Effects of alternatives in coalition bargaining

JOURNAL

OF EXPERIMENTAL

SOCIAL

20, 116-136 (1984)

PSYCHOLOGY

Effects of Alternatives S. S. KOMORITA University

in Coalition Bargaining

AND THOMAS P. HAMILTON

of Illinois,

Urbana-Champaign

AND DAVID A. KRAVITZ Universiiy

of Kentucky

Received August 31, 1982 A critical factor in bargaining and coalition formation is the alternative outcomes of the bargainers if an agreement cannot be reached. In some situations bargainers have individual alternatives while in other situations their alternatives must be negotiated with others. The purpose of this study was to contrast the effects of one-person and two-person alternatives on coalition outcomes. The second purpose of the study was to contrast the predictions of four theories of coalition formation: bargaining theory, equal excess model, Shapley value, and a special case of equity theory. The results indicate that one-person alternatives enhance the bargaining strength of the stronger players more than two-person alternatives. The predictions of the equal excess model and the Shapley value were more accurate than the predictions of bargaining theory and equity theory. However, the greater accuracy of the equal excess model and the Shapley value may be restricted to situations in which the bargainers have one-person rather than twoperson alternatives.

In many real-life situations a group or organization is faced with a situation in which rewards (or resources) must be allocated to all members of the group. In some situations unanimous agreement is required, and if one or more members are not satisfied with any of the proposals to allocate the group prize, they may threaten to defect and join another group. If the defection of some members results in a smaller prize for those who remain, there will be strong pressures to concede a larger share to those who threaten to defect. In evaluating such threats to This study was supported by a grant from the National Science Foundation (BNS 79111103) to the ftrst author. Requests for reprints should be sent to S. S. Komorita, University of Illinois, Department of Psychology, 603 E. Daniel St., Champaign, IL 61820. 116 OO22-1031/84 $3.00 Copyright All rights

0 1984 by Academic Press, Inc. of reproduction in any form reserved.

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OF

ALTERNATIVES

117

defect, the credibility of the threats must be considered and, in particular, the consequences of defection, not only for the group but for the threateners as well, must be evaluated. In this type of situation, a critical factor in bargaining and coalition formation is the alternatives of the bargainers if an agreement cannot be reached. In Thibaut and Kelley’s (1959) exchange theory, for example, such alternatives are called the party’s “comparison level for alternatives” (CLalt), and the bargainer who has the smaller CLalt is expected to make larger concessions. In some situations bargainers have individual alternatives that are certain, while in other situations their alternatives must be negotiated with others, and are uncertain. It would be desirable, therefore, to contrast the effects of one-person and two-person alternatives on coalition outcomes. This was one of the main purposes of this study. One-Person

vs Two-Person Alternatives

The bargaining situation described above is simulated by a class of games called superadditive. A superadditive game is one in which the value of any two disjoint (nonintersecting) subsets of the N players is equal to or greater than the sum of the values of the two subsets. To illustrate this class of games, consider Game I, one of the games that was used in this study, v(A) = v(B) = v(C) = 0; v(AB) = 120; v(AC) = 90; v(BC) = 60; and v(ABC) = 165; where A, B, and C denote the three players and v( ) denotes the value of each possible coalition. It can be seen that the value of the grand coalition (ABC) is greater than the sum of values of any disjoint subsets, and the value of each two-person coalition is greater than the sum of the values of the individual players. Thus, this game is superadditive. Game I is called a quota game, a class of games in which there is a vector of weights wi associated with the players such that for any pair of players i and j, v(Q) = wi + wj, where v(h) denotes the value of the two-person coalition of players i andj. The weights (wi and wj) are called the quota values of the players, and for players A, B, and C, the quota values for the above game are 75, 45, and 15, respectively.’ To illustrate the effects of two-person alternatives in coalition bargaining, consider the bargaining process in Game I. In negotiating the division ’ A strict definition of a quota game requires that r(ABC) = Cw,, and for the above game, v(ABC) should equal the sum of the quota values, or 135. Thus, a quota game, by the strict definition, is superadditive. Medlin (1976) has shown that the likelihood of the grand coalition varies directly with its value. In the present study it was essential for the purpose of the study to provide a strong incentive to form the grand coalition; hence, its value was set at a sufficiently high level to assure its occurrence with high probability.

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KRAVITZ

of reward in the grand coalition, it is reasonable to assume that expectations in the two-person coalitions will be used to justify each person’s share in ABC. However, these expectations (alternatives) will be uncertain. The joint expectations of each pair (defined by the values of the twoperson coalitions) will be certain, but the individual alternatives will be uncertain because they must be determined by mutual agreement of each pair. To illustrate the effects of one-person alternatives consider the following three-person game, denoted Game II: v(A) = 75; v(B) = 45; v(C) = 15; v(AB) = v(AC) = v(BC) = 0; v(ABC) = 165. Note that the values of the one-person coalitions, hereafter denoted v(i), correspond to the quota values of the players in the three-person quota game (Game I). However, the values of all two-person coalitions are zero. This means that when the three players negotiate the reward division of 165 in the grand coalition, their respective demands will be based on v(i), their one-person values. These alternatives, of course, are not uncertain, as in the case of Game I. Another interpretation of the difference between the one-person and two-person alternative conditions is in terms of a difference in the decision rule in negotiating in the grand coalition. In Game I the two-person coalitions can be interpreted as majority coalitions that can impose zero payoff to the third person. In Game II, in contrast, each person has a veto in the grand coalition and can assure him/herself of the one-person payoff. In this sense, Game I can be interpreted as group decision-making under a unanimity rule. Theories of Coalition Formation The basic problem for theories of coalition formation is to predict which coalition is likely to form and the division of the prize value negotiated by the coalition members. Two descriptive theories that predict both of these response measures are the equal excess model (Komorita, 1979) and a revision (Komorita & Tumonis, 1980) of the bargaining theory (Komorita & Chertkoff, 1973) of coalition formation.* A basic assumption of both of these theories is that the power (bargaining strength) of a player is a function of the alternatives of the players. If two or more players are negotiating the division of the prize, both theories ’ Gamson’s (1961) minimum resource theory, Gamson’s (1964) minimum power theory, and Komorita’s (1974) weighted probability model are not applicable in Games I and II. They are restricted to simple games. The bargaining theory was originally proposed for simple games but Komorita and Tumonis (1980) proposed an extension of the theory for multivalued games such as Games I and II.

