COMPUTING CLASSICAL POWER INDICES FOR LARGE FINITE VOTING GAMES
Dennis Leech
No 579
WARWICK ECONOMIC RESEARCH PAPERS
DEPARTMENT OF ECONOMICS
COMPUTING CLASSICAL POWER INDICES FOR LARGE FINITE VOTING GAMES
by
Dennis Leech University of Warwick
Revised July 2000
Dr. D Leech Department of Economics University of Warwick Coventry CV4 7AL United Kingdom Tel: (+44)(0)24 76523047 Fax: (+44)(0)24 76523032 Email:
[email protected]
COMPUTING CLASSICAL POWER INDICES FOR LARGE FINITE VOTING GAMES by Dennis Leech, University of Warwick ABSTRACT Voting Power Indices enable the analysis of the distribution of power in a legislature or voting body in which different members have different numbers of votes. Although this approach to the measurement of power, based on co-operative game theory, has been known for a long time its empirical application has been to some extent limited, in part by the difficulty of computing the indices when there are many players. This paper presents new algorithms for computing the classical power indices, those of Shapley and Shubik (1954) and of Banzhaf (1963), which are essentially modifications of approximation methods due to Owen, and have been shown to work well in real applications. They are of most utility in situations where both the number of players is large and their voting weights are very non-uniform, some members having considerably larger numbers of votes than others, where Owen's approximation methods are least accurate. The suggestion is made that the availability of such improved algorithms might stimulate further applied research in this field.
JEL Classification numbers: C63, C71, D71, D72 Key words: Voting; Weighted Voting; Power Index; Weighted Majority Game; Empirical Game Theory.
COMPUTING CLASSICAL POWER INDICES FOR LARGE FINITE VOTING GAMES Many organizations have governance systems based on voting which are designed to give different amounts of influence in decision making to different members. For example the joint stock company gives each shareholder a number of votes in proportion to his ownership of ordinary stock; the shareholder body is designed to be a democratic decision-making group with each share having equal influence but with individual shareholders having different numbers of shares to reflect their relative capital contributions. Many international economic organizations have been set up on a similar principle, each country being allocated a number of votes reflecting its respective contribution, the most prominent examples being the Bretton Woods institutions, the IMF and World Bank. Federal political bodies which use the principle of weighted voting where the weights reflect populations rather than contributions include the European Union Council of Ministers and the US Presidential Electoral College, where the individual states' votes are cast as blocks of different sizes. As general, abstract voting systems, without reference to their different contexts, all these cases are formally similar and can be considered as weighted majority games. They contain considerable analytical interest because, when we consider their practical implications, by studying all theoretically possible voting outcomes, and how individual members' votes relate to them, then it turns out that the distribution of power is often different from what the designers intended. On the other hand, it is almost always assumed, by writers analysing the distribution of votes, that the power of a member is the same as his share of the votes. For example, it is often the case in discussions of the IMF, that a member with five percent of the votes is described as possessing five percent of the voting power, or that the United States with approximately 18 percent of the votes therefore has 18 percent of the voting power. Yet the proportion of decisions which may - at least theoretically - be taken by vote in which the member who has five percent may be pivotal in determining the outcome may not actually be five percent at all and the votes of the United States may in fact be capable of being decisive in
more or less than 18 percent of cases. Therefore it is untrue to claim that their respective shares of the total voting power are 5% and 18%. A simple example which illustrates the point clearly is that of a company with three shareholders, two having 49 percent of the shares each and the third with 2 percent. It is not useful to describe these figures as shares of the power each has in running the company because if the decision rule requires a simple majority of 50 percent of the votes, then any two shareholders are required to support a motion for it to pass. Any shareholder can win by combining with one other and therefore the one with 2 percent has exactly the same power as one with 49 percent. Therefore by considering all possible voting outcomes it becomes clear that each shareholder has equal power despite the disparity in their votes. Many such examples can be constructed and found in the real world in which the distribution of power among members of a weighted voting body - a member's power being his ability to join coalitions of others which do not have the required majority and make them winning - is not at all the same as the distribution of votes. Another, well known, example - which points up the practical importance of the subject of this paper - is that of the original Council of the European Economic Community. Between 1958 and 1972 the community had six member countries and the system of weighted voting which was used in the ruling council of ministers (known as qualified majority voting) allocated 4 votes each to France, West Germany and Italy, 2 votes each to Belgium and the Netherlands and one vote to Luxembourg. Looking at these figures one might assume that the system gave more power to the smaller countries than their size would warrant. For example, Luxembourg cast 5.88 percent of the votes with less than 0.2 percent of the population; it had 25 percent as many votes as West Germany with only 0.57% of its population. In fact, however, with the majority requirement being set at 12 votes, Luxembourg's one vote could never make any difference. It was not possible for it to add its vote to those of any losing group of other countries in such a way that together they would achieve the majority requirement and therefore its formal voting power was precisely zero. This is an extreme case, but a real one, which illustrates the analytical importance of looking at the possible outcomes of a weighted majority vote, as 2
