Mathematical Approaches to the Study of Power in Group Decision Making: a Review

Joel H. Steckel, Columbia University
ABSTRACT - This paper attempts to synthesize published mathematical approaches to the study of power in a variety of disciplines from the viewpoint of consumer research. We demonstrate the graph and game theory approaches, Coleman's theory of collective decisions, and weighted linear models. The focus is on explanation and discussion of why these approaches have not contributed greatly to consumer research. The paper identifies some directions for future research.
[ to cite ]:
Joel H. Steckel (1985) ,"Mathematical Approaches to the Study of Power in Group Decision Making: a Review", in NA - Advances in Consumer Research Volume 12, eds. Elizabeth C. Hirschman and Moris B. Holbrook, Provo, UT : Association for Consumer Research, Pages: 577-581.

Advances in Consumer Research Volume 12, 1985      Pages 577-581


Joel H. Steckel, Columbia University


This paper attempts to synthesize published mathematical approaches to the study of power in a variety of disciplines from the viewpoint of consumer research. We demonstrate the graph and game theory approaches, Coleman's theory of collective decisions, and weighted linear models. The focus is on explanation and discussion of why these approaches have not contributed greatly to consumer research. The paper identifies some directions for future research.


Purchasing behavior often involves group decision making. In consumer settings, the family is an important decision making unit (DMU) . In industrial purchasing, the "buying center" has been recognized as the relevant unit of analysis. Even decisions which appear to be individual in nature (both consumer and industrial buying)can involve group decision making in that the preferences of relevant others are recognized by the individual decision maker (Wind 1976).

Power is a major concept in group decision making. There are many situations in which individuals in a group may have conflicting preferences for potential outcomes; yet decisions do get made. Power is a concept which is often used to explain how one individual overcomes the resistance of others. It is often conceptualized as a force one applies to break down this resistance.

Three major research traditions in power have emerged; the experimental study (c.f. Hovland. Janis, and Kelly 1953), the case study (c.f. Rossi 1957), and the mathematical model (c.f. Shapley and Shubik 1954). From the perspective of consumer research, tile mathematical approach has contributed the least to the understanding of group choice. This paper reviews the major efforts in an attempt to find out why.

We begin by demonstrating the major mathematical approaches that have appeared in other literatures. From this discussion we then deduce that the two major obstacles have been data availability and a lack of conceptual understanding of the phenomenon in question. This is in direct contrast to the literature on individual choice which has long employed surveys and a multiattribute framework. We conclude the paper by pointing out some directions for future development.


This: section briefly describes four well known approaches to the measurement of power in other literatures. They are

1) Graph Theory

2) Game Theory

3) Coleman's Theory of Collective Action

4) Weighted Linear Models.

Graph Theory

The graph theory approach is based on the representation of the group and its 'authority' structure as a set of points with lines connecting some pairs of these points. The points represent the group members and the lines have specified direction which describe the binary relation 'is superior to'. An example, adapted from Kemeny and Snell (1962, p. 103), is shown in Figure 1.



In the figure, for example, A is a direct superior of B, C, and J; I has E, F, and J as direct superiors. In order for the 'is superior to' relation to make sense, three properties must be satisfied (Kemeny and Snell 1962):

(1) Irreflexivity: No one is his own superior.

(2) Asymmetry: A person cannot be superior to another while the other is superior to him.

(3) Transitivity: A superior of a superior is a superior.

These are illustrated in Figure 1 in the following contexts: A is not his own superior (irreflexivity); B is superior to D but D is not superior to B (asymmetry); and A is superior to C who is superior to E so A is superior to E (transitivity).

Harary (1959) introduced a measure of power which utilizes the concept of the level of superiority. This is the number of lines in the shortest path from the superior to the 'subordinate'. Thus in Figure 1, J is one level below A while H is two and G is three. Harary's measure, denoted by h (.) is equal to


where p is said to have nk subordinates at level k. h(p) represents the number of levels between p and his subordinates. In Figure 1 h(A)S 1:3 1 2:5 + 3:1 = 16. Similarly h(B)= h(E)= 5, h(D)= 2, h(F)= h(J)= 1, and h(G)= h(H)= h(I)= 0.