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OF ALTERNATIVES

assume that the players will base their demands on their respective alternative coalitions if an agreement cannot be reached. Both theories also predict that the expectations of the players will change over rounds or trials of coalition bargaining. The equal excess model. The equal excess model assumes that in the initial stage of coalition formation, each player will attempt to form the coalition that maximizes initial expectation, given by EyS = v(S)/s, i = 1, 2, . ..) s, where v(S) denotes the value of coalition S, and s denotes the number of players in coalition S. For Game I, the three-person quota Eic = game, Eh = 60-60, for players A and B, respectively; 45-45 for A and C; Eic = 30-30; and EiBc = 55 each for A, B, and C. Hence, the model predicts that A and B will both initiate offers to form the AB coalition, while player C will prefer and attempt to form the grand coalition. On subsequent rounds of negotiations, the model predicts that expectations of the players will change as EFS = max EiF1 + (l/s)[v(S)

- max EJ;l]

(1)

where Ek denotes the expectation of player i in coalition S on round r; max E i; 1 denotes player i’s maximum expectation in alternative coalitions (S # T) on round r- 1; and the summation is over the members of coalition S (j = 1, 2, . . . . s). Equation 1 specifies that each player in S will demand a share of the prize, v(S), that equals his/her best alternative on the previous round, and the excess, v(S) minus the sum of these alternatives, will be shared equally by the players. Iterations of Eq. (1) yield predictions over successive rounds (r) of bargaining, as shown in Table 1. It can be seen that on TABLE PREDICTIONS

OF EQUAL

EXCESS

MODEL

1

FOR GAME

I (THREE-PERSON

QUOTA

GAME)

Round of bargaining (r) Coalition AB AC BC ABCb

0 60-60 45-4.5 30-30 55-55-55

1 68-52 60-30 38-22 72-58-35

2 71-49 68-22 41-19 79-56-30

. . . . ... .

5 74-46 74-16 44-16 84-55-26

..

Asymptote

..

75-45 75-15 45-15 85-55-25

.

” Quota values of players A, B, and C are 75, 45, and 15, respectively. Values on round 0 represent initial (prenegotiation) expectations. Round 1 values are hypothesized to be the best estimate of payoff shares. ’ Predicted values in the grand coalition (ABC) are based on a revision of the original theory, which assumes that expectations in the two-person coalitions on the same round will be used to justify demands on the grand coalition.

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AND KRAVITZ

round 0 (prior to negotiations), the AB coalition is mutually preferred, and players A and B are expected to reciprocate offers to each other. The ABC coalition maximizes the expectation of player C; hence, C is expected to send an offer to form ABC. The predictions in the grand coalition in Table 1 are based on a revision of the original model. In the original model (Komorita, 1979), it was assumed that the players would use their initial expectations (,!$J in the two-person coalitions to justify their demands in the grand coalition on round I. However, in Game I, where v(i) = 0, bargaining in the grand coalition is a unique situation in which none of the players can threaten to defect to an alternative coalition that does not include one of the players who is being threatened. For example, player A cannot threaten B and C with the AB or AC coalitions because these coalitions include B and C. Thus, when no player has a single-person alternative, the bargaining process in the grand coalition may be qualitatively different in that no player-as an individual-has a credible threat (alternative), and it is plausible that two players will threaten the third (e.g., A and B might threaten C with the AB coalition). For these reasons, a revision of the original model is proposed here, and it will be assumed that on a given round, expectations in the two-person coalitions will be used to justify each person’s demands in ABC on the same round. The model also assumes that situational factors affect the round on which an agreement is likely to occur, and in particular, it is assumed that an agreement is likely in the early rounds if: (1) Subjects are naive with little familiarity and experience with the structure of coalition games; (2) Communication is restricted (e.g., subjects are only allowed to make offers and counteroffers); and (3) The incentive (size of reward) is small relative to the value of time. When all of these conditions are met, Komorita (1979) hypothesized that the round 1 predictions of the model would yield the most accurate estimate of the payoffs, and this hypothesis has been supported in a recent study by Komorita and Kravitz (1981). In Game II, where the alternatives are certain, if we substitute the single-person alternatives (75, 4.5, and 15 for players A, B, and C, respectively) in Eq. (l), we have: Ej,t,,o

= 75 + (l/3) [165 - (75 + 45 + IS)] = 85;

(2a)

E&ABCj = 45 + (l/3) [165 - (75 + 45 + 15)l = 55;

(2b)

E&ABCj = 15 + (l/3) [I65 - (75 + 45 + 15)] = 25.

UC)

If we iterate Eq. (1) over successive rounds of bargaining, it can be shown that the predictions of the equal excess model are invariant (& = KS). Thus, in Game II the equal excess model predicts no change in expectations over successive rounds of bargaining. Moreover, note

