first and then attempting to eliminate each other. Shubik prOposed a theory to account for the outcome of the truel situation which was based on the presupposition that participants in a truel will act rationally. Although the possibility of implicit coalitions was noted by Shubik, he restricted his theory so that it did not allow explicit coalitions. One of the major predictions of his theory was that in some truel situations the power inversion effect would be observed. As an illustration, Shubik offered the following example. Person A, person B, and person C were each allowed to fire one shot at one of the other two. Person A had an 80% chance of hitting the person that he chose to shoot, person B had a 70% chance of hitting the person that he chose to shoot, and person C had a 60% chance of hitting the person that he chose to shoot. The rationalistic point of view assumes that each individual wishes to survive and will therefore shoot at that individual who poses the greatest threat to his survival. Shubik adopted this point of view and computed the probabilities of survival for each individual in each of the six possible firing orders. The following mean chances of survival for each individual were obtained: A=.260, B=.488, and C=.820. It is apparent from these results that the strongest person has the least chance to survive and that the weakest person has the best chance to survive. This illustration of the power inversion effect supports the hypothesis that power inversion is predicted in some truel situations, and that those truel situations in which power inversion is predicted are a function of the relative strength of the individuals involved. Therefore, "in a noncooperative environment it apparently does not pay to be slightly stronger than the others for this invites action against oneself." (Shubik, 1954, p.45) Only two empirical studies (e.g., Willis & Long, 1967; Cole & Phillips, 1967b) have examined the truel situation. Although the Willis and Long study utilized the truel situation, the subjects Each had a 100% chance of hitting whichever player they chose to attack. Since the players were all of equal strength, an examination of the power inversion effect in the Willis and Long study was im- possible. Therefore, the Cole and Phillips study furnished the only empirical data dealing with power inversion in the truel. Cole and Phillips used Caplow's (1956) Type 5 triadic situation, i.e., A>B>C, A<(B+C), which means that player A had greater power than player B, player B had greater power than player C, and player A had less power than players B and C combined. An examination of the number of times each player was attacked and who attacked him on all moves of each game, particularly on the initial move of each game, revealed evidence for the power inversion effect. That is, it was found that (1) player A was attacked a significantly greater number of times by both players B and C than either players B or C were attacked by each other; and (2) player C was attacked signifi- cantly less by players A and B than they were attacked by each other. However, in the truel situation which was examined every player could successfully attack the player of his choice with probability one, while in the situation which was discussed by Shubik the probability of successfully attacking the player of his choice was different for each member of the triad. Furthermore, Cole and Phillips allowed coalitions to form which may have added some aspects to the situation which differentiated it from the pure truel. Coalition Formation: Relevant Theories and an Evaluation2 The study of coalition formation has taken two forms characterized primarily by the nature of the task which the subjects are required to perform. One form, represented by the studies by Mills (1954), Strodtbeck (1954), and Torrance (1955), presents a group discussion task in which a coalition is said to have formed if two subjects come to an agreement. The procedure utilized by Mills (1954) offers an excellent illustration of the type of group discussion task that has been used. Mills asked the members of a triad to pretend that they were on a "military review court sitting on the case of Billy Budd." (p. 658) Mills used two confederates to manipulate the situation and noted with whom the subjects agreed as an index of coalition formation behavior. The other form of coalition research, characterized by Caplow (1956), Vinacke and Arkoff (1957), and Gamson (1961a), deals with situations in which the subjects play a mixed-motive game. The data is examined to determine if a manipu- lation of the initial resources effects the outcome of the game, how the game is played, or both. The first type of coalition studies 2 O O O O 0 For a more exten31ve rev1ew of the coalition theories and research see Appendix C. (coalitions in a group discussion task) has been thoroughly re- viewed by Marie L. Borgatta (1961) and will not be discussed in this paper. The present paper will focus on the second form of coalition Studies, i.e., those dealing with mixed—motive games. Caplow's (1956, 1959) theory of coalition formation provided the impetus for the theories which have subsequently been pr0posed. Table 1 presents Caplow's eight triadic types and the coalitions which are predicted by the four major theories of coalition formation, i.e., Control Theory (CT), Minimum Resource Theory (MRT), Minimum Power Theory (MPT), and Anticompetitive Theory (AT). CT was developed by Caplow (1956, 1959). MRT, which was deve10ped by Gamson (1961a) is a refinement of CT. MPT is based on game theory and was proposed by Gamson (1964). AT was first prOposed by Gamson (1964) and was later revised by Phillips and Nitz (in press). The eight triadic types presented in Table 1 were based on the relative power of the members of the triad. For example, in the Type 5 triad, player A has more power than player B, and player B has more power than player C; but player A has less power than player B and player C combined. Subsequent research has defined power to be equivalent to the amount of resources controlled, and resources have been defined as weights assigned to each member of the triad such that some critical quantity of those weights is necessary to control the distribution of the payoff. Table l Caplow's eight triadic types and the coalitions which are predicted by the four major theories. Coalitions predicted by CT MRT MPT AT* Type of Triad A B 1 A=B=C ...... ........ any any any any any 2 A>B, B=C, A<(B+C)... BC BC any BC AB or AC 3 A(B+C), B=C ........ none --- none none none 5 A>B>C, A<(B+C) ...... AC or BC BC any AB or BC AC 6 A>B>C, A>(B+C)...... none —-- none none none 7 A>B>C, A=(B+C). ..... AB or AC —-— AB or AC AB AC 8 A=(B+C), B=C' ........ AB or AC -—— AB or AC AB or AC AB or AC * = the predicted coalitions when the payoff is equally divisible. B = the predicted coalitions when the payoff is not equally divisible. Control Theory Control theory defines the term control as the ability to domi— nate the other members of the triad. For example, if one member of a triad has some part in determining the outcome of the situation and another member does not, then the first member is said to have control3 over the second. Moreover, if two members of a triad have some part 3This type of control is similar to "fate control” (Thibaut & Kelley, 1959) in that once a coalition has formed, the fate of the isolate is determined. in determining the outcome of the situation, then the stronger member, i.e., the member with the larger resource weight, is said to have control over the weaker member, i.e., the member with the smaller resource weight. The major assumption of CT is that each member of the triad will attempt to gain control over both of the other members if possible. If he cannot control both of the other members, he will attempt to control only one. Therefore, in the triadic types which do not contain an all powerful member, i.e., a dictator, coalitions will form. Which coalition will form is determined by the relative power of the members of the triad. For example, consider the Type 5 triad. Since A>B>C, player A is indifferent with reSpect to B and C as coalition partners because he can gain control of both of the other members by forming a coalition with either player B or player C. However, player B must form a coalition with player C if he is to gain control over both of the other members of the triad. An appraisal of player C's power reveals that he must form a coalition to gain control over one of the other members of the triad. Therefore, player C has no preference between player A and player B. As a result CT predicts that either AC or BC coalitions will form in the Type 5 triad. Minimum Resource Theory MRT is based on the assumption that the ”parity norm" will be salient, i.e., "any participant will expect others to demand from a coalition a share of the payoff proportional to the amount of resources which they contribute to a coalition." (Gamson, 1961a, p. 376) As a result, each member of the triad will attempt to form the winning coalition to which he contributes the greatest share of the resources, 9 i.e., the "cheapest winning coalition." MRT also assumes that reciprocal partner preferences are necessary before a coalition will form. The combination of the parity norm and the requirement of reciprocal partner preferences results in predictions which are more precise than the predictions of CT. For example, the fact that in the Type 5 triad player A wishes to form a coalition with player C does not mean that AC coalitions are predicted in the Type 5 triad. Similarly, the fact that player B wished to form a coalition with player C does not mean that BC coalitions will form. The critical factor is which player does player C prefer as a coalition partner. Since player C would contribute a larger share of the resources to a BC coalition than he would to an AC coalition, and since both coalitions are winning coalitions, it is predicted that player C would prefer player B as a coalition partner. As a result, MRT predicts that BC coalitions will form in the Type 5 triad. A comparison of CT and MRT reveals that there are situations in which only MRT predicts the power inversion effect. For example, consider the Type 5 triad. Although CT predicts that the "weakest” player (C) will be included in all coalitions, it does not predict that the "strongest" player (A) will be excluded from all coalitions. In fact, CT predicts that player A will be included in 50% of the coalitions. Therefore, the "weakness is strength" effect is predicted, but the "strength is weakness" effect is not. Thus by definition CT does not predict the power inversion effect in the Type 5 triad. On the other hand, MRT predicts that both the "strength is weakness" effect and the ”weakness is strength" effect will be observed in the 10 Type 5 triad. Thias is, player A will not be the preferred coalition partner of either of the other two players, and player C will be the preferred coalition partner of both of the other two players. As a result, player A will not be included in any predicted coalitions, i.e., ”strength is weakness", and player C will be included in every coalition, i.e., "weakness is strength.” Therefore, the power inversion effect is predicted by MRT. Minimum Power Theory MPT is based on the assumption that the participants in a coalition situation utilize a rational strategy to maximize their control over the outcome. It is also assumed that the real power of each participant is accurately perceived by all of the participants in a given situation. The rational strategy which the subjects are assumed to utilize is based on the real power of the members of the triad. Real power as defined by MPT is different from power as de- fined by CT. That is, MPT utilizes the Shapely (1953) notion of pivotol power. In triadic coalition situations the pivotol power of an individual is equivalent to the number of winning coalitions which he may enter. If the notion of pivotol power is utilized, only three power relationships can occur in the triad: (1) all three members are of equal power, i.e., any two members can coalace and control the outcome, (2) one member is all powerful, or (3) one member has veto power. MPT predicts that if all members of a triad are of equal power, then all possible coalitions will be equally likely; if one member of a y-l bub fJ (2) FL) 1 .1311 tha as: ll triad is all powerful, no coalition will form; and if one member of a triad has veto power, he will be included in every coalition. An examination of the Type 5 triad reveals that all three members of the triad are of equal power. Therefore, MPT predicts