well as the nominal voting strengths, in considering voting power. The point holds with equal force when the member has a positive influence over voting outcomes and it raises the general question of the quantitative relationship between the number of votes a member can cast and the voting power they represent. This example also illustrates that the power of each member depends not only on his own votes but those of every other member and the number of votes required for a decision as well; it was the fact that there was no way a group could be formed without Luxembourg which had precisely eleven votes that ensured Luxembourg had no power. The same point arises also in the context of the corporation when we study the power of large stockholders. Obviously if there is a majority shareholder he has all the voting power and none of the other shareholders has any voting power at all. However, it is well known that if the largest shareholder has a very substantial minority holding his vote will often be decisive in a proxy fight or fully attended company meeting, even to the extent that he could be said to have working control of the corporation if his voting power were sufficiently large. For example it is almost universally accepted by writers who have studied corporate ownership and control that a single 20 percent shareholder faced with many small shareholders is very powerful indeed1 . This power is certainly not reflected in the number of its shares and may in fact be very close to that of a majority shareholder. Although necessarily less than 100 percent, it may be extremely close to it. The question has been studied by the use of power indices as measures of the ability of members to influence voting outcomes. As a branch of co-operative game theory the field of power indices may be thought to date from the publication of the seminal paper by Shapley and Shubik in 1954. However it has failed to achieve wide acceptance due to ambiguity because different power indices have been defined on the basis of different voting models. Different indices yield different results in practical applications and research has so far provided little insight into the comparative merits of each. This has meant that the field has remained at the frontier for almost fifty years. Good surveys are provided by Straffin (1994), Felsenthal and Machover (1998) and Lucas (1983)2 .
3
This paper is concerned only with the computation of the so called classical power indices proposed by Shapley and Shubik (1954) and by Banzhaf (1963)3 , both of which have been widely applied. Both indices are based on a common idea that a member's power rests on how often he can add his votes to those of a losing coalition so that it wins, but they differ in the way that such coalitions are counted. In consequence, where both indices have been used to analyse the same voting body, they have often been found to give different results. This has meant that in the absence of any objective data on the actual distribution of power by which to test the indices, it has not been possible to establish the value of the approach as an aid to constitutional design. The use of the indices leads to a further question as to which one to choose. This has led to the development of further indices as variants however they have contributed little to the solution of the problem and may have made it worse by adding further rival results without contributing anything to the central issue of finding suitable appraisal criteria. One factor which has held up the wider use of power indices to study real institutions has been the difficulty of computing them when the number of members is large. The International Monetary Fund4 for example has not far short of 200 members and a typical large company has many thousands of shareholders. The only methods available for such large finite games are approximation methods due to Owen (1972, 1975, also described in 1995) but in some cases they give relatively large approximation errors. This paper proposes new algorithms, modifications of those of Owen, whose approximation errors are negligible. The wider study of the use of power indices made possible by the availability of good algorithms might yield insights leading to a better understanding of the approach and ultimately its acceptance. In another paper (Leech (2000a)) I have used these algorithms to compare the performance of the indices in analysing shareholder power in a large cross section of companies, a case for which we do have appraisal criteria. The question of the relative properties and merits of the indices is not addressed in the current paper. The notation used and the power indices are defined formally in Section I. Section II describes the direct enumeration method of computation, its limitations and those of other exact methods. 4
Section III describes the approximate methods for large games due Owen before the proposed new algorithms, which combine elements of both, are described in Section V. Section VI concludes. I. Power Indices: Notation and Definitions I consider a weighted majority game of voting in a legislature with n members or players represented by a set N = {1, 2, . . ., n} whose voting weights are w1 , w2 , . . . , wn . The players are ordered by their weight representing their respective number of votes, so that wi ≥ wi+1 for all i. The combined voting weight of all members of a coalition represented by a subset T, T ⊆ N, is denoted by the function w(T), where w(T) =
∑
i ∈T
also be needed: h(T) =
∑
i ∈T
wi . A sum of squares function will
wi2 .