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that the predicted values for this game (85-45-25) coincide exactly with the asymptotic predictions of the model for Game I (see Table 1). The bargaining theory. The bargaining theory (Komorita & Chertkoff, 1973) was originally formulated for simple games in which the power of the players is manipulated by the assignment of weights (resources, votes). Two norms of justice are assumed to operate in a coalition situation: The equality norm which prescribes an equal split, and the parity norm which prescribes that the shares of coalition members should be directly proportional to their resource weights. It is also assumed that each player will appeal to that norm of justice that maximizes his/her sehare of the prize: The weak players (with small weights) are expected to demand equal shares, while the strong players (with large weights) are expected to demand shares based on the parity norm. When resource weights are not assigned, as in Games I and II, the parity norm is meaningless. Accordingly, Komorita and Tumonis (1980) proposed a proportionality norm which prescribes that shares should be divided in direct proportion to the alternatives of the coalition members, rather than on their resource weights. For example, suppose players A and B in Game I are negotiating the division of v(AB) = 120. Player B will demand an equal split (60-60), while player A will demand a division of the prize that is proportional to their expectations in alternative coalitions. Since expectations in alternative coalitions are 45 for A in AC and 30 for B in BC, proportionality with respect to these alternative expectations prescribes a 72-48 split, i.e., (45/75)(120) = 72 for A and (30/75)(120) = 48 for B. A person’s maximum expectation in a coalition, denoted E max3 is then the larger of the values, based on either the equality or proportionality norm, e.g., E,,,,, for A is 72, based on proportionality, while E,,, for B is 60, based on equality. As in the original theory, the revised theory predicts that on trial 1 the most likely agreement will be a “split-the-difference” solution between the prescriptions of the two norms of justice: a 66-54 split for A and B, respectively. At the asymptote (after an indefinite number of trials), the theory predicts that expectations will converge to a solution that is proportional to each member’s maximum expectation in alternative coalitions, denoted Ek,,. For the AB coalition, the Ekax values for A and B are 60 (in AC) and 34 (in BC), respectively. Proportionality with respect to these Ekax values yield asymptotic predicted values of 77-43 for A and B, respectively. For the two-person coalitions, E,,, are based on the equality norm or proportionality with respect to alternatives, whichever is greater. The trial 1 predicted values are based on a split-the-difference solution between the two E,,, values. At the asymptote, the predicted values for the two-person coalitions are based on proportionality with respect to ELaxr the maximum expectation in alternative coalitions.

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HAMILTON,

AND KRAVITZ

The predictions of the bargaining theory for the two-person coalitions are based on the revision proposed by Komorita and Tumonis (1980). In the case of superadditive games, where v(ABC) is greater than the values of the two-person coalitions, an additional assumption is necessary to derive predictions. Hence, we shall assume that expectations in the two-person coalitions serve as the basis for negotiations in the grand coalition, and that the players will appeal to and demand their respective maximum expectations in the two-person coalitions. Proportionality with respect to these Em,, values yield values of 67-56-42, for A, B, and C, respectively, and a split-the-difference between proportionality and equality yield trial 1 predicted splits of 61-56-48. At the asymptote we shall assume that payoffs converge to a solution that is proportional to these E max values (67-56-42). For Game II, proportionality with respect to the single-person alternatives, v(i), yield values of 92-55-18, for A, B, and C, respectively. The split-the-difference solution between proportionality and equality (trial 1 values) yield a predicted split of 73-55-37. Table 2 summarizes the predictions of the two theories for the two games. In the present study all groups played several games for one trial only. Hence, the predictions of the bargaining theory shown in Table 2 are trial 1 values. For the equal excess model, the values shown in Game I are round 1 estimates (for naive bargainers playing for small incentives); in Game II the round 1 and asymptotic estimates are identical. It can be seen that both theories predict that the variance of payoff shares (deviations from equality) will be greater when the alternatives are certain (Game I) than when they are uncertain (Game II). Moreover, the equal excess model predicts that the variances will be much larger in both conditions than does the bargaining theory. Other theoretical predictions. As shown in Table 2 the predictions of two other models were also tested: the Shapley value (1953) and a variant TABLE SUMMARY

Game I II

PREDICTIONS

2

OF THEORIES

FOR GAMES

I AND

II”

Equal excess”

Bargaining theory‘

Shapley value

Shapley-w (equity)

72-58-35 85-55-25

61-56-48 73-55-37

70-55-40 85-55-25

77-55-33 77-55-33

” In Game I, v(i) = 0 and the players have two-person (uncertain) alternatives. In Game II, v(o) = 0 and the players have one-person (certain) alternatives. b In Game I the predictions of the equal excess model are round 1 estimates and are based on a revision of the original model. In Game II the round 1 and asymptotic predictions are identical. ‘ Predictions of the bargaining theory are trial 1 estimates (split-the-difference between proportionality and equality).

EFFECTS OF ALTERNATIVES

123

of the Shapley principle, denoted Shapley-w.3 The Shapley value is a normative (prescriptive) theory and is restricted to superadditive games. According to the Shapley value, the power of player i is a function of what player i contributes to coalition S, i.e., the increment in the value of coalition S when player i is added to S. The Shapley value of player i, denoted &, is the average (unweighted mean) of such increments, over all possible subsets S of N, and is given by

h=C

(s - l)! (n - s)!

,,

[v(S) - 4s - -WI

where n denotes the number of players: S denotes all possible subsets of n; s denotes the number of players in S; v(S) denotes the value of S; v(S - {i}) denotes the value of coalition S when player i is deleted; and the summation is over all coalitions that contain player i. As shown in Table 2, for Game I, the Shapley values of the three players are 70-55-40. In Game II, in accordance with Roth’s (1977b) analysis of the Shapley value, we shall assume for theoretical purposes that v(c) = v(i) + ~6). Based on this assumption the Shapley values in Game II are U-55-25. Note that these predicted values in Game II are identical to the predictions of the equal excess model. Moreover, the predictions of the two models are very similar in Game I as well. Shupley-w. Several variants of the Shapley value have been proposed. These include Shapley and Shubik’s (1954) index of pivotal power; Gamson’s (1964) theory of minimum power; and Roth’s (1977a, 1977b) extension for risk averse and risk neutral players. The Shapley-w (Komorita, 1983) is another variant of the Shapley value, and we shall show that this extension is a special case of Adams’ (1963) equity theory and Homans’ (1960) concept of “distributive justice.” Let wi denote the “worth” of player i to the grand coalition, given by wi = v(G) - v(G - {i}) (4) where v(G) denotes the value of the grand coalition and v(G - {i}) denotes the value of the coalition when player i is deleted (defects, refuses to join). For Game I, substituting values of v(G) and v(i) in Eq. (4), we have wA

= v(ABC) - v(BC) = 165 - 60 = 105;

(54

’ There is considerable evidence to support the predictions of Aumann and Maschler’s (1964) bargaining set, a very promising normative model of N-person cooperative games. We did not include this model in this study because it predicts a range of outcomes when the value of the grand coalition exceeds the sum of the quota values of the bargainers, as in all four games used in this study.

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KRAVITZ

B = v(ABC) - v(AC) = 165 - 90 = 75;

(5b)

wc = v(ABC) - v(AB) = 165 - 120 = 45.

