4
Measuring Power in Voting Bodies* HARVEY W. KUSHNER and ARNOLD B. URKEN Wagner College and Stevens Institute of Technology
In analyzing power, the nature of one's conceptual framework can have important consequences for the conclusions one draws. For instance, if one defines power as an attribute or possession, consistent inferences may be drawn only so long as the relations of power are relatively stable or constant. However, once the relations of power change, a definition of power in terms of an attribute or possession may become misleading. A good example of this type of shift can be found in the literature on international relations. Several writers have noted that changes in relations have severely reduced the scope of power of nations armed with nuclear weapons. 1 Although these nations have increased their power in terms of the nuclear power attributed to them or the number of nuclear weapons they possess, they find that they can no longer infer that the increase in this source of power will be relevant in achieving their goals in the international system. 2 Alternatively, if one defines power as a relation, one's analysis of power may be limited by changes in the sources of power attributed to or possessed b y a given *We would like to thank Professor Steven J. Brarns for his valuable comments on earlier drafts of this paper. I Cf. Henry A. Kissinger, Nuclear Weapons and Foreign Policy (New York: Harper and Row, 1957), Chapter IV; Henry A. Kissinger, "Central Issues of American Foreign Policy," in
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set of actors. This limitation has been highlighted by Kissinger's discussion of the problem of dealing with an international environment in which nations can develop their power without relying on territorial expansion. 3 Given the limitations of these definitions of power, it is desirable to develop measures of power which enable us to analyze situations in which the logical limits of the possessive and relational concepts of power are rigorously analyzed. Hopefully, alternative modes of analysis will enable us to f'md order in what appears to be intractably complex situations. These analyses may make it possible to increase our knowledge of power according to Weber's definition: "the probability that one actor will be in a position to carry out his own will despite resistance, regardless of the basis on which this probability exists." 4 In this paper, we will systematically analyze power in certain voting situations in which the probability that an actor will exert power is circumscribed by the rules of a voting body. Three goals will guide our investigation. The first is to provide a critique of the analysis of senatorial power contained in Dahl's "The Concept of Power. ''5 The second goal is to replicate, where possible, Daht's study of senatorial power and to contrast our findings with Dahl's conclusions. Our analysis will be based on a posteriori probabilistic model of voting behavior proposed by Steven J. Brams. 6 The replication employs a computer program which allows an analyst to calculate an index of voting power for the members of a voting body of variable size. The third goal of this analysis is to evaluate the usefulness of Brains' measure and highlight some problems wl~ich limit its general validity as a measure of voting power, In order to achieve these goals we will first describe the Brains model. Then we will analyze and critique the measure of voting power used by Dahl in studying voting power in the U.S. Senate. After comparing the theoretical characteristics of the two indices, will wilt briefly describe the results of calculations based on the Brains model and contrast them with Dahl's findings. We conclude by evaluating our research on the usefulness of the index.
H. A. Kissinger, American Foreign Policy: Three Essays, (New York: W. W. Norton, 1969); and Stanley Hoffman, "Terror in Theory and Practice," in Stanley Hoffman, The State of War: Essays on the Theory and Practice of International Relations (New York: Praeger, 1965). 2 Kissinger, "Central Issues of American Foreign Policy," op. cir. 3 See footnote 1. 4 Max Weber, Economy and Society: An Outline of Interpretative Sociology, edited by Guenther Roth and Claus Wittich (New York: Bedminster Press, 1968), p. 53. 5 Dahl's analysis is included in Roderick Bell, David V. Edwards, and R. Harrison Wagner (eds.), Political Power: A Reader in Theory and Research (New York: Free Press, 1969). Dahl derived his analysis of senatorial power from "Influence Ranking in the U.S. Senate," an unpublished paper co-authored by Dahl, James G. March, and David Nasatir. This paper was presented at the Annual Meeting of the American Political Science Association, Washington, D.C., September, 1956. We would like to thank Professor March for providing us with a copy of the paper. 6 This model was proposed to overcome the criticisms of Dahl's analysis made in a seminar on power at New York University. See Steven J. Brams, "Reconstructing Coalition~ Formation Processes and Influence Structures from Roll-Call Votes" (New York University, 1972), privately circulated.