that all coalitions are equally likely in the Type 5 triad. Anticompetitive Theory AT was first suggested by Gamson (1964); however, it was based on the results of a series of studies by Vinacke and his students. (Vinacke, 1959, 1962; Bond & Vinacke, 1961; Uesugi & Vinacke, 1963; Vinacke, Crowell, Dien, & Young, 1966). Vinacke (1959) hypothesized that ''female subjects might be much less concerned with winning and more oriented toward social and ethical considerations —— i.e., try to avoid an aggressive display of power and attempt rather to be 'fair' to everyone.” (p. 344) This hypothesis and subsequent research lead Gamson (1964) to postulate that an ”anticompetitive norm" exists and that as a result "coalitions will form along the lines of least resistance in bargaining." (p. 90) Therefore, Gamson's AT predicts that coalitions will form between players who are equal in resources. An extension of AT (Phillips & Nitz, in press) suggests that the predictions of which coalitions will form should be based on the divisibility of the payoff as well as the initial distribution of resources. That is, if the payoff can be divided equally, the coalition which requires the least bargaining is between the members of the triad who have the smallest resource differences. On the other hand, if the payoff cannot be divided equally, the least bargaining 12 will be required for the coalition which includes those members of the triad who have the greatest resource difference. Therefore, if the payoff can be divided equally, coalitions will form between the members who are the most similar with respect to resources. If the payoff cannot be divided equally, coalitions will form between those members who are the most dissimilar with respect to resources. It is apparent that AT predicts that when the payoff is equally divisible AB and BC coalitions are equally likely in the Type 5 triad. On the other hand, if the payoff is not equally divisible, AC coalitions are predicted. Evaluation of Theories Reference to Table 1 indicates that the Type 5 triad (A>B>C, A<(B+C)) is the only triadic situation which differentiates between the four theories. That is, for the Type 5 triad CT predicts that AC and BC coalitions are equally likely, MRT predicts that BC coalitions will form, MPT predicts that all coalitions are equally likely, and AT predicts that if the payoff is equally divisible, AB and BC coalitions are equally likely and if the payoff is not equally divisible, AC coalitions will form. Most of the research on coalition formation has examined the Type 5 triad. As a result, the following comparison of the theories will concentrate on how well the data obtained from the research on Type 5 triads is pre- dicted by each of the four theories. However, where it is deemed important, the data from other triadic types will be reported. The early research on coalition formation in the triad l3 (Vinacke & Arkoff, 1957; Vinacke, 1959; Chaney & Vinacke, 1960; Bond & Vinacke, 1961) seemed to support CT. At the same time, there was a general refutation of MPT. However, the fact that there was a tendency for more BC coalitions than AC coalitions to form in the Type 5 triad could not be accounted for by CT. MRT, on the other hand, does account for this effect. Furthermore research subsequent to Gamson's (1961a) develOpment of MRT (Vinacke, 1962; Uesugi & Vinacke, 1963; Chertkoff, 1966; Vinacke, Crowell, Dien, & Young, 1966; Cole & Phillips, 1967a, in preparation; Phillips & Nitz, in press; Cole, Nitz, & Phillips, in preparation) has shown support for MRT. That is, most of the research has shown some support for the hypothesis that the parity norm is salient in the coalition situation. Moreover, a power inversion effect has appeared in almost every study. Thus MRT has proven to be the most reliable theory which has been proposed to account for behavior in the coalition situation. AT has been supported to some extent by most of the research which has examined feminine behavior in the coalition formation situation. (Vinacke, 1959, 1962; Bond & Vinacke, 1961; Uesugi & Vinacke, 1963; Cole & Phillips, in preparation; Cole, et. al., in preparation). The support for AT has been manifest in the strategy which seems to be employed by females. There is a consistant tendency for females to employ a strategy which results in a weaker power inversion effect than is obtained in the situations which use all male triads. Research which has attempted to test AT as it relates to masculine behavior has concentrated on the Type 2 and Type 3 triads 14 (Phillips & Nitz, in press). The results of the Phillips and Nitz study supported MRT over AT; however, an indication that an ”anti- competitive norm” may exist was noted. Since AT has received sufficient support to indicate that an anticompetitive norm is Operating in the coalition situation, it is necessary to account for the resulting effect. However, it is not felt that the effect of the anticompetitive norm is strong enough to necessitate a theory based on it. It is proposed that a better method of acknowledging the existence of an anticompetitive norm would be to extend MRT so that it accounts for the possibility that there are individuals who do not wish to strictly maximize their share of the reward. Instead, they desire to divide the reward as "fairly" as possible. Therefore, in those triads which include one or more individuals who play a strategy based on the anticompeti- tive norm, the coalitions which form may not be a function of the parity norm. They may be a function of the anticompetitive norm and in some cases, both norms combined. For example, consider the Type 5 triad in which the strategies of player A and player B are dictated by the parity norm and in which player C's strategy is a function of the anticompetitive norm. MRT predicts that players A and B will both prefer player C as a coalition partner. If the payoff can be divided equally, player C would prefer player B and a BC coalition would be predicted. However, if the payoff can not be divided equally, player C would prefer player A and an AC coalition would be predicted. In many situations this could account for those coalitions 15 which occur and are not predicted by MRT. There are two major criticisms of MRT which are evident to the present writer. One criticism of MRT which can be made of most psychological theories, is that the predictions are too strong. This results from the fact that if MRT is strictly interpreted, only BC coalitions are predicted to form in the Type 5 triad. However, there have been no studies which have reported only BC coalitions. Two attempts to rectify this criticism, i.e., Chertkoff (1967) and Shelly and Phillips (1966) have pr0posed to do so by developing a mathematical model. Both the Shelly and Phillips model and the Chertkoff model are based on the assumption that the probability that a simultaneous reciprocal contact will occur is an important variable to consider when determining which coalitions will form. Both models have had some success but neither the Chertkoff model nor the Shelly and Phillips model was able to satisfactorily account for the coalitions which form. The second major criticism of MRT is that the boundary condi- tions are too narrow. This is indicated by the fact that there are variables that weaken the power inversion effect which have not been taken into account by MRT. In order to extend MRT so that it will cover a wider range of situations, it is necessary to examine those variables. One procedural variable which has weakened the power inversion effect is the use of cumulative score. (Vinacke, 1959, 1962; Kelley & Arrowood, 1960; Cole & Phillips, in preparation). The studies which have examined cumulative score had the subjects play a series 16 of games in which the score for each subject at any point in the series of games was the amount of the payoff that he had accumulated. The winner was the player who accumulated the most points over all games. It is suggested that the amount of the payoff that each subject had accumulated at any point in the series of games must be considered when the relative strengths of the members of the triad are determined. In all games after the first game the strength of each member of the triad is a combination of his assigned resources plus his accumulated rewards. Therefore, in the first game the power inversion effect is manifest by the propensity to form coalitions between the two members of the triad who are weakest with reSpect to assigned resources. However, as the series of the games progresses, the tendency is for coalitions to form between the two players who are behind in accumulated score (Vinacke, 1959, 1962). This indicates a continuance of the power inversion effect with the strength of each member of the triad being determined by his accumulated rewards rather than his assigned resource weight. Therefore, MRT can account for the effects of cumulative score by redefining the concept of resources. Previously a resource has been defined as a weight assigned to each player in a game such that the distribution of the payoff is de- termined by some critical quantity of that weight. It is suggested that a given weight which is assigned to an individual player is a value on some resource dimension. The total resources of the players in.a given situation consist of values on a combination of resource dimensions, i.e., multiple resource dimensions. Therefore, one of 17 the key variables which MRT must consider is the concept of multiple resource dimensions. One manipulation which has removed the power inversion effect from the coalition situation is found in a study by Chertkoff (1966). Chertkoff used the "political convention paradigm"4 for the study of coalitions in the triad. However, he manipulated the probability of future success by varying the probability that a coalition would win the election after it had won the nomination. In addition he made the payoff contingent upon winning the election. The Type 5 triad was examined in all conditions, i.e., A>B>C, A<(B+C). In all three experimental conditions, if player B or C won the nomination, he had a .5 probability of success in the national election. If player A won the nomination, his probability of winning the national election was .5, .7, or .9, depending upon the condition. A control condition was used in which the probability of future success was not a feature. In the control condition the power inversion effect was reported. However, in the conditions in which player A had a .5 or a .7 probability of future success the power inversion effect was weakened, 4In the political convention paradigm each subject plays the part of a delegation chairman at a political convention. Each subject is assigned a given number of votes at the beginning of the convention. For each subject the object is to acquire as many "jobs" for his faction as he can. To acquire "jobs" the subjects must form a coalition which can win the nomination. The number of "jobs" that each subject acquires if he has entered into a winning coalition depends on the division of the "jobs" that the two subjects who formed the coalition agree upon. If a subject is not in a winning coalition, he receives no "jobs" for his faction. (Gamson, 1961b) l8 i.e., the distribution of preferred coalition partners was not different from what would be expected by chance. In the condition in which player A had a .9 probability of future success the power inversion effect was nonexistant. That is, a "strength is strength” effect was reported. Moreover, as the probability of future success increased the parity norm became more salient. This was indicated by the tendency for the player with the highest probability of future success to demand and receive a larger share of the payoff. An attempt to explain why the results vary when the probability of future success is manipulated leads to a differentiation of probabilistic and deterministic situations. Probabilistic situations are those in which no unit or combination of units will win with probability one. Deterministic situations are those situations in which some unit or combination of units will win with probability one. Prior to Chertkoff's study the theories and research had all focused on the deterministic situation. However, Chertkoff's study extended the research on coalition formation to the probabilistic situation. Subsequent to Chertkoff's study only one study (Vinacke, Lichtman, & Cherulnik, 1967) has examined the differential effects resulting from the probabilistic and deterministic nature of the coalition for— mation situation. The major finding of the Vinacke, et. al., study ‘was that there is a propensity