(5c)

W

Suppose each player demands wi, what each contributes to the grand coalition. Since the sum of the wi values exceeds the value of the grand coalition by 60 points (225 - 165), one or more players must concede from wi. If we assume that the players concede equally (20 points each) from their respective wi values, it can be shown that the predicted values are identical to the asymptotic solution of the equal excess model (8555-25). In contrast, if we assume that the players concede proportionally from the wi values, it can be shown that this assumption corresponds to the model

+(wi)= 2. I v(G) where I denotes Shapley-w for player i; wi denotes the “worth” of the player i, as per Eq. (4); v(G) denotes the value of the grand coalition; and the summation is over all players (j = 1, 2, 3, . . . . n). Based on Eq. (6), the Shapley-w values in Game I are 77-55-33, as shown in Table 2. In Game II, we shall assume that v(i) = v(i) + v(j), so that wi = v(G) - vb) - v(k). For example, wA = v(ABC) - v(B) - v(C) = 165 - 45 - 15 = 105, which is identical to wA in Game I. It can be shown that the wi values of the three players are identical in the two games; hence the predicted payoff shares are also identical, as shown in Table 2. The intriguing feature of the Shapley-w model is that the wi values can be interpreted as “inputs” in Adams’ (1963) equity theory and Homans’ (1960) “distributive justice” formulation. Both theories assume that an individual’s “output” (share of the reward) should be directly proportional to their respective input (contribution). Equation (6) shows that the Shapleyw is simply a special case of these theories in a coalition bargaining situation. Summary of hypotheses. Table 2 summarizes the predictions of the four theories that were contrasted. The Shapley-w model predicts that payoff shares should not differ in the two types of games (one-person vs two-person alternatives), while the other three theories predict that payoff shares should differ. The Shapley value and the predictions of the equal excess model are remarkably similar, but differ from the predictions of the bargaining theory.4 4 It would also be possible to present both types of alternatives, v(i) and v(i). Such a condition was not manipulated in this study because when both single-person and twoperson alternatives are available to the players, all four theories predict that the outcomes should be identical to those in Game II.

EFFECTS OF ALTERNATIVES

125

In addition to the type of alternatives, the bargaining procedure was also manipulated. In half of the triads, subjects bargained in a face-toface situation, while the other half bargained by means of written offers. Stryker (1972) has hypothesized that bargaining in a face-to-face situation inhibits the competitive tendencies of subjects; hence, variability of payoff splits (deviations from equality) should be smaller with face-to-face bargaining than bargaining by means of written offers. In contrast, the equal excess model implies that restrictions on communication and information among the bargainers should facilitate an agreement on the early rounds of bargaining. Since player A’s payoff share is predicted to increase while player C’s share is predicted to decrease over rounds of bargaining, the equal excess model implies that variability of payoff splits should be larger with face-to-face bargaining than by means of written offers. Thus, the results of the two bargaining conditions should provide a critical test of the two opposing hypotheses.5 METHOD Subjects The subjects were 96 male undergraduate students enrolled in introductory psychology classes. Participation in the experiment partially fulfilled a requirement of the course.

Design Three variables were manipulated: (1) Alternatives: one-person vs two-person, denoted v(i) and v(ij) conditions, respectively; (2) Bargaining procedure: face-to-face vs written offers; and (3) Games: four games varying in players’ quota values. Each triad played four games, all of the same type and with the same procedure, so that the overall design was a 2 X 2 x 4 ANOVA with repeated measures on the last factor. Eight triads were assigned to each of the four (2 x 2) between-subject conditions. In all games the value of the grand coalition was 165. Table 3 shows the four games in the v(i) and v(b) conditions. Note that Game 1 is the same game we used previously to illustrate the predictions of the four theories. Also note that in all four games the values of the two-person coalitions in the v(o) condition equaled the sum of the values of corresponding single-person coalitions in the v(i) condition. For example, in the v(i) condition of Game 2, v(A) = 90, v(B) = 30, and v(C) = 15. In the v(a) condition for this game v(AB) = 120, v(AC) = 105, and v(BC) = 4.5. Hence, v(h) = v(i) + v(j). The four games differ in two respects. Games 1 and 2 differ from games 3 and 4 in the magnitude of the quota values (alternatives): The sum of the quota values in Games 1 and 2 (135) is 50% larger than the sum of quota values in Games 3 and 4 (90). Games 1 and 3 differ from Games 2 and 4 in the ratio of quota values of the players: (5: 3: 1) in Games 1 and 3 and (6:2: 1) in Games 2 and 4, for players A, B, and C. respectively. Half of the triads in each condition ’ The equal excess model assumes that agreements under unrestricted communication will occur on later rounds than agreements under restricted communication. However, the model does not specify precisely which round should provide the most accurate predictions under these conditions. Although the bargaining procedures used in this study (face-toface vs written offers) differ in the amount of communication allowed, the round 1 estimates were used for theory tests in both bargaining conditions because of this ambiguity in the model’s predictions.

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HAMILTON,

FOUR GAMES

OF

AND KRAVITZ

TABLE 3 v(i) AND v(ij) CONDITIONS~

4)

VW

Game?

A

B

C

AB

AC

BC

1 2 3 4

75 90 50 60

45 30 30 20

15 15 10 10

120 120 80 80

90 105 60 70

60 45 40 30

’ v(r) denotes one-person (certain) alternatives; v(ij) denotes values of two-person (uncertain) alternatives. * In all four games, the value of the grand coalition (ABC) was 165.

played the four games in a randomized reverse order.