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L Description o f the Brains Model The Brains model is based on a set ofaposteriori probabilities derived from roU-call voting data. The design and application of the model are illustrated in the following hypothetical example. Table 1: Hypothetical Voting Body Member
Rolt Calls
Code
1
2
3
4
5
1
Y
Y
Y
N
N
Y = Yes
2
N
Y
N
X
N
N = No
3
Y
A
N
N
N
A = Abstain
4
N
Y
N
Y
Y
X = Absent
5
Y
N
Y
N
Y
Y
Y
N
N
N
Majority Outcome
Table 1 represents a hypothetical voting body of five members voting on five roll calls. In this case, the decision-rule is assured to be a simple majority (at least three out of five with a quorum of three members). In such a case, three different measures of voting power may be defined: individual voting power, pairwise probabitistic power (PDIFF), and weighted pairwise probabilistic power (E(PDIFF)). Individual Voting Power We define the individual voting power of a member i of a voting body to be the probability that his votes agree with the majority outcomes across a set of roll calls, i.e., P(i). For example, in Table 2 the individual voting power for member 1 is 5/5 = 1.00, since member 1 agrees with the majority on all five roll calls. Pairwise Probabilistic Power In contrast to P(i), the second and third measures of voting power represent the influence that pairwise agreement or disagreement with other members has on a voter's chances of being on the winning side. For this purpose, we define a conditional probability which gauges the extent to which two members i and j sustain the majority outcome, given that they agree fAG) with each other [ (P(i,j[AG i ~) ] ; and conditional probabilities that, given that i and j disagree (DG) with each o'~her, i votes with the majority [P(i]DG.i,j') ], or j" votes with the majority [P(j IDGi i) ]. Given these conditional probabilities, we can define a measure which represent~ the difference that agreement, as contrasted with disagreement, makes on each member's chances of being on the winning side.
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Table 2: Individual Power Values Member
Individual Power
Abstentions
Absences
0.80 0.75 1.00 0.40 0.40
For member i, this probabitistic difference (PDIFF) is
PDIFF il j = P(i,jlAGi, j),
P(ilDGi,j) '
and for member j,
PDIFFjli = P(i,jlAGi.j) - P(jlDGi,j)To illustrate,
for members t and 2
PDIFFII 2 =
2 1 1 (~ - ~ ) = ~ =
PDIFF2'It =
(
2
I
=
i
.5
.5
In this example, the difference that l's agreement with 2 and 2's agreement with 1 makes on the probability that each member agrees with the winning outcome is the same. Specifically, on the two roll calls on which these members agree, they agree with the majority outcome in both instances, giving a probability of agreement of 2/2 = 1. Conversely, on the two roll calls on which members 1 and 2 disagree, each agrees with the majority outcome on one roll call, giving a probability of agreement of 1/2. For both actors, the probabilistic difference between voting together is therefore 0.50. Weighted Pairwise Probabilistic Power In the previous section we saw that voters 1 and 2 agree on two roll calls and disagree on two. Yet the PD1FF measure does not take account of the frequencies with which two members agree or disagree with each other over a set of roll calls. In order to account for this factor, the third measure incorporates unconditional probabilities of agreement and disagreement [P(AGi,j) and P(DGi,j) ] as weights in the PDIFF measure to give an expected value
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for PDIFF [E(PDIFF) ]. For member i,
E(PDIFF)il j = P(AGi,j)P(i,jlAGi, j) - P(DGi,j)P(ilDGi,j), and for member j,
E(PDIFF)jli = P(AGi,j)P(i,jlAGi, j) - P(DGi,j)P(jlDGi,j). To illustrate, the E(PDIFF) of member 1 in combination with member 2 is
1!2
025
and for member 2 in combination with member 1 is
211
025
from our hypothetical example in Table 1. In this example, the weighting factors of 1/2 reduce the value of the advantage which each member gains from agreeing with the other from the previous 0.50 to 0.25. Since members 1 and 2 agree with each other on two out of four roll calls in Table 1, their weighting factor for agreement [ P(AG1,2) ] is equal to 2/4 = 1/2. Similarly, since agreement and disagreement with the outcome are events which are mutually exclusive ancl exhaustive, P(llAG1,2) + P(2IDG1,2) = 1. It follows, then, that the weighting factor for disagreement for these two actors [P(DG1,2) ] is equal to 1 - 1/2 = 1/2. Interpretation of the PDIFF and E(PDIFF) Measures PDIFF and E(PDIFF) represent different measures of the difference that one member's agreement makes to another member's probability or expected probability of being on the winning side, i.e., voting with the majority. To illustrate these measures we give in Table 3 all the values of PDIFF and E(PDIFF) for our hypothetical voting bocly presented in Table 1. From the PDIFF and E(PDIFF) values in Table 3, we can determine for any member i the PDIFF or E(PDIFF) advantage (or disadvantage) that agreement with all other members j would bring. To illustrate this advantage (or disadvantage), let us examine the values for member 4 produced by the PDIFF measure. For member 4 the PDIFF that member 1 would bring to member 4 is .75, that member 2 would bring to member 4 is .67, that member 3 would bring to member 4 is 1.00, and that member 5 would bring to member 4 is -.50. Consequently, member 4's most:helpful partner according to this probabilistic measure is member 3. However, the expected probabilistic measure, E(PDIFF), may yield different consequences. For instance, for member 4 the E{PDIFF) that member 1 would bring to member 4 is 0, that member 2 would bring to member 4 is .50, that member 3 would bring to member 4 is .25, and that member 5 would bring to member 4 is -.40. According to the E(PDIFF) measure, member 4's most