The voting decision rule is defined in terms of a quota, q, by which a coalition of players represented by subset T is winning if w(T) ≥ q and losing if w(T) < q. It is customary to impose the restriction q ≥ w(N)/2 to ensure a unique decision and that the voting game is a proper game. A power index is an n-vector whose elements denote the respective ability of each player to determine the outcome of a general vote. The index for each player is defined in terms of the relative number of times that player can influence the decision by transferring his voting weight to a coalition which is losing without him but wins with him. This is referred to as a swing5 for player i. Formally a swing for player i, is defined as a pair of subsets, (,Ti, Ti + {i}) such that Ti is losing, but Ti + {i} is winning. In terms of voting weight,.Ti is a swing if q - wi ≤ w(Ti) < q. The power index for player i is defined as the relative frequency or probability of swings for i with respect to a coalition model where, in some sense, each possible coalition is treated equally; if coalitions are regarded as being formed randomly then the index is a probability. The two indices however, employ different probability models and are mathematically distinct.
5
The Shapley-Shubik index is the probability that i swings (or is "pivotal" in the terminology of Shapley and Shubik) if all orderings of players are equally likely. Thus, given a particular swing for a member, the index is the number of orderings of both the members of the coalition Ti and the players not in Ti relative to the number of orderings of the set of all players N: every reordering is counted separately. The index is the probability of a swing for the player within this probability model. For a given swing for player i, the number of orderings of the members of the subset Ti and its complement (apart from player i ), N-Ti-{i}, is t!(n-t-1)! where t is the number of members of Ti and n is the total number of players, members of N. The total number of swings for i defined in this way for this coalition model is
∑ t!(n − t − 1)!. The index, φ , is this number as a i
Ti
proportion of the number of orderings of all players in N, φi =
∑ Ti
t!(n − t − 1)! . n!
(1)
If all orderings are equiprobable, it is the probability of a swing. The Banzhaf index, on the other hand, treats all coalitions Ti as equiprobable: players are arranged in no particular order; they might be arranged alphabetically, or in any other arbitrary order. A member's power index is then the number of swings expressed as a proportion of either the total number of coalitions (the probability of a swing), or of the total number of swings when for all players. The number of swings is then ∑ 1 . The two versions of the index are defined by Ti
expressing this number over different denominators. The Non-Normalized Banzhaf index (or Banzhaf Swing Probability), β i', uses the number of coalitions which do not include i , the number
6
of subsets of N–{i}, 2n-1, as denominator, and therefore it can be written as: β i' =
∑ 1 /2
n-1
T
.
(2)
i
The Normalized Banzhaf Index, β i, uses the total number of swings for all players as the denominator in order that it can be used to allocate voting power among players: βi =
∑1 / ∑ ∑1 . Ti
i
(3)
Ti
The normalized indices sum to unity over players:
∑β
i
= 1.
i
In the discussion of computation of the Banzhaf index below it is only necessary to consider the details of computing the swing probability version, (2), since β i = β i'/ ∑ βi ' . i
II. Computing the Indices by Direct Enumeration and Other Exact Methods Several methods are available to compute the Shapley-Shubik indices, with simple modifications for the Banzhaf indices: Direct Enumeration; Generating Functions (Mann and Shapley (1962)); Monte Carlo simulation (Mann and Shapley (1960)); Multilinear Extensions (Owen (1972, 1975)); MLE Approximation (Owen (1972, 1975)). The simplest approach is Direct Enumeration, which consists of searching over all possible coalitions and applying the fundamental definitions of the indices directly by counting swings. I have implemented this method by using an algorithm which finds each subset of players once only, then for each subset it finds all swings and updates expressions (1) and (2) repeatedly. That is, for a given subset S ⊆ N the procedure finds all Ti ⊆ S-{i} such that (Ti, Ti + {i}) is a swing. This is relatively straightforward but expensive in computer time. It is only feasible for small and medium values of n. Experience suggests it is practical for values of n up to about 24 but it very quickly becomes prohibitively slow beyond that because computing time is an exponential function of n6 .
7
The method of generating functions of Mann and Shapley (1962) and Owen's (1972) multilinear extensions method are alternative exact methods which are usually regarded as more suitable for small (or medium sized) games than for large games7 . This paper propose a mixed approach which combines direct enumeration, described above, with the approximation methods due to Owen (1972, 1975). These latter methods have been used for large games in a number of studies8 . However, they are based on the use of the central limit theorem to get approximations to expressions (1) and (2), using the normal distribution function, based on assumptions of probabilistic voting. In order to describe the algorithms proposed in this paper, it is first necessary to describe those of Owen. III. Owen's MLE Approximation Algorithms Expression (1) for the Shapley-Shubik index can be rewritten by noting that the term inside the summation is a beta function: t!(n − t −1)! = n!
1
∫ x (1− x) t
n − t−1
0
dx
(4)
The integrand on the RHS of (4), xt(1-x)n-t-1 , can be regarded as the probability that the (random) subset Ti appears, when x is the probability that any member joins Ti , assumed constant and independent for all players j, j ∈ N - {i}. Summing this expression over all swings gives the probability of a swing for i. Let us call this probability ƒi(x):
∑
ƒi(x) =
xt(1-x)n-t-1 .