Harary's measure has several limitations. First, it is not necessary that if P is superior to Q then h(P) > h(Q 1. In Figure 2, for example, h(A)t h(B)= 6 despite the fact that A is superior to B. As Kemeny and Snell (1962) point out, this is rectified if we change the definition of "level" from the number of lines in the shortest path to the number in the longest. Then h (A) - 10 and h(B)- 6. Second, Harary's measure (even with Kemeny and Snell's modification) is best suited only to groups that can be inferred to have formal or informal superior-subordinate relationships. It does not apply as well to informal groups with informal processes.



Game Theory

A large number of power indices have arisen in the game theory literature (c.f. Banzhaf 1965; Brams 1975, Deegan and Packel 1978, 1983; Dubey 1975; Lucas 1983; Shapley and Shubik 1954). Each of these efforts depends on certain assumptions about coalition formation within a group and the importance of an individual to such coalitions. The central concept in these measures is that of a winning coalition, one which is large enough to ensure its desired outcome. The list of these coalitions can be quite long for a given group. Therefore, an efficient summary concept is the minimal winning coalition, one that contains no other winning coalitions as proper subsets.

The two most well-known game theoretic indices are those of Shapley and Shubik (1954) and Banzhaf (1965). Shapley and Shubik (1954; n. 788) Propose the following scheme:

There is a group of individuals all willing to vote for some bill. They vote in order. As soon as enough members have voted for it, it is declared passed and the member who voted last is given credit for having passed it. Let us choose the order of the members randomly. Then we may compute how often a given individual is pivotal. This latter number serves to rive us our index.

A computational procedure is implicit in the above discussion write down all n! orders of the" group members. In each ordering, underline the individual whose addition makes the set of members up to and including himself a winning coalition. The Shapley-Shubik index 0. for individual i is equal to the fraction of times that i is that pivotal member

Consider the following example,adapted from Straffin (1983, p. 259), of a group with Sour members A, B, C, and D. The minimal winning coalitions are {AB}. {BC}. and {CD}. There are 24 possible orderings of the members. These are listed below with the pivotal member underlined:








0" = 4/24 = 1/6, 0B = 8/24 = 1/3, 0C = 8/24 = 1/3, 0D = 4/24 = 1/6 .

Our second power index was originally proposed by Banzhaf (1965). He argues (p. 331):

The appropriate measure of a legislator's power is simply the number of different situations in which he is able to determine outcome. More explicitly, in a case in which there are" legislators, each acting independently and each capable of influencing the outcome only by means of his votes, the ratio of the power of legislator X to the power of legislator Y is the same as the ratio of the number of possible voting combinations of the entire legislature in which X can alter the outcome by changing his vote, to the number of combinations in which Y can alter the outcome by changing his vote.

Implicit in this excerpt is the following operational procedure. Write down all the winning coalitions. In each one underline the swing members, those whose defection would cause the coalition no nonger to be winning. If we let ni be the number of winning coalitions for which i is a swing member, then the Banzhaf power index Bi is given


In the previous example, the winning coalitions are all subsets of the group which contain {AB}, {BC}, or {CD}. They are



where the swing members have been underlined. It is trivial to show now that

B" = 1/6, BB = 1/3, BC = 1/3, and BD = 1/6.

In this example, the Banzhaf and Shapley-Shubik indices agree. This is by no means always true. Straffin (1983) presents a series of examples where the differences are quite startling. The difference between the two indices is analogous to the difference between permutations and combinations. Shapley-Shubik reasoning considers the probability that a member will be "pivotal" if all permutations of the members are equally likely while Banzhaf reasoning hinges on the probability of a member being "swing" if all combinations are equally likely.

A great deal of ingenious mathematics has emerged from the study of these two indices (c.f. Dubey 1975a,b; Dubey and Shapley 1979; Papayanopoulos 1973; Roth 1977). This work investigates sets of axioms which imply each index, asymptotic properties of the indices as the group gets arbitrarily large, and topological representations of these indices. The direct relevance to consumer research of much Of this work is unclear. The axioms are usually stated in algebraic terms with little attention to behavior, and consumer research is usually concerned with finite groups.