order; the other half played the four games in the

Procedure In all conditions, the three subjects were seated in armchair desks arranged in a circle facing each other. On each desk was a set of printed instructions, a pencil, and a card identifying their position in the game (marked H, P, or S). Subjects were asked to read the instructions while listening to a tape recording of them. In the instructions, subjects were asked to assume that they were presidents of firms considering mergers with other businesses to increase their profits. In the v(i) conditions, subjects were informed that they could negotiate to form the three-person merger (coalition) and divide the 165 points, or accept the number of points each could earn alone (single-person alternative). Unanimous agreement was required to form the three-person merger, so that if one person decided to choose his alternative, all were forced to do so. In the v(b) condition, they were informed that they could negotiate to form either the three-person or the two-person coalitions and divide the specified number of points. In all conditions subjects were instructed to maximize their point totals (profits) over all games. The points accumulated over all games would be converted to prizes worth up to $3 (ball-point and felt-tip pens and pencils) at the end of the experiment. Subjects were informed that they would remain in the same bargaining position throughout the experiment. However, it was stressed that the amount of prizes each person received would not depend on the number of points earned in relation to other members of the group. Instead, their point totals would be compared with the point totals of other persons who had held the same bargaining position in previous groups. Thus, although subjects held different bargaining advantages through the experiment, it did not affect the amount of prizes each received. This incentive scheme was used to reduce pressures toward equal splits and increase the individualistic tendencies of the subjects (cf. Mumighan, Komorita, & Szwajkowski, 1977). Practice trials. Triads in all conditions were given the same practice trials in which a two-stage bargaining procedure was used: (1) In the “tentative offer stage,” each subject was asked to make nonbinding written offers to form a coalition. These written offers included the shares (amount of points) for each person in the proposed coalition. These nonbinding offers were then projected onto a screen with an overhead projector so that all subjects could see the offers made by each person. The group was then given up to 3

EFFECTS OF ALTERNATIVES

127

min to discuss these offers. The tentative offer stage ended when either 3 min had elapsed or when one or more persons submitted “Offer Response Forms,” which initiated the second stage of the process. (2) In the “offer response stage,” subjects were asked to accept or reject the “tentative offers” of the first stage. They could reject all proposalsincluding their own-but they could not accept more than one of the proposals. To ensure that the grand coalition would form with high probability, a stricter procedure of forming coalitions other than the grand coalition was used. In forming the grand coalition, a “binding agreement” was reached (coalition formed) if all three members accepted the same proposal. In contrast, in forming the two-person coalitions in the v(ij) condition, a “binding agreement” was reached if the same proposal was accepted on two consecutive rounds of bargaining. Moreover, subjects in this condition were advised to avoid letting a two-person coalition form that would exclude them, and that they should make an appealing offer to one of the others, so as to inhibit a “binding agreement” on the next round. Similarly, in the r(i) condition, for the single-person alternative to become a “binding agreement,” one of the subjects had to propose and accept his single-person alternative on two consecutive rounds. Subjects in this condition were advised that if one of the others threatened to take his alternative on a given round, and if they preferred the grand coalition, they should make an appealing offer to the threatening person on the next round. If no “binding agreement” was reached on a given round of bargaining, the two-stage procedure was repeated until a “binding agreement” was reached. Three practice games (trials) were presented in all conditions. The three practice games were of the same type, v(i) or r(Q), as the test games, but involved different coalition values. Procedure manipulation. In the face-to-face condition, triads bargained in the four test games under the same procedure as the practice games: make tentative offers; display and discuss offers; respond to offers (accept/reject). In the written offer condition, after the three practice trials, subjects were moved behind wooden partitions to prevent any communication between them during the test trials. The bargaining procedure was identical to that used in the practice trials with the single exception that the free-discussion period was eliminated.

RESULTS

Over the 128 instances (32 triads x 4 games), the grand coalition formed in 123 (96%) of the cases. An examination of the 5 cases in which the grand coalition did not form revealed no systematic relation to any of the independent variables. Hence, in all subsequent presentation of the data, the results are based on unweighted means analyses. Effects of Independent Variables To assess the effects of the independent variables on the payoff splits in the grand coalition, separate analyses were conducted on the following dependent measures: shares of persons A, B, C, and the difference between the shares of persons A and C, denoted (A - C). The last measure (A - C) was used as an index of variability of payoff shares. Since these measures are not independent, the .Ol level of significance was used in all tests to control for Type I error. Moreover, the homogeneity of treatment difference variance assumption was not satisfied for any of the measures; hence, the Greenhouse-Geisser (1959) correction, an extremely conservative test, was used. The results of these analyses indicated

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PREDICTED

Game 1 2 3 4

AND

Equal excess* 85-55-25 100-40-25 75-55-35 85-45-35

OBSERVED

AND KRAVITZ 4

PAYOFFS

IN v(i)

CONDITION’

Bargaining theory’

Shapley-w (equity)

Observed meand

73-55-37 82-46-37 73-55-37 82-46-37

77-55-33 88-44-33 66-55-44 71-50-44

87-55-23 103-39-23 80-55-30 93-42-30

” Predictions of the Shapley value are identical to predictions of the equal excess model in this condition. ’ The round 1 and asymptotic predictions of the equal excess model are identical in this condition. ’ The predictions of bargaining theory are trial 1 values (all groups played each game only once). ’ Data are based on 16 triads (pooled over bargaining procedure).

significant main effects of both Alternatives and Games for all four measures, and a significant interaction between these two variables on all measures except for player C’s payoff share. Surprisingly, neither the main effect of procedure, nor its interactions with any of the other variable, was significant for any of these measures. The last column of Tables 4 and 5 shows the mean splits in the grand coalition for the v(i) and v(g) conditions, respectively. These means show that the main effects of Alternatives can be attributed to the fact that Person A achieved a much larger mean share in the v(i) condition (91) than in the v(d) condition (71). Conversely, Persons B and C achieved larger mean shares in the v(b) condition (58, 37) than in the v(i) condition (48, 26). These results suggest that having one-person alternatives enhance TABLE PREDICTED

Game 1 2 3 4

Equal exces@ 72-57-35 79-49-39 67-57-42 71-51-43

AND

OBSERVED

5

PAYOFTS

IN v(g)

CONDITION”

Bargaining theory’

Shapley value

Observed meand

61-56-48 63-53-49 61-56-48 63-53-49

70-55-40 77-48-40 65-55-45 70-50-45

71-59-36 71-59-35 71-57-37 71-55-39

” Values of Shapley-w (equity theory) are identical to those shown in Table 4 for condition. b Values represent round 1 predictions of equal excess model. ’ Values represent trial 1 predictions (each game played once). ’ Data are based on 16 triads (pooled over bargaining procedure).

v(i)