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helpful partner is member 2. Thus, the PDIFF measure makes member 4's most helpful partner member 3 whereas the E(PDIFF) measure makes member 4's most helpful partner member 2. Table 3: PDIFF and E(PDIFF) for Hypothetical Voting Body Pair (i,j)
PDIFFi~
E(PDIFF)ilj
PDIFFj Ii
E(PDIFF)j[ i
(1,2)
0.50
0.25
0.50
0.25
(1,3)
1.00
0.75
0.00
0.50
(1,4)
0.25
0.40
0.75
0.00
{1,5)
-0.33
0.00
-0.67
0.40
(2,3)
1.00
0.67
0.00
0.33
(2,4)
-0.33
0.25
0.67
0.50
(2,5)
-0.75
-0.75
-0.25
-0.25
(3,4)
0.00
-0.50
1.00
0.25
(3,5)
0.00
0.00
1.00
0.50
(4,5)
-0.50
- 0.40
-0.50
0.40
In contrast to the relative advantage one member can achieve by voting with another member, there may be a possible disadvantage when two members vote together. For example, for member 4 the PDIFF and E(PDIFF) values that member 5 would bring to member 4 are -0.50 and -0.40, respectively. These negative values indicate that agreement between members 4 and 5 is disadvantageous to being on the winning side. Power Ordering We can determine a rank ordering of voting partners for each member of our hypothetical voting body. (see Table 4). We will refer to this rank ordering as the "power ordering." The power ordering represents the ranking of each member vis-a-vis all other members. This ordering indicates which member is the most helpful partner in bringing the member in question a greater probability (or expected probability) of being on the winning side. To illustrate this point and other consequences, we have reproduced in Table 4 the PDIFF values and power orderings for member 4. the E(PDIFF) values and power ordering for member 4 are expressed in Table 5. According to the PDIFF values in Table 4, member 4's most helpful partner is member 3, whereas the expected probabilistic measure E(PDIFF) makes member 2
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Table 4: PDIFF Values and Power Orderings for Member 4 of Hypothetical Voting Body Pairs
PDIFF Values
Powder Or&ring
(4,1)
0.75
(4,3)
(4,2)
0.67
(4,1)
(4,3)
1.00
(4,2)
0.50
(4,5)
(4,5)
-
Table 5: E(PDIFF) Values and Power Orderings for Member 4 of Hypothetical Voting Body Pairs
E(PDIFF Values
Power Orderings
(4,1)
0.00
(4,2)
(4,2)
0.50
(4,9)
(4,3)
0.25
(4,1)
(4,5)
- 0.40
(4,5)
the most helpful ally. Clearly, the two probabilistic measures may yield different power orderings for each member. For either measure, the power orderings may not reflect reciprocal choices among potential partners. This phenomenon occurs if member i is the most helpful partner to member j, but member j is not the most helpful partner to member i.
11. Dahl's Analysis of Power Before considering Dahl's measure of voting power, it is necessary to explicitly state and justify the criteria used in evaluating the validity of a power measure. Measuring Voting Power A valid measure of voting power should guage the relationship between an actor's efforts to exert power and the results of his actions. In voting bodies, this relationship may be defined in terms of an actor's preference and the effect of his decision on the outcome of the collective decision. Generally, the most valid measure of voting power would take account of all ways in which a voter may exert power. It is obvious that an actor may exert power by voting "yes" or "no," but it is also true that a voter may use an abstention" or "absence" to exert power. In the United Nations, for example, a member of the General Assembly may abstain from
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a roll call in order to prevent the formation of a winning coalition on an "important question" that requires a two-thirds majority for approval. Although a voting system may not include all four alternatives, a general measure of voting power must be applicable to voting bodies which employ all of the alternatives. Dahl's Measure o f Power In "The Concept of Power," Dahl deveiops two measures of voting power: an individual power index and a pairwise power indes. In both indices, Dahl's criterion for exerting power is that a voter's preference be sustained by the "yes" vote of the majority of the voting body. The goal of Dahl's analysis is to determine which actors are most influential in moving other actors to vote with them to carry a motion. In crediting an actor with exerting power, Dahl omits the possible exercise of power through voting " n o " (N), "abstain" (A), or "absent" (X). This omission invalidates both of his power indices. Moreover, Dahl's "individual power index" does not meet Dahl s own cnterton for conceptuahzmg power. According to D a h , power should be considered as a "relation among people." Yet the individual power index does not measure the interaction of one actor with another. Instead, the index merely gauges the coincidence of the voter's "yes" vote with the affirmative outcomes of the voting body. In taking account of voter interaction, it may not be necessary to assume that preferences are