to "gamble" in the probabilistic situa- tion, i.e., there is a tendency to play without forming a coalition. Dioreover, the propensity for a subject to play the game alone increases as the probability that he will win alone increases. As a result of the research which has examined the probabilistic l9 situation, it is apparent that behavior in the probabilistic situation differs from behavior in the deterministic situation. Therefore, any theory which offers a comprehensive explanation of behavior in the coalition formation situation must consider the probabilistic situation as well as the deterministic situation. Problem Research on coalition formation has consistently reported the power inversion effect in those situations for which it is predicted. However, as has been noted, Chertkoff (1966) replaced the power in- version effect with the "strength is strength" effect by making the payoff contingent on chance. That is, rather than designing the situation so that any coalition would win, Chertkoff designed the situation so that each coalition had a different probability of winning. A consideration of the possible effects of the probabilistic and deterministic aspects of the coalition situation indicated that two types of coalition situations must be accounted for. The first type was the deterministic situation and was defined as those situations in which some unit or combination of units will win with probability one. The second type was the probabilistic situation and was defined as those situations in which no unit or combination of units will win with probability one. The present study extended the theories on coalition formation from the deterministic situation which has been the focus of previous coalition theories to the probabilistic situation which has been virtually ignored. Extension of MRT As a base for a theory that is applicable to the probabilistic situation the assumptions and definitions of MRT were adepted. In essence this means that the parity norm and the requirement of reciprocal partner preferences are presumed to be in effect in the probabilistic situation as well as in the deterministic situation. 20 21 However, the "cheapest winning coalition" is not a useful concept in the probabilistic situation, since, in the probabilistic situation there is no "cheapest winning coalition." This follows from the definition of the probabilistic situation which rules out any a_priori "winning coalitions" as defined by Gamson (1961a), i.e., a coalition with sufficient strength to dictate the terms for the distribution of the payoff. Since there can be no "winning coalition", there can be no "cheapest winning coalition." The major assumption which was added to MRT so that it would predict behavior in the probabilistic situation was that in all situations the members of a triad will form the coalition which maximizes their chance of winning. Therefore, in the probabilistic situation the parity norm, although salient, has little effect in determining which coalition will form. Rather, the uncertainty of the situation fosters a need for security which will be designated the "security norm". Since any coalition will win with probability one in the determi- nistic situation, the members of the coalition will be secure, i.e., will win, regardless of which coalition is formed. In the deterministic situation the only manifestation of the security norm is that the members of the triad will form coalitions. The coalition that is preferred will be a function of the parity norm. As a result, if for any two possible coalitions, the probability of their winning is one, each player will prefer the coalition to which he contributes the largest share of the resources. The probability of winning is what determines the strength of 22 the players in a probabilistic situation and no player or coalition is guaranteed to win. Therefore, the security norm will be salient in the probabilistic situation and will influence preferred coalitions. If coalitions have different probabilities of winning, each player will prefer that coalition which maximizes his chance of winning. In order to maximize his chance of winning each player must form a coalition with the stronger of the other two players. Thus, it is apparent that the strong member of the triad will be included in all preferred coalitions in the probabilistic situation while the weak man will be the least preferred coalition partner. This means that in the probabilistic situation the "strength is strength" effect is predicted as Opposed to the power inversion effect which is predicted in the deterministic situation. Even though the security norm will be salient in the probabi- listic situation with respect to the preferred coalition, the parity norm will remain salient for the division of the payoff. This hypothesis is made because of the reasoning that the greater preference for the strong member of the triad as a coalition partner will give him a better bargaining position. Therefore, he will demand and receive the largest share of the payoff. _flypothesis As a result of the data which have been collected in the past few years, it was predicted that in the deterministic situation which allowed coalitions the power inversion effect would be observed. However, it was predicted that in the probabilistic situation which allowed coalitions the "strength is strength" effect would be observed. 23 That is, in the probabilistic situation each member of the triad will wish to form that coalition which maximizes his chance of winning. Therefore, he will prefer that coalition which maximizes the strength of the coalition relative to the isolate. As a result the stronger of the other two members of the triad will be the preferred coalition partner and the " strength is strength" effect will appear. A further prediction was that in both the deterministic and probabilistic pure truel situations, and the coalition situations in which no coalitions formed, the power inversion effect would be observed. This followed from the assumption that the players would attempt to eliminate those players who had the best chance to win, since by this strategy each player would maximize his chance of winning. As a result the weaker attack choice5 would be attacked the least and the stronger attack choice6 would be attacked the most, i.e., power inversion. The final prediction was that the parity norm would be salient in both situations which allowed coalitions. Specifically, the hypotheses tested in the present research were: (1) In the probabilistic coalition formation situation each player prefers the stronger of the other two players as a coalition partner, resulting in AB coalitions being the most frequent. 5The weaker attack choice for player C is player B, and the ‘weaker attack choice for players A and B is player C. 6The stronger attack choice for player A is player B, and the stronger attack choice for players B and C is player A. 24 (2) In the deterministic coalition formation situation each player prefers the weaker of the other two players as a coali- tion partner, resulting in BC coalitions being the most frequent. (3) In the pure truel situations and those games in which coalitions were permitted but not formed the power inversion effect would appear in the following manner. First, there would be a prOpensity for each player to attack the stronger of the other two players, and second, player A would win significantly fewer games than player B, and player B would win significantly fewer games than player C. (4) The parity norm would be evident in both the probabi- listic and deterministic situations which allowed coalitions. That is, the division of the payoff would tend to approximate the ratio of the resources contributed to the coalition by its members. Method Subjects. The subjects were 255 male undergraduate volunteers who received course credit for their participation. In addition, they were informed that they would have a chance to win $10 if they parti- cipated in the experiment. Apparatus. The apparatus for the game was 45 white poker chips; three wooden tokens, one inscribed with an A, one inscribed with a B, and one inscribed with a C; and a die. Plain white 3" X 5" scratch pads were used for all communication, and clipboards were used to allow the communication to remain secret. Game Design. The structure of the basic game was as follows. Each of the players began the game with 15 chips. The position that each subject played was determined by chance, i.e., each subject drew a token (marked either A, B, or C) from a hopper and played the game in the position indicated on the token. The rules of the game required that each player take away a given number of chips from one of the other two players on each move. Player A took away 4 chips, player B took away 3 chips, and player C took away 2 chips. All of the chips that were removed were taken out of the game and did not belong to any of the players. Each player wrote down the letter of the player he wished to attack and gave the message to the ex— perimenter on each move. At no time was a player's choice known 'before every player had made his choice, i.e., the moves in the game Ivere simultaneous. The players were told who had attacked whom after the experimenter had received the choices from all of the players. Procedure. Four conditions were examined: (Dc) a deterministic 25 26 game which allowed coalitions, (D) a pure truel deterministic game, (Pc) a probabilistic game which allowed coalitions, and (P) a pure truel probabilistic game. The same basic truel game was used to keep the conditions as similar as possible while varying enough to allow the game to be defined as either deterministic or probabilistic. Eighty-five triads participated in the experiment. Five triads (one Dc, one D, two Pc, and one P) were discarded either because the subjects had previous knowledge about the experiment or because two of the subjects in a triad knew each other. Therefore, there were twenty triads in each condition. Each triad played one game. Communication was limited, i.e., all messages were written and directed toward the experimenter. A pre-game questionnaire was administered to provide a measure of the subjects acceptance of the implied power structure. (See Appendix A for the details of the pre-game questionnaire) In the Dc and D conditions, each player took chips away on every move. In the Pc and P conditions, whether a player took chips away or not was determined by chance, i.e., each player rolled a die after he had decided which player to attack, if the die came up even he took the chips away, if the die came up odd he did not take the chips away. This meant that each player had a 50% chance (If taking chips away on each move in the Pc and P conditions. In the Dc and Pc conditions each subject was given a chance to forum a coalition prior to the actual play of the game. They were askxed to indicate on a form provided (see Appendix A), whether they 27 wished to form a partnership or not, and if so with whom. If two of the subjects indicated a reciprocal preference, they were given two minutes in another room to reach an agreement on how to divide the payoff. The payoff was a $10 prize which the winner or winning coalition in each game had a chance to win in a raffle. If no agreement could be reached in that two minutes, the game was played as a truel. If an agreement was reached, the partnership started the game with 15 chips and the power to take away the number of chips which was equal to their combined power, i.e., an AB partnership could take away 7 chips, an AC partnership could take away 6 chips, and a BC partnership could take away 5 chips. This meant that in the Dc condition any partnership was guaranteed to win. In the Pc condition, although a partnership did have a better chance to win than a lone player, a partnership was not guaranteed to win. For example, if an AB partnership formed in the Pc condition it had a 50% chance of removing 7 chips while C, the isolate, had a 50% chance of removing 2 chips. The expected number of trials for an AB partnership to eliminate player C was 3 while the expected number of trials for C to eliminate an AB partnership was 8. In all four conditions the game was over when only one player (Ir partnership had chips remaining. The winner or winning partner- stiip received a chance to win $10 in a raffle. After the game was completed all of the subjects were re- qiuasted to fill out a post-game questionnaire. (See Appendix A) 28 In addition, to ascertain whether there was any effect due to the subjects perception of the experimenter, the subjects were asked to evaluate the experimenter on a 19 scale semantic