The entire game theoretic research stream on power stems from political science and consequently has been based upon legislative bodies and voting groups. The winning and minimal winning coalitions in these are well specified often by either majority rule, weighted majority rule, or an electoral college type rule. Despite this, n-person games are a reality in much organizational decision making where such rules are not evident.

If we were to apply this research to organizational buying behavior we would need a way to determine which of the 2n coalitions are in fact winning. This remains a barrier to implementing the theory. Some previous research has shown that ad-hoc groups with an odd number of members tend to form winning coalitions based on simple majorities (Riker 1962; Shull, Delbecq, and Cummings 1970). While this may be useful in predicting group choice (Steckel 1984), the symmetry of the finding provides little insight into assessing an individual's power.

Coleman's Theory of Collective Decisions

In 1966, James S. Coleman published his now classic paper, "Foundations for a Theory of Collective Decisions". The foundations were later built upon in a book (Coleman 1973). Coleman approaches power from the standpoint of social exchange theory (Blau 1964; Homans 1974; Thibaut and Kelley 1959).

Two assumptions of social exchange theory are that inter-dependence is an inherent feature of group formation and exchanges are intrinsic aspects of any group relationship. Therefore, a given individual will support a second one in one decision in exchange for the second's support at some other time. Power then would relate to an individual's ability to obtain favorable consequences in the decisions most important to him. This line of reasoning leads Coleman to examine power as a property relating to a complete system of decisions rather than to group structure or a single decision.

The major mathematical constructs in Coleman's theory are two matrices, control and interest. The control matrix g shows the amount of control each group member has over each decision. The entry cij is the direct control that member j has over decision i. The sum of cij over i is less than or equal to one. It is less than one if the decision outcome is partly controlled by unknown factors. The interest matrix t has entries x.; that reflect the interest of each individual in each decision. The xi are functions of absolute utility differences for outcomes. The motivation for the interest matrix springs from the notion that control of a decision is irrelevant if the outcomes do not mean anything to anybody.

It is clear, then, that an individual's power, or ability to obtain desired outcomes, depends not only on what he controls, but on what others who have interest in what he controls control themselves and can give in exchange. It follows then that the value of controlling a given decision is dependent on the interests of others in that decision and their power. On the other hand, individuals are powerful if they control actions with a high value. Therefore, neither value nor power can be calibrated without regard to the other. Coleman writes (p. 627):

The value of control over action i is equal to the sum of interests of i of each actor times the total power of that actor. In turn, the total power of actor j is equal to the power of j over each action k times the value of action k, summed over all actions k.

Mathematically, these relationships are summarized by


where V a vector of values for the various actions (decisions). The power of the jth group member is E ckj vk, or equivalently the powers can be summarized by C' V.

Consider the following example taken from Coleman (1973, p. 83). There are two members and two decisions with control and interest matrices as shown in Figure 3. The system V = X C V is also shown on the following page.




If we add the constraint that v1 + v2 = 1, the solution is

v1 = 4/7, v2 = 3/7 .

This approach is different from the others in that power relates to a system of decisions and cannot be viewed in the context of single one. Families and buying centers do make series of decisions. Coleman (1973) does proceed to drive a measure for 'partial' control of a decision assuming that the participants perform strategic exchanges of control for events that have more interest for them. However, this still depend.c- on a series of decisions.

What we must ask is whether consumer research is more interested in global system-wide measures of power or the relative power of the participants in particular decisions. Most of the behavioral research on family decision making (e.g. Davis 1970) and organizational buying behavior (e.g. Silk and Kalwani 1982) would suggest the latter. But, one should note that, this view has led to very specific research and may, in the long run, inhibit the development of a general theory.

Weighted Linear Models

The approach most familiar to consumer researchers involves weighted linear models (WLM's). These models conceptualize a group value (e.g. attitude, utility, or choice probability) as being a weighted sum of corresponding individual values where the weights are indicators o' power. Consumer research has used WLM's in modelling family (Curry and Menasco 1979) and organizational buying behavior (Choffray and Lilien 1980). In addition, decision analysts (Keeney and Kirkwood 1975), political scientists (March 1966), and psychologists (Davis 19734 have all used them.