EFFECTS OF ALTERNATIVES

129

the bargaining strength of the “stronger” persons, at the expense of the weaker. In order to examine the main effect of Games in detail, additional analyses were conducted.6 For the v(i) condition, Table 4 shows that the main effect of Games can be attributed to the fact that Person A achieved larger shares in Games 1 and 2 (sum of quota values equals 135) than in Games 3 and 4 (sum of quotas equals 90). In contrast, Persons B and C achieved smaller shares in Games 1 and 2 than in Games 3 and 4. Similarly, Person A achieved larger shares in Games 2 and 4 (ratio of 6:2: 1) than in Games 1 and 3 (ratio of 5: 3: 1). Conversely, the mean shares of Person B were in the opposite direction, and the mean shares of Person C were unaffected by ratio of quota values.’ What is surprising is the fact that these effects of games are unique to the v(i) condition. In the v(g) condition, Table 5 shows that splits are almost invariant across games. This difference in effects of games in the two conditions accounts for the significant interaction between Alternatives and Games. One plausible explanation for the absence of variability in the v(u) condition is that a group norm for a fair (reasonable) division of rewards developed in the first or second game, and this norm may have been used as a basis for reward division in all subsequent games. This hypothesis is plausible because one of the functions of social norms is to minimize tension and conflict when there is disagreement among group members, and in a bargaining situation the use of norms reduces the need to haggle and bargain tough. To eliminate this possible confounding effect of order of play, mean payoffs on the first game only were analyzed. Since half of the triads in each condition played Game 2 first while the other half played Game 3 first, a 2 X 2 x 2 (Alternatives x Procedure x Games) ANOVA was conducted. However, the results of this analysis yielded the identical ’ Separate ANOVA’s, treating Games as a 2 x 2 factor (sum x ratio of quota values). were conducted on the outcomes of persons A, B, and C in the v(i) condition. These tests crossed two between-subjects factors (Procedure x Game Order) and two within-subjects factors (Sum and Ratio of quota values of the four games) in a 2J design. The results of these analyses indicated that the main effects of sum of quota values were significant at the .Ol level for all three players (A, B, and C). The main effects of Ratio of quota values were significant at the .Ol level for player A and B’s outcomes. but not for player C’s outcomes. ’ The observed reward divisions in the v(i) condition reveal a remarkable consistency: In each of the four games it appears that the reward division is three-tenths of the way between equal division of the excess and proportional division of the excess (proportional to alternatives). We are indebted to Dave Messick for observing this linear relation, but we shall not pursue this matter for two reasons: (I) We have no explanation for this finding: and (2) this linear relation was not observed in a previous study using a similar paradigm (Komorita, Lapworth. & Tumonis, 1981).

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pattern of significant effects: main effects of Alternatives and of Games and the interaction between these two factors (all significant at .Ol level). Tests of Theories Tables 4 and 5 show the predictions of the four theories for the v(i) and v(g) conditions, respectively. As a measure of predictive accuracy, the square root of the mean square error (RMSE) was used, RMSE

= j/C

(Pi - Oi)2/3,

where Pi and Oi denote the predicted and observed share of bargainer i, and the summation is over the three bargainers in each triad. Values of RMSE were assessed for each theory in each triad, and Table 6 shows the mean RMSE values for the v(i) and v(b) conditions. Since bargaining procedure did not have significant effects on payoff shares, data for the face-to-face and written offer conditions were combined, and for each pair of theories, a 2 x 2 x 4 (Alternatives x Theories x Games) ANOVA was conducted.8 As in the analyses of payoff splits, to minimize Type I error, alpha level was set at .Ol and the Greenhouse-Geisser correction was used in all tests. The last row of Table 6 summarizes the significant main effects of theories, where grand means with a common subscript indicate that they do not differ significantly at the .Ol level. It can be seen that the predictions of equal excess model and Shapley value were more accurate than the predictions of bargaining theory and Shapley-w. In addition, there were several significant interactions (.Ol level) between theories and alternatives. Equal excess and Shapley value are more accurate in the v(i) than in the v(ij) conditions, and this is one of the main reasons that these two theories are more accurate than bargaining theory and Shapley-w. Indeed, Table 6 shows that in the v(ij) condition, it is difficult to differentiate the four theories in terms of their accuracy. In the v(i) condition, however, equal excess and Shapley value are clearly more accurate than bargaining theory and Shapley-w. There were also significant Theory x Games and three-way interactions. In the v(i) condition, equal excess and Shapley value are more accurate in Games I and 2 than in Games 3 and 4, but the reverse is true for bargaining theory. Thus, equal excess and Shapley value are more accurate when the players have one-person alternatives and when the sum of these one-person alternatives (135) is relatively equal to the value of the grand coalition (165) then when the sum is much smaller (90). ’ A single 2 x 4 x 4 (Alternatives x Theories x Games) analysis was not used because the Q’s for the error mean square would be highly spurious. In the 2 x 2 x 4 analyses, in contrast, the main effect of Theories is comparable to the t test for correlated means.

EFFECTS

VALUES

Grand meanb

TABLE 6 RMSE FORFOUR THEORIES’

Equal excess

Condition

v(i)

OF

131

OF ALTERNATIVES

Bargaining theory

Shapley value

Shapley-w

(equity)

I 2 3 4 Mean

2.18 2.67 5.58 6.62 4.26

11.02 14.65 4.35 8.08 10.03

2.18 2.67 5.58 6.62 4.26

8.04 10.70 11.98 16.33 11.76

1 2 3 4 Mean

7.56 10.97 9.83 8.85 9.30

10.87 11.76 11.56 10.55 11.18

7.77 1I .22 10.43 9.08 9.63

8.54 15.22 10.30 9.14 10.80

6.78,

10.61,

6.95,

11.28,

U RMSE denotes square root of mean of squared deviations (error) between predicted and observed values. Data have been pooled over two bargaining procedures. b Values with a common subscript do not differ significantly (.Ol level).