differential. (See Appendix A) The subjects were also asked to complete the "Survey of Interpersonal Values” (Gordon, 1960). This was utilized to determine whether there were any effects that were due to personality differences. Results Results Relevant to Hypothesis 1 Hypothesis 1 predicted that in the Pc condition each player would prefer the stronger of the other two players as a coalition partner resulting in AB coalitions being the most frequent. Table 2 presents a who-to-whom matrix for offers in the Pc condition. An analysis by chi-square supported the hypothesis that the stronger of the other two players would be the preferred coalition partner in the probabilistic situation (X2 = 8.43, p<.05). Further support for the ”strength is strength" hypothesis was observed in the significant difference in number of offers received by each player. (X2 = 11.29, p<.Ol). Player A received more offers than player B and player B received more offers than player C. Since only three coalitions were formed (2 AB coalitions and 1 BC coalition), a test of the second part of hypothesis 1, i.e., AB coalitions will be the most frequent, was not feasible. Results Relevant to Hypothesis 2 Hypothesis 2 predicted that in the Dc condition, each of the players would prefer the weaker of the other two players as a coali- tion partner, resulting in BC coalitions being the most frequent. Table 2 presents a who—to-whom matrix for offers in the Dc «condition. An analysis by chi—square indicated that there was no Siignigicant difference from chance expectancy with respect to who prxeferred whom as a coalition partner (X2 = 3.25, .50>p>.30). Thiriflner refutation of the power inversion hypothesis was observed in 29 30 the distribution of offers received, i.e., the distribution of offers received did not differ from chance expectancy. An examination of type of coalitions formed revealed 6 AB, 2 AC, and 2 BC coalitions. Al- though an accurate statistical evaluation of the situation is not feasible because of the small number of coalitions that were formed, it seems clear that the power inversion hypothesis was not supported. The data in Table 2 suggested that there was a tendency for the preferred coalition partner to be the same in the Pc and Dc conditions. Since the Pc and Dc conditions were independent, the coalition partner preference in the Dc condition was compared with the coalition partner preference in the Pc condition by computing an F-ratio of the chi— squares (Lindquist, 1956, p. 39). The F-ratio implied that there was no significant difference in preferred coalition partner in the Pc and Dc conditions (F=2.59, NS). Chi—squares to test the hypothesis that each player makes the same number of offers in the Pc and Dc conditions revealed a significant difference between players in the Pc condition (X2 = 7.23, p<.05), but no significant difference between players in the Dc condition (X=.37, .90>p>.80). An F-ratio of chi- squares indicated that there was a significant difference between the distribution of number of offers made in the Pc condition and the distribution of number of offers made in the Dc condition (F=9.62, p<.05). Although there was no significant difference between the total number of offers made in the Dc condition and the total number of offers made in the Pc condition, there was a significant difference between total number of coalitions formed in the Pc condition and total number of coalitions formed in the Dc condition (X2=3.88, p<.05). 31 Table 2 Who—to—whom matrix for offers in the Pc and Dc conditions Pc Dc Recipient Recipient 2 2 A B C Tl X A B C Tl X A 0 5 0 5 5.00** O 8 3 11 2.27 Offerer B 7 0 2 9 2.78 8 0 13 .69 10 7 0 17 .65 8 o 14 .29 8.43** 3.25 T2 17 12 2 31 16 14 8 38 *p<.01 ** xi = 7.23** xi = .37 p<°05 1 1 xi = 11.29* xi — 2.73 2 2 xi = 1.67 3 tests the hypothesis that each player makes the same number of offers. (df=2) 13‘... X tests the hypothesis that each player received the same number of offers. (df=2) HN tests the hypothesis that the total number of offers in A“... the Dc condition equals the total number of offers in the Pc condition. (df=l) 32 Results Relevant to Hypothesis 3 Hypothesis 3 predicted that in the pure truel situations and in those games in which coalitions were permitted but did not form, the power inversion effect would appear in the following manner. First there would be a prOpensity for each player to attack the stronger of the other two players; and second, player A would win significantly fewer games than player B, and player B would win significantly fewer games than player C. Table 3 presents the frequency with which attacks were directed toward the stronger and weaker player for conditions Dc, D, Pc, and P on each trial. To test hypothesis 3 adequately would have required that the relative strengths of each member of each triad be determined for each trial. To accurately determine the relative strengths of each member of each triad on a trial by trial basis would have required that the number of chips that the players had remaining on a given trial be taken into consideration. Since such an analysis would have been prohibitive, the analysis of the data on a trial by trial basis was conducted as if the number of chips that the players had re- maining on any given trial had no effect on the relative strengths of the players. Therefore, the analysis of the attack data for initial trial was the only analysis which was based on an accurate assess— ment of the diSparity of relative strengths. For all other trials the initial strength of each player was taken as his strength on that trial. An examination of the data in Table 3 revealed a significant power inversion effect on the initial trial in each condition, i.e., an... HCHNH. 33 .chHquaou om was on onu dH pocHmeo mums mGOHuHHmoo snow uoc vHuanoHns mUMHuu mmofiu %Hao H .mlm mHmHuu do mGOHqucoo m dam om ozu CH memo onu mH meMuum mo GOHuanHume onu umsu mHmonuomm: onu mumou UNN .m HMHHu co mGOHquaoo m pew .om .Q asp CH meow mnu mH mxomuum mo GOHuDQHuumHo onu umnu mHmmnuomhn onu mumou QNN .meHuu know umHHm man do chHUHwaoo HHm cH meow man mH mxomuum mo :oHunnHuume onu umnu mHmmnuom%£ mnu mumou me "muoz Hump «o. oq. Hw.H co. uoNM Numo 9V“ 0 " wwwvahi 3 Mm” 24 84. N. madman mo.va« w o .N II II mm. II 2 II *¢N.q we. «NN.¢ .xoq.m.. NN.m ,sz.q NN N e N NH HN mH oN NN HN mN NN Hoxmoz m a m S 2 AN mm 3 mm am 2 mm umwcobm II II aoq.o II nw.H. mm. , oo.H «mm.q oo.m om. xxwo.n...nm.N «amoqw,*««mN.qH ..... NM. H N H m m w mH MH wH NN 0H 0N mH NH umxmoz um N H m m 0H HH HN 0N om mN mm Hm mm mm Hmwcouum mm. «wo.q. 00.. .«oq.m. «a~o.w .NN 0 NH NN HN mH memos a m S mm mm 3 Hmwcohm 1|. .NHq. mH. ...mm.m1«««mH.0H NN N HH «H OH q HoMMo3 on e 2 3 ON 3 8983.0. «H mH NH HH 0H. m ..... w n. ..o m. q. n N H (HwHua .HHmHuu comm do GOHuHuaou sumo you ampthm uoxmoB m mHQMH paw Hmwcouum map oum3ou poHUOHHv mum mxumuum SUHSB nuHB kocmsvmum 34 the stronger player was attacked more often than the weaker player. A trial by trial analysis revealed that there was no significant power inversion effect on any trial after the initial trial in the Dc condition. In the D condition there was a significant power inversion effect on trials 2 and 4. In the Pc condition there was support for the power inversion hypothesis on trials 2, 4, 7, and 12. In the P condition the power inversion hypothesis was supported on trials 3, 4, and 6. To further test the power inversion hypothesis a sign test was used to examine all trials for each condition. If the strong man was attacked more than the weak man a plus was assigned to the trial. In the Dc and D conditions, there were five pluses in five trials. The probability that this would have occurred by chance was less than .05. In the Pc condition there were 12 pluses in 14 trials (p<.01), and in the P condition there were 9 pluses in 11 trials (p<.01) To compare the distribution of attacks between the four con- ditions, a chi-square was computed on all trials in which the eXpected attack frequencies were greater than 5. (see Table 3) This meant that all four conditions were compared on the first four trials, conditions D, Pc, and P were compared on trial 5, and condi- tions Pc and P were compared on trials 6—9. With the exception of the initial trial (X2= 6.28, .10>p>.05) the chi-squares revealed no indication that there was any difference between conditions with Irespect to the distribution of attacks. When the initial attack data for the combined Dc and Pc conditions were compared with the initial 35 attack for the combined P and D conditions, a significant difference was observed (X2= 4.95, p<.05, df=l). The second part of hypothesis 3, i.e., that player A will win fewer games than player B and that player B will win fewer games than player C, was partially supported by the data. Table 4 presents the number of games won by each player in each condition. An analysis by chi—square revealed support for the "weakness is strength" hypothesis in the Dc condition (X2= 8.33, p<.02), however, the power inversion effect was not reported in any condition. An examination of Table 4 indicated that there mignt be a difference between coalition and no-coalition situations in the distribution of games won. Therefore, F-ratios of the chi-squares were computed between the Dc and D conditions, the Dc and Pc conditions, the Pc and P conditions, and the D and P conditions. The resulting F-ratios revealed a significant difference between the Dc and D conditions (F= 69.42, p<.01) and no other significant differences. The significant difference between the Dc and D condition and the lack of a significant difference between the Po and Dc conditions or the D and P conditions was taken as support for the suggestion that a difference may exist between the conditions which allowed coalitions and those that did not allow coalitions. As a result the data from the Pc and Dc conditions were combined and the data from the P and D conditions were combined. A chi—square on the combined data indicated that there was a non-significant tendency for the power inversion effect tn) appear in the truel situations which allow coalitions (X2= 5.82, .(15p>.05). There was an in— sufficient amount of data to compute a chi-square for Pc-final payoff. Results Relevant to Perceived Relative Power To ascertain whether the subjects perceived player A to be the strongest player before the game began, each subject was asked to fill out a pre-game questionnaire. The questionnaire asked two questions: (1) "Which player do you think has the best chance to win?", and (2) "Which player would you choose to be, if you had your choice?" A significant positive correlation between the answers to the two questions in all conditions was obtained (r=.79 in Dc and D conditions, .59 in the Pc condition, and .80 in the P condition, p<.0005 for all conditions). Therefore, only question number 1 was used to determine which player was preferred prior to the game. A similar question on the post-game questionnaire was analyzed to determine which player was preferred after the game had been played. Table 6 presents the frequency with which player A was the preferred position before and after the game was played. Analysis by chi-square indicated that player A was also the preferred position after the game had been played in conditions Dc, Pc, and P. Since the post-game data were not independent of the pre—game data, a three factor analysis of variance design with repeated measures Vfias employed (Weiner, 1962, p. 337) to test the pre-game post—game (finanges in position preference. Table 7 presents the summary table for the:analysis of variance. There was a significant effect due to the 40 Table 6 Frequency with which A was the preferred player or one of the other players was the preferred player in each condition before and after the game was played. pre-game post—game 2 2 A Other X A Other X Dc 50 10 67.50* 34 26 24.07* D 42 18 36.30* 21 39 3.20 Pc 56 4 86.70* 36 24 39.20* P 46 14 50.70* 31 29 22.76* *p<.001 note: Expected cell frequencies are A=20, Other=40 for the pre-game data and A=15, Other=45 for the post-game data. The differential expected frequencies result from the fact that a "no preference" choice was offered on the post-game questionnaire and was not offered on the pre—game questionnaire. 