The models are identical in structure to the linear multiattribute models. The individual weights used in individual decision making for the group are analogous to attribute weights for the individual. Unfortunately, researchers have not met with as much success in estimating the individual weights as they have with the attribute ones. Davis and March both attribute this to unrealistic data requirements. In particular, Davis (1973, P. 115) writes:

... experiments on group decision making have not employed a task appropriate to the application of the social decision scheme estimation procedure... In most cases, a single group decision or at most two or three decisions have been observed during the experimental session. Moreover, the task or social issue sometimes varies so much from decision to decision, it is difficult to imagine that the social decision scheme to be estimated remains generally constant.

March (1966) does present a number of variations on the basic model in which the weights vary from decision to decision. In one of them, the force depletion model, power is considered a scarce resource. The exercise of power depletes that resource. We can write that the (unnormalized) weight of the ith individual in the jth decision in a series, Wij is equal to

wio (1 - xi1) (1 - x>i2) ... (1 - xi, j-1)

where the xik represent the proportion of i's total power exercised in the kth decision, w. is the total 'weight' available to i 10 at the beginning. An individual will then choose to act (i.e. choose values of the X.k) depending on his interest in the various 1 actions. In this respect the force depletion model is very similar to Coleman's theory. While March's variations may add many realistic features, they tremendously exacerbate the already present data problems.

In addition, WLM's have the reverse problem of graph and game theory. They are especially well suited to informal decision making. It is the formal process where they fall short in capturing the phenomena.


We have already cited some of the problems in applying each approach to consumer research. Two problems, however, are common to all approaches; data requirements and the lack of a conceptual understanding of exactly what power is.

Data and estimation problems have already been discussed in conjunction with weighted linear models. Unfortunately, the problems are worse in the other approaches. Appropriate data for the WLM are hard to collect but at least we know what they are. It is not nearly as clear what types of data would be necessary to determine Coleman's control matrix, winning coalitions, or 'superior' relations unless they were explicit in a formal decision making process (e.g. majority rule). Informal processes would be extremely difficult to deal with.

A second more serious factor is a lack of conceptual understanding of the phenomena. No two parties agree on what we should be interested in. Is power a property of the group antics total activities as Coleman contends or is it meaningful only with respect to a specific decision? Advocates of the graph theory approach would describe power as it relates a formal hierarchy while game theorists would say power can be described by an individual's membership in winning coalitions. This inconsistency is in itself a cause for concern. As Bierstedt (1950) notes, the more things a term can be applied to the less precise is its meaning.

Even those authors who specify the context of the term do not precisely define it. The weighted linear model, game, and graph theory approaches all describe some index, parameter, or number. Power is then defined to be that quantity with little regard for conceptual clarity. This is a strong hinderance to the true understanding of behavior.


Implicit in the above problems are a set of opportunities for resolving the data problems and conceptualizing the phenomenon.

In order to solve the data problem one must devise procedures for either collecting more observations per group or aggregating data across groups. Promising work has begun in both directions. Krishnamurthi (1983) increased the effective number of observations by using conjoint analysis on household partners' job preference data (spouse or spouse equivalent), both individually and jointly, in order to test hypotheses on joint decision making. A simple extension of his procedure would be to apply conjoint analysis to the joint preference data with individual preferences as the independent variables instead of attributes. The resulting part worths would be akin to the weights of a WLM. In contrast, Steckel, Lehmann, and Corfman (1984) show how data can be aggregated across groups in a WLM by making the weights functions of personal and situational variables.

Progress in the conceptual area is not as encouraging. There are "innumerable theories, clarifications, and reclarifications" (Bacharach and Lawler 1981, p. 10). Those that emphasize probabilities (e.g. Kaplan 1964, Weber 1947) may lend themselves best to mathematical analysis. In any case, a behaviorally significant mathematical contribution cannot be made to the study of power until someone presents a mathematically tractable, meaningful definition of the concept.


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