consciously traded. For certain purposes, it may be sufficient to consider agreement or disagreement among actors as "symbolic transactions."7 Another problem with Dahl's measurement of voting power concerns the evaluation of the data generated by applying his pairwise measure. Daht's analysis produces numbers which indicate the relative influence each member of a voting body has on every other member. These numbers are arranged in power orderings which indicate the relative influence of actors from the viewpoint of each member of the voting body, 1 to n. Consequently, one is faced with resolving the classical problem of inter~personal comparison of utilities before one can validly compare the power of voters, Comparing the Indices 1) While the grams' measure is based on inferences which are rigorously deduced from a model, Dahl's measure is not based on systematic derivations from a general model. 9 2) The Dahl and grams "individual power" indices measure the coincidence of individual preference and majority outcome, not the effect of individual choice on the outcome of the voting body. None of the Dahl and grams conditional measures take account of the full range of alternatives a voter may use to exert power. The Dahl index employs "yes" majority votes to measure the power of an actor's preferences while the grams index uses " y e s " and " n o " votes o f the maiority as a criteron o f voter power. 10 In other words, the Brams measure is more inclusive than Dahl's measure, but it lacks general validity because it does not take account of such other voting options as abstention and absence. Still, it is not clear how one could refine the measures to '
'
°
"
"
"
l
7 DaM, op. c#., p. 202. The notion of a "symbolic transaction" is developed in Steven J. Brams and Michael K. O'Leary, "An Axiomatic Model of Voting Bodies," American Political Science Review, 64 (June, 1970), pp. 449-470, especially footnote 34. 8 el. William H. Riker and Peter Ordeshook, Introduction to Positive Political Theory (Englew0od Cliffs: Prentice Hall, (1972). Dahl does not explain how he deals with his problem. el. Dalai, op. cit., p. 212. 9 Ibid., p. 212. 10Ibid., pp. 212-3.
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incorporate natural definitions of the full range of voter alternatives. 3) While both indices generate power ordering data for members, neither one includes a systematic method for producing a group ordering. It should be noted, however, that Brains has proposed using digraph analysis to determine and compare the structures of influence relationships among all members of a voting body. This approach extends the use of the concept of "symbolic transactions" to interpret the behavior of voters as if they acted to maximize their probability or expected probability of voting with the majority. 11
IV. Comparing the Results of Applying the Brams and Dahl Indices As suggested in Section III, the "individual power" indices of Daht and Brains do not actually measure power as a "relation among people." As a consequence, it is not interesting to compare the detailed results of the computations based on each measure. Nevertheless, since Dahl and Brams employ different assumptions in measuring "individual power," it is worth recording that, as one would expect, the measures produce different results. Specifically, according to Dahl's measure of influence-rank, Senators Hayden and Magnuson are tied as the most influential actors in the area of foreign policy (1946-1954). But Brams' individual power index indicates that Magnuson is the most influential actor. In the area of tax and economic policy, DaM's list shows that Senator George has the most influence. In contrast, the Brains index indicates that Senator Knowland is most influential. Like the individual power indices, the conditional power indices of DaM and Brains give different results. According to DaM's list of pairwise influence rankings in the area of foreign policy, Senator George is the most influential actor. However, the Brams PDIFF and E(PDIFF) measures indicate that Senator Magnuson is the most influential actor, tn the area Of tax and economic policy, Dahl lists the most influential actor as Senator George. In contrast, the PDIFF'measure shows that Millikin is most influential. When Dahl combines foreign policy with tax and economic policy, he again concludes that George is the most influential senator. But the PDIFF measure suggests that Ferguson is the most influential senator and the E(PDIFF) measure shows Magnuson to be the most influential actor.
IV. Conclusion This analysis of the DaM and Brains measures of voting power indicates that the Brains measure is a more valid measure of power in voting bodies. Nevertheless, in cases in which abstention or absences are formal or de facto voting alternatives, the Brams index becomes invalid because it does not account for the full range of an actor's choices. Moreover, the utility of the Brains index is limited by the problem of analyzing the power orderings it generates. t l Brams, op. cir.: Steven J. Brams, "Measuring the Concentration of Power in Political Systems," American Political Science Review, 62 (June 1968), pp. 461-475; and "The Structure of Influence Relationships Jal the International System," in James N. Rosenau fed.), International Politics and Foreign Policy: A Reader in Research and Theory (New York: Free Press, 1969), pp. 583-59%