41 deterministic vs. probabilistic manipulation (F= 3.59, p<.05), and the coalition vs. no-coalition manipulation (F= 10.59, p<.01) as well as a significant difference between the pre-game and post-game data (F= 77.70, p<.001). There were no significant interactions between these'variables. Table 8 presents the means and standard deviations for the data used in the analysis of variance. An examination of Table 8 reveals that (1) there was a tendency to prefer player A more in the pro- babilistic situation than in the deterministic situation both before and after the game was played, (2) there was a tendency to prefer player A more in the coalition situation than in the no-coalition situation both before and after the game, and (3) there was a tendency to prefer player A more before the game than after the game in all conditions. Results Relevant to Individual Differences To ascertain the effects of the personality variables that were measured, a product-moment correlation was computed between each of the six scales on the personality inventory and the dependent variables in each of the four eXperimental conditions. Three out of 156 correlations computed were significant, i.e., in the Pc condition "recognition" was negatively correlated with winning (r= -.37, p<.003) as well as with.amount won (r= -.34, p<.009) and "independence" was positively correlated with the amount won (r= .27, p<.039). Table 9 presents the correlations between the dependent variables and the individual difference measurements for all conditions. The small number of significant correlations between the dependent variables and 42 Table 7 Summary of the analysis of variance on the preferred playing position before and after the game is played. Source SS df MS F Between Subjects 63.97 239 A(Deter. vs. Prob.) 1.01 1 1.01 3.95* B(Coal. vs. no-coal.) 2 7O 1 2.70 10.59** AB .08 l .08 .29 Subject within groups 60.18 236 .26 Within Subjects 44.00 240 C(Pre. vs. Post) 10.80 1 10.80 77.70*** AC .01 1 .01 .06 BC .00 l .00 .00 ABC .21 l .21 1.50 C X Subj. Within groups 32.98 236 .14 *p<.05 **p<.01 ***p<.001 Table 8 Means and standard deviations for the pre and post game preferences of player position Pre—Game Post—Game Deterministic Probabilistic Deterministic Probabilistic E SD 3? SD x SD E SD Coalition .833 .373 .933 .251 .567 .494 .600 .490 No Coalition .700 .458 .767 .423 .350 .476 .517 .500 a N = 240 60 replications per cell 43 the personality traits which were measured was accepted as an indication that those individual differences which were measured had a very slight effect on either the strategies which were employed in the game or the outcome of the game. A further indication of the lack of a meaningful effect due to individual differences was obtained from the analysis of the 19 scale semantic differential with which the subjects were asked to evaluate the experimenter. (See Appendix A) A principal—axis factor analysis with a varimax rotation (Williams, 1967) collapsed the 19 scales to three factors, i.e., Factor 1, sociability; Factor 2, capability; and Factor 3, emotionality. (See Appendix B) An examination of the correlation of each of the three factors with the dependent variables (Table 9) revealed that 1 out of 78 corre- lations was significant, i.e., Factor 3 correlated negatively with who was attacked in the D condition (r= -.29, p<.03l). Because of the virtual lack of observed effects due to individual differences, there was no attempt to correct for those effects in the analyses of the data. 44 Table 9 Correlations between the dependent variables and the individual difference measurements in the Dc, D, Pc, and P conditions Personality Experimenter """"" Inventoryfi""" "'Eva1uation** .......... .1 S "‘C" R""I"“B"‘ L""1""2 __3 Player -.08 .20 .05 —.24 -.O6 .10 -.15 .08 .00 Preferred partner .01 .16 .00 -.03 -.O6 -.05 -.12 -.10 -.22 Desired share .02 .13 .01 -.02 -.07 -.05 -.11 -.ll -.21 Expected share .01 .18 .01 -.05 -.06 -.07 -.13 -.07 -.23 Dc Final share -.02 -.03 .00 .09 .02 -.06 .00 .00 -.12 Attacked -.03 —.O3 -.10' .19 .01 -.10 .10 .13 -.17 Coalition -.08 .09 .05 .02 -.15 .06 -.05 .07_-.l7 WonHLost -.Ol -.06 .14 -.02 -.06 .02 -.08 -.01 .00 ... Amount won ........ e.02.e.01...15.-.O4.-.07 ..04 -.07,-.03,..09 Player .01 -.10 .08 -.04 .02 -.07 .15 .23 -.04 D Attacked -.1O -.10 -.ll -.11 .13 .00 -.29d-.24 .01 WonHLost -.13 -.13 -.14 .ll .14 .06 -.04 -.09 .07 ..Amount won ..—.13.—.13.—.14...11 .14 ..06 —.04 -.09 .07 Player .18 -.15 -.O6 .02 .06 .06 -.O4 -.22 .11 Preferred partner .05 -.05 -.21 .04‘ .22 -.08 -.12 -.20 -.06 Desired share .03 -.03 -.21 .04 .22 -.O7 -.11 -.19 -.07 EXpected share .03 -.07 -.22 .06 .24 -.08 -.12 -.21 -.04 Pc Final share -.07 .21 -.21 .06 .12 -.ll -.08 .05 -.21 Attacked -.09 .03 .07 .03 -.17 .14 -.03 .00 -.01 Coalition -.13 .13 -.09 .04 .05 -.01 .01 .03 -.21 Won-Lost -.06 .20 -.37a .25 .21 -.25 .05 -.03 .17 Amount won. -.04 H.12.e.34b_.279..20-.25 .07 -.o4...22 Player .15 -.14 .07 .16 -.15‘-.03 -.07 -.10 -.07 Attacked .01 .08 .02 -.02 -.04 -.01 .13 .20 -.04 WOn-Lost .01 -.O7 -.07 .05 .08 .08 .05 .13 -.05 .....Amount.won ......... .01.r.07.r.07 ..05,..08...O8. .05 .13,e.05 51 P<-603 2-tailed *S=Support I=Independence **Factor l= Sociability 1’ P<:009 2-tailed C=Conformity B=Benevolance Factor 2= Capability C3 PB, B=C), and A<(B+C). In Type 3, two of the members of the triad are initially stronger as well as equal while the third member is weaker (A=B, B>C). In Type 4, two of the members are equal and their combined strength is less than the strength Of the third member (A>(B+C), B=C). The Type 4 triad has been described as a dictator situation (Gamson, 1961a). In Type 5, no two members of the triad are of equal strength but the combined strength of any two members is greater than the strength Of the third (A>B>C, A<(B+C)). In Type 6, no two members of the triad are of equal strength, but one member has a greater initial strength than the other two combined (A>B>C, A>(B+C)). Like the Type 4 triad, the Type 6 triad represents a dictator situation. Based on the initial assumptions, CT makes the following predictions about which coalitions would form in each Of the six triadic situations: Type 1, any; Type 2, BC; Type 3, AB Or AC; Type 4, none; Type 5, AB or AC; and Type 6, none. Caplow (1959) extended CT so that it covered two new situations. Type 7 was the situation in which all members are unequal, but one 96 member is equal to the other members combined. (A>B>C, A=(B+C)). Type 8 is the situation in which one member is equal to the other members combined and two of the members are equal (A=(B+C), B=C). In both of these situations, the most powerful individual (A) is said to have veto power (Gamson, 1961a). CT predicts that in both Type 7 and Type 8 situations AB or AC coalitions will be equally likely. One Of the assumptions of CT is that the member of the triad who has the least power is Often preferred as a coalition partner by both of the other members of the triad. As a result he becomes the "chooser" (Caplow, 1959); and as the chooser, he becomes the strongest member of the triad in a functional sense. This phenomenon is referred to as the "weakness is strength" effect. Caplow (1959) suggested that depending on the circumstances one of two assumptions about what is taking place in a coalition situation is correct. The two assumptions are: (l) "the 'chooser' in a triad seeks the maximum advantage or minimum disadvantage of strength relative to his coalition partner," or (2) "the 'chooser' in a triad seeks to maximize the strength of the coalition in re— lation to the excluded member.” (p. 492) It is apparent that the two assumptions result in different predictions. For example, in the Type 5 triad, if assumption number one is correct, CT predicts that BC coalitions will be prevalent. However, if assumption number two is correct, AC coalitions will prevail. Although the two assump- tions result in differential predictions, Caplow offered no suggestions as to the circumstances which would determine the assumption to be applied. Nevertheless, the two assumptions have proven to be useful 97 in subsequent theories. Minimum Power Theory. Immediately following Caplow's proposal Of CT it was suggested that an equally viable theory Of coalition formation could be based on the assumption that the participants in a coalition situation utilize a rational strategy to maximize their control over the outcome (Vinacke & Arkoff, 1957). This theory which was developed within the framework of game theory was later referred to as "minimum power theory" (Gamson, 1964). One of the major assumptions of ”minimum power theory" (hereafter designated MPT) is that the real power of each participant is accurately perceived by all of the participants in a given situation. Moreover, real Lpower as defined by MPT is different from power as defined by CT. That is, MPT utilized the Shapely (1953) notion of pivotol power. In three person mixed-motive situations the pivotol power of an individual is determined by the number Of times he can form a winning coalition. If the concept of pivotol power is applied to triadic situations, only three power relationships can occur: (1) all three members can be of equal power, i.e., any two members can coalace and control the outcome; (2) one member can be all powerful, i.e., the dictator situation; and (3) one member can have veto power. MPT predicts that in the situation in which all three members are equal in power every coalition will form, and in the situations in which one member has veto power the coalition will include the member with veto power. Minimum Resource Theory. Minimum resource theory (hereafter designated MRT) had its beginnings with Caplow's (1956) original CT and was later developed to its present state by Gamson (1961a). 98 Gamson limits the sphere Of applicability of MRT to situations which meet the following conditions: 1. There is a decision tO be made and there are more than two social units attempting to maximize their share Of the payoff. 2. NO single alternative will maximize the payoff to all participants. 3. NO participant has dictatorial powers, i.e., no one has initial resources sufficient to control the decision by himself. 4. NO participant has veto power, i.e., no member must be included in every winning coalition. (p.347) Providing that the situation falls within the sphere Of applicability, the following information is required in order to predict which coalition will form: (1) the relevant resources and what pr0portion of those resources are controlled by each player; (2) the payoff for each condition, Specifically, the payoff that is associated with the winning coalition; (3) the prOpensities to form coalitions with the other players regardless Of initial resources controlled; (4) the effective decision point, i.e., the amount of resources which are sufficient to control the decision as to how the rewards will be distributed. TO understand the basic predictions of MRT it is necessary to define a "minimal winning coalition” and theitheapest winning coalition." “A”minimal‘winningycoalition is a coalition such that the defection of any member will make the coalition no longer winning. The cheapest 'winning coalition is that minimal winning coalition with resources closest to the decision point." (Gamson, 1961a, p. 376) : Although the preceding definition could be interpreted to mean that only one 99 cheapest winning coalition can exist per triad, the present paper adOpts the interpretation that each member Of the triad has a cheapest winning coalition. The basic postulate Of MRT is the "parity norm", and is stated in the form of a general hypothesis: Any participant will expect others to demand from a coalition a share of the payoff proportional to the amount Of resources which they contribute to a coalition. (Gamson, 1961a, p. 376) Combining the maximization boundary condition and the parity norm results in the prediction that each player will choose to form the coalition which maximizes the ratio of his resources to the total resources of the coalition. However, a further boundary condition is that the total payoff must remain constant regardless Of which coalition forms. Thus, MRT predicts that the cheapest winning coalition will be preferred, if the total payoff is held constant. The final assumption of the theory is that reciprocal strategy choices are required before a coalition will form. Reciprocal strategy choices are present only in those situations where two of the individuals are required members in each others cheapest winning coalition. For example, consider a situation in which the distribu- tion of resources is A=4, B=3, and C=2. In this situation, for A the cheapest winning coalition would be an.AC coalition, for B the cheapest winning coalition would be a BC coalition, and for C the cheapest winning coalition would be a BC coalition. It is apparent that in this case B and C have reciprocal strategy choices and a BC coalition would be predicted by MRT. 100 When.MRT is applied to Caplow's eight triadic types, the following coalitions are predicted: Type 1, any; Type 2, BC; Type 