DISCUSSION AND CONCLUSIONS

The results of this study clearly indicate that in a bargaining situation whether players have one-person or two-person alternatives significantly affects the outcomes negotiated by the bargainers. Player A (the strong person) was able to negotiate a greater share in the v(i) condition, where alternatives were certain, than in the v(u) condition, where alternatives were uncertain. One plausible explanation of this effect is that in the v(i) condition, player A’s threat to form alternative coalitions (t,o justify his demand) depended on the cooperation of one of the weaker players, a player he was threatening. In the v(i) condition, in contrast, his threat to take his single-person alternative did not depend on others, and his threat may have been more credible and persuasive than in the v(c) condition. Another plausible explanation of this effect is the difference in consequence to player A if an agreement in the grand coalition could not be reached. In the v(g) condition, players B and C could form a coalition against him, and he would receive zero payoff, whereas in the v(i) condition, he was assured of his single-person alternative. Thus, in negotiations in the grand coalition, player A may have been more risk averse in the v(o) than in the v(i) condition. In either case (explanation), under the conditions of this study, the results suggest that the absence of one-person alternatives attenuates the bargaining strength of the stronger person(s) in the group.

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This hypothesis is supported by the fact that there was virtually no variability of the payoffs in the v(q) condition. We initially attributed this result to the effect of group norms, and there is theoretical and empirical support for this explanation (Thibaut & Kelley, 1959; Thibaut & Faucheux, 1965; Thibaut & Gruder, 1969). However, since our supplementary analyses revealed only marginally significant effects of game order (see footnotes 3 and 5), support for this explanation is slim. A more plausible explanation is that the threat of one-person alternatives is more credible, and provides greater bargaining power, then the threat of two-person alternatives; thus, the presence of one-person alternatives should have greater impact on coalition outcomes. This hypothesis, moreover, is supported by the fact that all of the theories predict greater variability among games in the v(i) condition than in the v(ij) condition (cf. Tables 4 and 5). With regard to the validity of the four theories that were contrasted, the predictions of equal excess model and Shapley value did not differ significantly and both were more accurate than bargaining theory and Shapley-w in the v(i) condition. The Shapley value is simple but elegant, and there is a rigorous axiomatic basis for the theory (Lute & Raiffa, 1957; Rapoport, 1970). The main weaknesses of the theory are that it is restricted to superadditive games, and it does not yield predictions about which coalitions are likely to occur. These weaknesses are not shared by the equal excess model: It is applicable in both nonsuperadditive and superadditive games and predicts which coalitions are likely to form. Its main weakness is that it predicts a range of outcomes over rounds of bargaining, and in its present form, it does not specify the exact round on which an agreement is likely. Despite this weakness, the equal excess model seems to be one of the more promising general models of coalition formation (Komorita & Tumonis, 1980). The bargaining theory is also one of the more promising theories of coalition formation (cf. reviews by Murnighan, 1978; Miller & Crandall, 1980). For the v(i) condition the trial 1 estimates of the bargaining theory were less accurate than the predictions of equal excess model and Shapley value. However, the asymptotic predictions of the theory yield more accurate predictions than the trial 1 estimates. In the present study the trial 1 predictions were used because each triad played each game only once. For the asymptotic values, the mean RMSE values are 7.83 and 9.31 for the v(i) and v(g) conditions, respectively. With reference to the RMSE values shown in Table 6, these asymptotic values do not differ significantly from the values for equal excess model and Shapley value. The asymptotic predictions of the bargaining theory are based on proportionality with respect to each person’s Em,, in the two-person coalitions, while the trial 1 values are based on a split-the-difference solution between proportionality and equal shares. There are two plausible explanations

EFFECTS

OF ALTERNATIVES

133

for the greater accuracy of the asymptotic values. First, when there are two bargainers, one demanding proportionality and the other demanding equality (as conceptualized in the original theory), splitting the difference may be a reasonable estimate of the final outcome. However, when there are three or more bargainers, as in this study, splitting the difference may not yield a good estimate of the outcome. For example, if a majority of the bargainers demands proportionality, the proportionality rule (the asymptotic prediction) may be adopted by the group. A second plausible explanation is based on an assumption of the equal excess model: the trial 1 values may apply to completely naive bargainers, unfamiliar with the structure of coalition games, while the asymptotic values may apply to more sophisticated, experienced bargainers. This interpretation of the difference between trial 1 and asymptotic estimates, borrowed from equal excess model, raises an interesting question regarding the difference in outcomes for naive and sophisticated bargainers. In the v(i) condition, the round 1 and asymptotic predictions of equal excess model are identical. In the v(g) condition, however, the asymptotic values are considerably different from the round 1 values shown in Table 5. They are identical to the predicted values in the v(i) condition (see Table 4). Thus, equal excess model predicts that the outcomes in the v(i) and v(o) conditions should be identical for sophisticated, experienced bargainers. The main implication of this observation is that the relative validity of equal excess model and the Shapley value can be assessed by comparing outcomes in the ~($1 condition using sophisticated bargainers. In Game 1, for example, equal excess model predicts that the shares should be (85-55-25) while the Shapley values are (70-55-40). Fortunately, there are some data collected by Medlin (1977) that provide such a comparison. In Medlin’s study, subjects were given 3 hrs of instructions, practice, and training prior to the test games; hence, they may be considered to be “experienced” bargainers, and the asymptotic values of the equal excess model are more accurate than the Shapley value. Indeed, Medlin concluded that “the Shapley value model is rejected as inadequate” (p. 48) and thus, Medlin’s data provide additional support for the validity of the equal excess model. We indicated earlier that Shapley-w, the fourth theory that was contrasted, is a special case of Adams’ (1963) equity theory and Homans’ (1960) concept of “distributive justice.” The results of this study indicate that the Shapley-w is less accurate than both equal excess model and the Shapley value. In defense of equity theory, it might be argued that our assumption that wi, the “worth” (contribution) of person i, when added to the grand coalition, is not a valid measure of input. If so, equity theory must clarify and specify more exactly what type of measures can be, and cannot be, treated as inputs. The fact that both bargaining theory and Shapley-w are based on a proportionality norm may provide some