3, AB or AC; and Type 5, BC. Type 4 and Type 6 are not applicable because they contain a dictator, and Type 7 and Type 8 are not applicable because one member has veto power. One asPect of MRI which differentiates it from CT is that in addition to the "weakness is strength” effect which is predicted by CT, MRT predicts a "strength is weakness" effect. That is, the weaker Of the other two members of the triad is always the preferred coali- tion partner. Since the strongest member of the triad is never a preferred coalition partner, he becomes the weakest member Of the triad in a functional sense, i.e., "strength is weakness." The reasoning on which the differentiation between the "strength is weakness" effect and the "weakness is strength" effect is based follows. MRT predicts that the strongest member of the triad will never be preferred as a coalition partner by the "chooser". Contrary to this, according to CT the strongest member of the triad will be the preferred coalition partner of the chooser 50% Of the time. Therefore, in the relevant triadic situations MRT predicts that the strongest member Of the triad will not be included in any of the coalitions while CT predicts that he will be included in 50% of the coalitions. The "weakness is strength" effect is manifest in both CT and MRT due to the fact that the weakest member Of the triad is a member Of all predicted coa- litions for both theories. Anticompgtitive'Theory. Anticompetitive theory (hereafter designated AT) was first suggested by Gamson (1964), however, it was 101 based on the results of a series of studies by Vinacke and his students. (Vinacke, 1959, 1962; Bond & Vinacke, 1961; Uesugi & Vinacke, 1963; Vinacke, et. al., 1966). Vinacke (1959) hypothesized that "female subjects might be much less concerned with winning and more oriented towards social and ethical conditions —- i.e., try to avoid an agressive display of power and attempt rather to be 'fair' to everyone." (p. 344) This hypothesis and subsequent research led Gamson (1964) to believe that an "anticompetitive norm" exists and that as a result coalitions will form along the lines of least resistance in bargaining." (p. 90) Therefore, Gamson's AT predicts that coalitions will form between players who are equal in resources. An extension Of AT (Phillips & Nitz, in press) suggests that the predictions as to which coalitions will form should be based on the divisibility of the payoff as well as the initial distribution of resources. If the payoff is equally divisible the coalition which requires the least bargaining is between the members of the triad who have the smallest resource differences. If the payoff is not equally divisible, the least bargaining will be required for the coali— tion which includes those members of the triad who have the greatest resource difference. Therefore, if the payoff is equally divisible, the predicted coalition will form between the members who are the most similar with respect to resources; and if the payoff is not equally divisible, the predicted coalition will form between those members who are the most dissimilar with respect to resources. ”Utter4COnfusion Theory. The last theory or "antitheory" to be 102 discussed was proposed by Gamson (1964). He chose to call it "utter-confusion theory." Gamson suggested that there was a need for this theory to deal with those situations in which the partici- pants are in a state Of confusion and are not able to make rational decisions. As an example, he cites political conventions where such things as "missed telephone calls" can influence which coalitions will form. Although Collins and Raven (in press) seem to agree with Gamson, the present writer feels that the suggested need for utter- confusion theory is unfounded. It is admitted that chance happenings can influence the formation Of coalitions. However, an implied boundary condition in the previously mentioned theories is that the subjects understand the "rules of the game" and "Utter-confusion Theory" is based on the assumption that the subjects do not understand the "rules of the game." Therefore, the Optimal conditions for testing Utter-confusion Theory would be to give the subjects no information at all but to simply ask them whom they would like to form a coalition with. In such a situation, Utter—confusion Theory predicts random choice, and it is possible that its predictions may be supported if such an experiment were run. Such a situation is, however, psychologically uninteresting. Furthermore, even though the theories to date do not account for all of the data, the present writer believes that to rely on such theories as Utter-confusion Theory would unnecessarily retard the progress of coalition research. .Eyaluations of Theories Much research has been done in an attempt to test the theories 103 which have been discussed in the immediately preceding section of this paper. Before the relevant research is examined in an attempt to determine which theory best explains the data, Caplow's eight triadic types and the coalitions that are predicted by each theory will be summarized. The predictions Of the theories are delineated in Table 1. An examination of Table 1 reveals that the only triadic type that differentiates between the four theories is the Type 5 triadic situation, i.e., A>B>C, A<(B+C). In the Type 5 situation, Control Theory predicts that AB and BC coalitions are equally likely, MRT predicts BC coalitions will form, MPT predicts that all coalitions are equally likely, and anticompetitive theory predicts that if the payoff is equally divisible AB or BC coalitions will be equally likely and if the payoff is not equally divisible AC coalitions will form. Because of this unique feature Of the Type 5 triad and the fact that virtually all of the relevant research has examined it the present review of the literature will concentrate on the data obtained from the examination Of the Type 5 triad. However, where it is deemed important, data from other triadic types will be utilized. The utilization of the Type 5 triad is not meant to imply that the other triadic types are not equally as important or that the Type 5 triad has been the only triadic type examined. It does mean that the present writer believes that a greater understanding of coalition :research.can be reached by concentrating on the Type 5 triad rather than considering the data for all of the eight triadic types which have been delineated . 104 Table l Coalitions predicted by the four theories of coalition formation in the triad for the eight triadic types delineated by Caplow ..... Coalitions.Predicted_by...., CT MRT MPT ' * ‘ ' ' AT* , Type of Triad, .. . ..,. _ ... ....... A... B l A=B=C ............. any any any any any 2 A>B, B=C, A<(B+C).. BC BC any BC AB or AC 3 A1Bd-C), B=C . . . . . . none --- none none none 5 A>B>C, A<(B+C) .... AC or BC BC any AB or BC AC 6 A>B>C, A>(B+C) .... ‘ none --' none none none 7 A>B>C, A=(B+C) .... AB or AC -- AB or AC AB AC 8 A=(B+C), B=C ...... AB or AC -- AB or AC AB or AC AB or AC *A = the predicted coalitions when the payoff is equally divisible. B = the predicted coalitions when the payoff is not equally divisible. 105 Minimum Power Theory. In general MPT has been refuted by the coalition research which has been conducted to date. (e.g., Vinacke & Arkoff, 1957; Vinacke, 1959, 1962; Phillips & Nitz, in press; Cole & Phillips, in preparation) That is the prediction that all coalitions are equally likely in the Type 5 triadic situation has not been reported. Kelley and Arrowood (1960) suggested that if the procedure were simplified and more fully explained, the subjects would acquire an adequate understanding of the true power relationship within the triad. As a result, they would play a more rational strategy. Therefore, they designed an experiment based on the pachisi paradigm2 in which all triads played a lengthy series of 2Since the pachisi paradigm has subsequently been used ex- tensively in coalition formation studies, a detailed description of the paradigm and the procedure for using it as taken from Vinacke and Arkoff (1957) follows: "Apparatus for the experiment consisted of a modified pachisi board. Only the exterior lanes of the board were used, and the spaces of these lines were numbered consecutively making a total Of 67 spaces. The object Of the game was to reach 'home' first. The winner was awarded a prize of 100 points. In the event of a coalition, the prize was shared in a manner agreed upon by the allies. A single die, cast by the experimenter, was used. Each player's move was determined by the weight inscribed on the counter he drew from a hopper; he was entitled to move forward the number of spaces equal to his weight times the number shown by the die. All of the players started from the same home base and moved simultaneously, that is, every player moved each time the die was thrown. At any time during the game, any player, in return for a promise of a specified portion of the prize, could form an alliance with any other player. In this case, the allies immediately pooled their strengths and proceeded to a position equal to their combined acquired Spaces; in further throws they moved forward according to their combined weights (times the die). Once an alliance was formed, it was considered [Hermanent for that game. Any player could concede defeat when his position appeared hopeless." (p. 408) 106 games in the Type 5 triadic situation only. The power relationship within each triad was kept constant across all games. Kelley and Arrowood argue that due to the simplified nature of the instructions and of the overall game some subjects perceived the real power structure very early. This was suggested by the fact that the number Of the various types of coalitions in the first three "7“ trials combined did not differ significantly from what would be 1 predicted by MPT. However, there are two aspects of the Kelley and ! Arrowood procedure which accounts for the results without recourse to Q MPT. These two aspects are (1) each subject remained in the same f” power position throughout the experiment, i.e., the power relation- ship was permanent, and (2) the subjects were instructed to accu- mulate as many points as possible over trials, i.e., the payoff was cumulative. The combination of these two aspects adds a second resource dimension to the Situation. That is, on the first trial the subjects have only the initial resource weights, i.e., one resource dimension, on which to base their strategy. However, on all subsequent trials the strategy is based on the initial resource weights, i.e., the first resource dimension, plus the amount Of payoff that each subject has accumulated, i.e., the second resource dimension. Therefore, at the beginning of any trial after the first the relative power Of each player would be determined by what had happened on the previous trials. The suggestion that a second resource dimension was salient after the first trial is supported by studies which have examined the effects of cumulative score in the coalition situation (Vinacke, , 107 1959, 1962; Bond & Vinacke, 1961; Uesugi & Vinacke, 1963; Cole & Phillips, in preparation). Vinacke (1959) obtained data which indicated that over three games the two weakest players with respect to initial resources tend to ally against the stronger. However, there was a tendency for the two strongest members of the triad to coalace in the third game. Further support for the suggestion ;~+1 that the use of cumulative score adds a second resource dimension results from a study by Vinacke (1962). He Observed that in the f first game Of the cumulative score condition, BC coalitions were prevalent in the Type 5 triadic condition; however, by the third game there were more AB coalitions formed than BC coalitions. Moreover, in a condition which utilized a discrete payoff, BC coalitions were formed Significantly more Often than AB coalitions in each of the first three games. A comparison of the cumulative score and discrete payoff conditions in the third game indicates that the cumulative score does have an effect. However, the effect is not highly visible if the data for the first three games is combined as was done by Kelley and Arrowood. The results of the Cole and Phillips study (in preparation) indicate that the effect of cumulative score is even more pronounced if the subjects remain in the same power position over games. That is, by the second game in the Cole and Phillips study there were more AB coalitions formed than AC or BC coalitions. Since Kelley and Arrowood utilized the permanent power relationship, their results are more likely to be an artifact Of the experimental design than a . 92 91 cums ‘hnl i 108 consequence of the fact that the subjects perceived the true power relationship and played a rational strategy. AS a result Of the Kelley and Arrowood study, Vinacke, Crowell, Dein, and Young (1966) designed an experiment which was proposed to determine "the influence Of information about Opposing views of the power pattern on the behavior Of triads." (p. 181) Specifically, they wanted tO know if giving the players a knowledge Of the true power relationship in the triadic situation would result in their f using the rational strategy which is predicted by MPT. Z ng- . g Since Kelley and Arrowood had critized the original pachisi »~» paradigm for being too complicated, Vinacke, et. a1. attempted to simplify the procedure. However, they did not simplify the procedure tO the extent that Kelley and Arrowood had. For instance, based on the results Of previous research (e.g., Vinacke, 1959, 1962), Vinacke, et. a1. decided that keeping the players' weights constant over games would result in a loss of interest in the game and a tendency for the players who were behind in any game after the first game to form a coalition. For this reason, resource weights were randomly assigned before each game. Moreover, discrete rather than cumulative score was used. The method which.Vinacke, et. al. used to simplify the Situation was to inform some of the subjects that the true power Of each player was equal. The games were played in two sections. The first three games 'were.replications of the original pachisi paradigm and were referred to as the learning period. The last three games (the second section) ‘were played under the.information condition. Prior to the second 109 section, randomly selected subjects were given instructions which informed them Of the two alternative strategies: (1) any two players will win if they form a coalition, hence the initial power makes no difference, and (2) the two weaker players can defeat the stronger player if they form a coalition. "In one set of triads, these eXplanations were given to one member, in a second set to two members, and in a third set to all three members." (p. 183) The results refuted MPT in all conditions except the condition which examined all male triads and in which all three members of the triad received the information about the true power relationship. Since virtually all of the research to date has refuted the predictions of MPT, it is felt that MPT Should be rejected as a major theory of coalition formation. However, it is not felt that it Should be forgotten. That is, there are aspects of coalition formation.research which indicate that some of the assumptions of MPT may prove useful in develOping an acceptable theory of coalition formation. For instance, the assumption that the subjects play a rational strategy may prove to be an important part Of any new theory which is deve10ped. Moreover, the structure Of MPT, i.e., the quantitative approach, would result in a more precise presentation of any theory which is prOposed. Control Theory vs. Minimum ResOurce Theory. The early research on coalition formation in the triad (Vinacke & Arkoff, 1957; Vinacke, 1959; Chaney & Vinacke, 1960; Bond & Vinacke, 1961) seemed to support CT. However, the tendency for Significantly more BC coalitions than.AC coalitions to form in the Type 5 triad could 110 not be accounted for by CT. MRT on the other hand, does account for this effect. That is, MRT predicts that BC coalitions will be prevalent in the Type 5 triad. Therefore, while CT served its purpose as a pioneer theory in coalition formation it has been replaced by MRT. HOwever, a recent extension of CT by Chertkoff (1967) suggests that CT is still very much a part Of coalition theory. Chertkoff prOposed that the predictions of CT Should take reciprocal partner preferences into account. Therefore, the f probability that two players will contact each other is an im— portant variable when predictions as to who will form coalitions are made. For example, consider the Type 5 triad. CT predicts that player A will contact player B 50% of the time, and player C 50% of the time; player B will contact player C 100% of the time and player A 0% of the time; and player C will contact player A 50% of the time and player B 50% of the time. As a result the probability that an AB coalition will form is .5 x 0, therefore, no AB coalitions are predicted; the probability of AC coalitions forming is .5 x .5, therefore, AC coalitions are predicted 25% of the time; and the probability Of a BC coalition forming is 1.0 x .5, therefore BC coalitions are predicted 50% of the time. If the subjects are allowed to repeat this process until reciprocal contacts are made, the final distribution of coalitions would be in a ratio of 2 BC coalitions to each AC coalition and there would be no AB coalition. While this does not accurately predict the distribution of coalitions which have been reported in the various studies, it is as accurate as MRT. Therefore, to determine which theory best fits the coalition lll situation the process by which coalitions are formed as well as the outcome of the coalition situation must be examined. TO examine the process of coalition formation Shelly and Phillips' (1966) assumption that there are two disjoint segments (the social contact segment and the bargaining segment) within the coalition process, will be utilized. TO simplify the comparison of CT and MRT, only the Type 5 triad will be examined. The question to be answered is: "What are the differences between the predictions i of MRT and CT within each segment of the coalition formation process. ; If CT is supported in the contact segment player A will contact players B and C equally often, player B will contact player C only, and player C will contact players A and B equally Often. However, if MRT is supported, the "strength is weakness” effect Should be observed in the contact phase, i.e., each player should contact the weaker of the other two players. The comparison of predicted results in the bargaining segment of the coalition process reveals no apparent differences between MRT and CT. Since all of the contact data after the initial contact is complicated by extraneous factors such as who each player has contacted previously and why a coalition has not formed, only the initial contact data will be examined. Two studies report the initial contact data in the Type 5 triad, i.e., Chertkoff (1966) and Cole and Phillips (in preparation). Both Chertkoff and Cole and Phillips report that in the Type 5 triad each player initially contacts the player who has the smaller resource weight significantly more Often than the player who has the larger resource weight, i.e., "strength is weakness." Therefore MRT 112 is supported with respect to the contact segment of the coalition formation process in the Type 5 triad. Further support for MRT over CT is reported in a series of studies conducted by Phillips and his students, (Cole & Phillips, 1967a; Phillips & Nitz, in press; Cole, Nitz, & Phillips, in preparation.) The relevant studies have used a'political decision questionnaire"3 to examine the contact segment of the coalition formation process and have concentrated on the Types 1, 2, 3, 4, and 8 triadic situations. Therefore, the data which they have 3The political_decision questionnaire (PDQ) allowed subjects to indicate their initial preference for a coalition partner and was worded as follows: "Assume you are the manager for a candidate in a political party convention. There are a total of 300 votes among the delegates, and at least a majority (151) Of these are re- quired to win the nomination. Your man, Candidate A, has votes pledged to him, Candidate B has votes, and Candidate C has votes. Which of the other two candidates, B or C, will you approach first to try to make a deal?" B C (circle one) The preceding item was the first item on each form. There were ten items in all. Candidate A was assigned 33 different voting values, however, only the situations in which A had more than 75 votes were reported. On all items either B or C was assigned a number of votes which was equal to A's and the third candidate was assigned a number of votes so that A + B + C = 300. . 0....) 113 obtained adds to the support of MRT which has been reported in the Type 5 triad. That is, with the exception of the Type 1 and Type 4 triadic Situations those situations in which contacts have been examined report a significant ”strength is weakness" effect. In the Type 1 triad all members are equal and there can be no "strength is weakness” effect by definition. The Type 4 triad contains a F““‘ dictator so it is not covered by MRT. . . ‘ To further examine the merits of MRT, the hypothesis that the parity norm will be salient was examined. Various studies have examined the saliency of the parity norm by comparing the number of ru'::_-, .-.. H“. ,. ... times the payoff is split on a 50/50 basis with the number of times the payoff is Split so that the larger share Of the payoff is obtained by the member Of the coalition who contributed the larger Share of the resources. (e.g., Vinacke & Arkoff, 1957; Vinacke, 1959; Cole & Phillips, in preparation). In general they have reported that the parity norm is salient. However, while the number of 50/50 splits is small in those coalitions in which the players are not equal, it is also true that the number of times that an exact parity Split is Obtained is small. Thus, the parity norm does seem to effect which coalition is preferred but the division of the payoff does not accurately reflect the parity norm. As a result of the comparison of MRT and CT, MRT has emerged as the most acceptable of the two theories. However, one other theory, i.e., anticompetitive theory (AT), remains to be examined. Anticompgtitive Theory vs. Minimum Resource Theory. AT has 114 been supported to some extent by most of the research which has examined behavior in the coalition formation situation (Vinacke, 1959, 1962; Bond & Vinacke, 1961; Uesugi & Vinacke, 1963; Cole, Nitz, & Phillips, in preparation). This support for AT has been manifest predominantly in the strategy which is employed by females. That is, females tend to consistently employ a less competitive strategy than males. Although the feminine strategy is consistently different than the masculine strategy, it is not statistically different. As a result, the predictions of MRT have continued to account for the process and outcome of the relevant coalition situations even though they have been weakened somewhat by the feminine strategy. Chaney and Vinacke (1960) and Amidjaja and Vinacke (1965) examined the effects Of need for achievement and need for nurturance on the strategy employed in a coalition situation. The predictions Of MRT were supported regardless Of the motivational factors involved. However, an effect similar to what has been noted between males and females was Observed between males high in need for nurturance and males high in need for achievement. Those males who were high in need for nurturance played a somewhat less competitive strategy than those males who were high in need for achievement, although the resulting strategy was not significantly less competitive. The effects attributed to need for achievement and need for nurturance in the male triads were not Observed in the female triads (Amidjaja & Vinacke). This adds more support for the hypothesis that there is a 115 systematic difference in the strategies employed by the two sexes. Moreover, it suggests that the effects of personality variables need to be examined more carefully. That is, measures which better examine the personality characteristics of the competitive and anticompetitive strategies Should be deve10ped. Geiss (1964) studied the coalition situation with respect to how it was played by high machiavellians as compared to medium and low machiavellians. The data indicated that high machiavellians, i.e., individuals who are willing and able "to use guile, deceit, and other Opportunistic strategies in interpersonal relations in order to manipulate others," (p. l) are successful in their attempts to manipulate others. Since Geiss does not report the coalition data, MRT and AT cannot be compared. However, Dawson and Phillips (unpublished data) report that high machiavellians prefer to form the weakest coalition even when it is not a winning coalition. An examination of verbalized strategies (Cole & Phillips, 1967a) reveals that many subjects do not consider the fact that a coalition has no way of obtaining some of the votes which are promised to the isolate. Therefore, the high machiavellians, being more competitive, may attempt to form the coalition to which they contribute the most votes with the plan of subsequently attempting to seduce votes from the isolate. By such strategies a losing coalition can be transformed into a winning coalition and the high machiavellians would receive more of the payoff than they would by forming a winning coalition in the first place. 