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insights into why they yielded inaccurate predictions. Bargaining theory is based on proportionality with respect to maximum expectations in alternative coalitions (Ek,,), while Shapley-w is based on proportionality with respect to wi (see Eq. (4)). In contrast, equal excess model and the Shapley value are based on the differences between the wi values (wi Wj). Thus, it is equally plausible that wi is a valid measure of input, but the functional relation between inputs and outputs (shares of the bargainers) may be a difference rather than a ratio relation. This hypothesis is consistent with Harris’ (1976) linear equity formula, and is supported by a study by Komorita and Kravitz (1979). As a final point, the nonsignificant effects of bargaining procedure deserves comment. The fact that face-to-face bargaining yielded virtually the same outcomes as bargaining by written offers suggests that faceto-face bargaining does not necessarily reduce the competitive tendencies of subjects, as hypothesized by Stryker (1972). In addition to possible effects on motivational tendencies, there is another important difference between the two types of procedures: the amount of information available to the bargainers. Stryker’s hypothesis was proposed in the context of Gamson’s (1961) “reciprocity paradigm,” where subjects made initial choices of a coalition partner. If a pair of subjects reciprocated choices, they were led to another room to negotiate the division of the prize. If they reached an agreement, the game terminated: if they could not reach agreement, the procedure was repeated. The important point is that the remaining subjects, those who did not reciprocate choices, were not aware of the offers and counteroffers of the negotiating pair. In a study by Komorita and Meek (1978), Gamson’s procedure and a procedure similar to the written offer condition of the present study were contrasted, and markedly different results were obtained. However, their contrast is confounded with the amount of information available to the subjects. In the present study, subjects in all conditions had complete information about all offers made by all subjects, e.g., in the written offer condition, all offers were displayed on a screen visible to all three bargainers. Thus, the results of the present study suggest that it is not face-to-face bargaining per se that evokes differences in outcomes (a motivational factor), but the amount of information available to the bargainers (a cognitive factor). REFERENCES Adams, J. S. Toward an understanding of inequity. Journal of Abnormal and Social Psychology, 1963, 67, 422436. Aumann, R. J.. & Maschler. M. The bargaining set for cooperative games. In M. Dresher, L. S. Shaplay, & A. W. Tucker (Eds.), Advances in game theory. Princeton, N.J.: Princeton Univ. Press, 1964. Chertkoff, J. M. Sociopsychological theories and research on coalition formation. In S. Groennings, E. W. Kelley, & M. Leiserson (Eds.), The study of coalition behavior. New York: Holt, Rinehart & Winston, 1970.

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Gamson, W. A. A theory of coalition formation. American Sociological Review, 1961, 26, 373-382. (a) Gamson, W. A. Experimental studies of coalition formation. In L. Berkowitz (Ed.), Advances in experimental social psychology (Vol. 1). New York: Academic Press, 1964. Greenhouse, S. W., & Geisser, S. On methods in the analysis of profile data. Psychometrika, 1958, 24, 95-112. Harris, R. J. Handling negative inputs in equity theory: On the plausible equity formulae. Journal of Experimental Social Psychology, 1976, 12, 194-209. Homans, G. C. Social behavior: its elementav forms. New York: Harcourt, Brace & World, 1961. Komorita, S. S. A weighted probability model of coalition formation. Psychological Review, 1974, 81, 242-256. Komorita, S. S. An equal excess model of coalition formation. Behavioral Science, 1979, 24, 369-381. Komorita, S. S. Coalition bargaining. In L. Berkowitz (Ed.), Advances in experimental social psychology (Vol. 17). New York: Academic Press, 1983. Komorita, S. S., & Chertkoff, J. M. A bargaining theory of coalition formation. Psychological Review, 1973, 80, 149-162. Komorita, S. S., & Kravitz, D. A. The effects of alternatives in bargaining. Journal of Experimental Social Psychology, 1979, 15, 147-157. Komorita, S. S., & Kravitz, D. A. The effects of prior experience on coalition behavior. Journal of Personality and Social Psychology, 1981, 40, 675-686. Komorita, S. S.. Lapworth, W. C., & Tumonis, T. M. The effects of certain vs. risky alternatives in bargaining. Journal of Experimenral Social Psychology, 1981, 17, 525544. Komorita. S. S., & Meek, D. D. Generality and validity of some theories of coalition formation. Journal of Personality and Social Psychology, 1978. 36, 392-404. Komorita, S. S., & Tumonis, T. M. Extensions and tests of some descriptive theories of coalition formation. Journal of Personality and Social Psychology. 1980, 39, 256-268. Lute, R. D., & Raiffa, H. Games and decisions. New York: Wiley, 1957. Medlin, S. M. Effects of grand coalition payoffs on coalition formation in three-person games. Behavioral Science, 1976, 21, 48-61. Miller, C. E., & Crandall, R. Experimental research on the social psychology of bargaining and coalition formation. In P. B. Paulus (Ed.), Psychology ofgroup injuence, Hillsdale, N.J.: Erlbaum, 1980. Mumighan. J. K. Models of coalition behavior: Game theoretic, social psychological, and political perspectives. Psychological Bulletin, 1978, 85, 1130-l 153. Murnighan, J. K., Komorita, S. S., & Szwajkowski, E. Theories of coalition formation and the effects of reference groups. Journal of Experimental Social Psychology, 1977, 13, 166-181. Rapoport, An. N-person game theory. Ann Arbor: Univ. of Michigan Press, 1960. Roth, A. E. Bargaining ability, the utility of playing a game and models of coalition formation. Journul of Mafhematical Psychology, 1977, 16, 153-160. (a) Roth, A. E. The Shapley value as a von Neumann-Morgenstern utility. Econometrica. 1977, 45, 657-664. (b) Shapley, L. S. A value for n-person games. In H. W. Kuhn & A. W. Tucker (Eds.). Contributions to the theory of games (Vol. 2). Princeton, N.J.: Princeton Univ. Press, 1953. Shapley, L. S., & Shubik, M. Method for evaluating the distribution of power in a committee system. American Political Science Review, 1954, 48, 787-792. Stryker. S. Coalition behavior. In C. G. McClintock (Ed.), Experimental social psychology. New York: Holt. Rinehart & Winston. 1972.

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Thibaut, J., & Faucheux, C. The development of contractual norms in a bargaining situation under two types of stress. Journal of Experimental Social Psychology, 1965. 1, 89102. Thibaut, J., & Gruder, C. L. Formation of contractual agreements between parties of unequal power. Journal of Personality and Social Psychology, 1%9, 11, 59-65. Thibaut, J., & Kelley, H. H. The social psychology of groups. New York: Wiley, 1959.