116 The connection between the studies which examined sex differences (i.e., Vinacke, 1959, 1963; Bond & Vinacke, 1961; Uesugi & Vinacke, 1963; Cole, et. al., in preparation) and those which examined personality differences (i.e., Chaney & Vinacke, 1960; Geiss, 1964; Amidjaja & Vinacke, 1965) is apparent. That is, the saliency of the anticompetitive norm may be the result of r“‘\ underlying personality variables rather than sex differences per. se. Therefore, AT may apply to males depending upon their personalities. In a recent study designed to compare the predictions of ..1-guy. .- 1.1,\ “as MRT, MPT, and AT, Phillips and Nitz (in press) obtained results which err” supported MRT. However, some support for a weakened version of AT was reported, i.e., in those Situations in which the payoff is not equally divisible'the probability Of contacting a given person is reduced by some factor if that person is Similar in terms of resources." (p. 10) The weakened version of AT is a modification of MRT. However, it is important because it allows for the suggestion that an anticompetitive norm may exist. Further support for MRT over AT is found in a study by Cole and Phillips (1967a). Since there were some subjects who did not follow the predictions Of MRT, they administered the PDQ and then asked the subjects to verbalize the reasons why they chose the candidate that they did. An analysis of those verbalizations supported MRT over AT. More than 85% of all strategies were based on the strong-weak dimension rather than the same different dimension, and better than half of the strategies verbalized were the ideal 117 MRT strategy. Since AT has found sufficient support to indicate that an anticompetitive norm may be operating in the coalition Situation, the effects of the anticompetitive norm need to be accounted for. However, it is not felt that the effects of the anticompetitive norm are strong enough to necessitate a Special theory to explain it. It is proposed that a better method Of acknowledging the existence of an anticompetitive norm would be to extend MRT. MRT should take into account that there are individuals who do not wish to maximize their share of the reward. Instead they attempt to divide the reward as ”fairly" as possible. Therefore, in those triads which include one or more individuals who play a strategy based on the anticompetitive norm the predicted coalitions may not follow from the parity norm. In many situations this could account for those coalitions which occur and are not predicted by MRT. Criticisms of MRT. There are two major criticisms of MRT which are evident to the present writer. One criticism, which is true of most psychological theories, is that the predictions are too strong. For example, if MRT is strictly interpreted only BC coalitions are predicted in the Type 5 triad. A search of the coalition literature failed to uncover any studies which Observed only BC coalitions in the Type 5 triad. The second criticism is that the boundary conditions of MRT are too narrow. By this it is meant that there are too many valid coalition situations which are outside the scope of MRT. It?“ «ma—l 3L in-“ ‘ 118 There have been two attempts to rectify the first criticism, i.e., Chertkoff (1967) and Shelly and Phillips (1966). Both attempts have proposed to do so by developing a mathematical model. The basis for both models is the differential propensity of each member of a triad to prefer each of the other members of the triad as a coalition partner. Chertkoff's model has already been discussed with respect to CT for which it was Offered. Moreover, it was shown that even though MRT makes less accurate predictions than Chertkoff's revision Of CT, the assumptions upon which MRT is based are supported by the data concerning the process of coalition formation. Therefore, Chertkoff's model has been refuted. Shelly and Phillips (1966) base their model on the proba- bility of a simultaneous reciprocal contact as computed from the data. In order for a simultaneous reciprocal contact to occur two members Of the triad must contact each other on the same trial. A trial is defined as "that unit of the contact segment of the process in which each person makes one contact." (p. 1) Subsequent de- velopment Of the model has resulted in predictions which are more accurate than the predictions of MRT in the situations in which simultaneous reciprocal contacts are required. However, the state Of the model requires further development. Even though the two models which have been proposed are not adequate, it is suggested that before an adequate theory Of coalition formation is developed the contact probabilities of each member of the triad will have to be considered. Fly—""7?“ _‘;T:. ‘ - .‘-s.~ 119 The second criticiSm of MRT, i.e., that the boundary conditions are too narrow, results from studies which have examined coalition Situations outside of the sc0pe of MRT. Studies which have fallen outside of the boundary conditions of MRT (e.g., Kelley & Arrowood, 1960; Stryker & Psathas, 1960; Psathas & Stryker, 1965; Chertkoff, 1966) have consistently failed to report results similar to those rwmfl studies which have remained within the boundary conditions of MRT. Since the studies are outside of the scope of MRT, it is not _"'."K._". H" “.11-.5; L 4 surprising to discover that MRT does not accurately predict their ‘3.) 3' \ results. However they must be examined prior to any further attempt to develop a theory of coalition formation which will not be subject to the criticism that its boundary conditions are too narrow. The study by Kelley and Arrowood (1960) falls outside Of the scope of MRT due to the fact that after the first game has been played the power of each member of a triad is determined by more than one resource dimension. That is, the power of each in— dividual is determined by the amount of the payoff which he has accumulated as well as the assigned resource weight, i.e., there are two relevant resource dimensions. As has been stated previously, this interpretation has been supported by research, i.e., Vinacke (1959, 1962), Vinacke, et. a1. (1966), Cole and Phillips (1967b). The key concept to be considered in such a situation is the concept Of multiple resource dimensions. A resource has been defined as a weight assigned to each player in a game such that the distribution of the payoff is determined by some critical quantity of that weight. 120 However, the weight assigned to an individual player is in reality a value on some resource dimension. Moreover, the total resources of the players in a given situation may be composed of more than one resource dimension which are not necessarily assigned to the players. Thus, the concept of multiple resource dimensions must be considered if the criticism of narrow boundary conditions is I‘m“ to be overcome. Stryker and Psathas (1960) and Psathas and Stryker (1965) fell outside of the scope of MRT because they introduced contention between the appropriate subjects by informing them that they had or been enemies of long standing, and could not form a coalition. In one condition, contention was introduced between the weak man and one of the strong men; in another condition, the contention was between the two strong men; and in another, there was no contention. The introduction of the contention factor makes it impossible to interpret the Stryker and Psathas studies within MRT. Even so, it is suggested that the relevance Of the contention factor should be carefully examined before it is considered in an attempt to increase the boundary conditions Of MRT. Of the studies which fell outside of the scope Of MRT, only one (Chertkoff, 1966) reported results which were strikingly different from the predictions of MRT. Chertkoff used the political 121 convention paradigm4 for the study of coalitions in the triad. How- ever he added a second resource dimension by manipulating the proba- bility of future success. This was accomplished by varying the probability that a coalition would win the election after it had won the nomination. The Type 6 triad was examined in all conditions. In all three experimental conditions players B and C had a .5 probability Of future success in the national election. Player A's probability of future success varied between .5, .7 and .9 depending on the condition. A control condition was used in which the probability of future success was not a factor. In the control condition the "strength is weakness" effect was reported. However, in the conditions in which player A had a .5 and .7 probability of future success, the power inversion effect was weakened, i.e., the distribution of preferred coalition partner was not different from chance expectancy. In the condition in which player A had a .9 probability of future success the power inversion effect was replaced by the "strength is strength" effect, i.e., AB coalitions were preferred. Moreover, as the probability of future success increased the parity principle 4In the political convention paradigm each subject plays the part of a delegation chairman at a political convention. Each subject is assigned a given number of votes at the beginning Of the convention. For each subject the object is to acquire as many "jobs” for his faction as he can. To acquire jobs the subjects must form a coalition which can win the nomination. The number of jobs that each subject acquires if he has entered into a winning coalition depends on the division Of the jobs that the two subjects who formed the coalition agree upon. If a subject is not in a winning coalition, he receives no jobs for his faction. (Gamson, 1961b) 122 became more salient. This is indicated by the tendency for the player with the highest probability of future success to demand and receive a larger Share of the payoff. An attempt to explain why the results vary when the probability of future success is manipulated leads to a differentiation of the probabilistic and deterministic5 situations. Prior to Chertkoff's F“"”= study the theories and research had all focused on the deterministic situation. However, Chertkoff's study extended the research on coalition formation to the probabilistic situation. As a result any attempt to develop a theory of coalition formation must take the 2"!“ ' ”Brig. .- “SUI-I probabilistic situation as well as the deterministic situation into account. Subsequent to Chertkoff's study only one study has examined the effects of the probabilistic and deterministic aspects Of the coalition situation. (Vinacke, Lichtman, & Cherulnik, 1967). The major finding Of the Vinacke, et. a1. study was that there is a propensity to gamble on winning all of the payoff in the probabilistic situation, i.e., there is a tendency to play without forming a coalition. Moreover, the propensity for a participant to play 5Probabilistic situations are those in which no unit or combination Of units will win with probability one. Deterministic situations are those situations in which some unit.Or combination of units will win with probability one. 123 the game alone increases as the probability of that participant winning alone increases. In conclusion it is proposed that MRT has the basic assumptions which are necessary for a theory of coalition formation. However, for MRT to adequately predict behavior in all coalition formation situations the scope of MRT must be increased by considering gems three factors. The first factor requires an acceptance of the . presupposition that the anticompetitive norm is salient for certain individuals. Therefore, the possibility Of the anticompetitive norm effecting the process and outcome of the coalition formation -—~ F" situation must be considered. Secondly, MRT must redefine the concept of resources to include multiple resource dimensions. The third variable to be considered is that the probabilistic and deterministic nature Of a given coalition formation situation must be accounted for. Although future research may point out other variables which must be considered, at the present time an extended version Of MRT which considers the anticompetitive norm, the concept Of multiple resource dimensions, and the probabilistic vs. deterministic aspects Of a given coalition Situation, will cover those situations which by standards of current research, are of major importance. ‘ P r .. '4 v ‘ p 1 A‘ WEB 2 3 1968 "lmnmummmwuwummmmt##1## 4953