(5)
Ti
Integrating x out of (5) gives the index, because, substituting (4) into (1) gives: φi =
∑ ∫ x (1− x) 1
Ti
=
∫
t
0
n − t−1
dx =
∫
1 0
[ ∑ xt(1-x)n-t-1 ] dx Ti
1 0
ƒi(x) dx .
(6)
We can evaluate φi approximately using a suitable approximation for ƒi(x). In large games with
8
many small weights, and no very large weights, this can be done with reasonable accuracy using suitable probabilistic voting assumptions and the normal distribution. Assuming each player j ≠ i votes the same way as i with probability x, independently of the others, defines a random variable, vj with the following dichotomous distribution: Pr(vj=wj) = x, Pr(vj=0) = 1 -x, Pr(vj ≠wj and vj ≠ 0) = 0. Therefore its first two moments are: E(vj) = xwj ,
Var(vj) = x(1-x)wj2 , all j.
The total number of votes cast by players j in the same way as that of player i is a random variable vi(x) =
∑ v . Then v (x) has an approximate normal distribution with moments: j
i
j∈N−{i}
E(vi(x)) = xw(N-{i}) = µi(x), say, and Var(vi(x)) = x(1-x) h(N-{i}) = σi(x)2 . Then the required probability, ƒi(x) = Pr[q - wi ≤ vi(x) < q],
(7)
can be obtained approximately using the normal distribution function, Φ(.) by evaluating the expression: ƒi(x) = Φ(
q −µ i (x) q −µ i (x) − wi ) - Φ( ). σi (x) σ i (x)
(8)
The Shapley-Shubik index in (6) is approximated by numerically integrating out x in (8). The Banzhaf index is obtained by setting x = 0.5 in (8). This method has been used in a number of studies but its accuracy depends on the validity of the normal approximation. In many real world weighted voting bodies the approximation is not good and consequent computation errors are large because of the wide range of variation of the voting weights wi.
9
IV. Extended Owen Algorithms for Large Finite Games For games where n is too large for exact methods to be efficient (or feasible), and where the distribution of weights is too skewed for Owen's algorithms to be sufficiently accurate, alternatives can be used which combine the essential features of both approaches. The general procedure is as follows. The players are divided into two groups: sets of major players M = {1, 2, . . . , m} and minor players N - M. The value of m is chosen for computational convenience, along a tradeoff between accuracy and efficiency. A general rule would be to choose m as large as possible while computing time is not too great. The algorithm finds all subsets of M in the same way as does the Direct Enumeration algorithm. Given a particular subset, S ⊆ M, it then evaluates the approximate conditional swing probability for each player making the standard assumptions about random voting by minor players only, conditional on S. The probability of the swing is then obtained as the product of the probability of the formation of S, by random voting by major players, and that of the conditional swing. The index is obtained by summing these joint probabilities over all the subsets. There are two cases to consider: (1) player i is a major player, i ∈ M; (2) i is a minor player, i ∈ N - M. (1) Major Players It is necessary to search over all subsets of M which do not include player i; any subset of which i is a member cannnot define a swing and the swing probability associated with it is definitionally zero. For each such subset consider the probability of its forming and the probability of its being a swing for i. Suppose S is a subset of M - {i}. We let the swing probability be ƒi(x) as before. This can be written as: ƒi(x) = Pr(swing for i) =
∑ S
10
Pr(S)Pr(swing for i|S)
Defining the conditional probability of a swing given S as the function gi(S, x), and the probability of selecting S randomly by the function p(s, m-1, x), we can write: ƒi(x) =
∑
p(s, m-1, x)gi(S, x).
S
The first factor inside the summation is: p(s, m-1, x) = xs(1-x)m-s-1. To find the second factor, define the random variable: vi(x) =
∑
vj ,
j∈N− M
where vj is as before, to represent the random number of votes cast by the minor players. So,
E(vi(x)) = xw(N-M) = µi(x),
and
Var(vi(x)) = x(1-x)h(N-M) = σi(x)2 .
Using these moments and the normal approximation to the distribution of vi(x), we can obtain the required probability as: gi(S, x) = Pr[q - w(S) -wi ≤ vi(x) < q - w(S)] q − w(S) −µ i (x) q − w(S) − w i −µ i (x) ) - Φ( ). (9) σi (x) σ i (x)
= Φ( Therefore, we can write ƒi(x) =
∑
S⊆ M −{i}
xs(1-x)m-s-1 gi(S, x).
(10)
The required index is then: φi = =
∫
1 0
ƒi(x) dx =
∑ ∫
S⊆ M −{i}
1 0
∫
1 0
[
∑
S⊆ M −{i}
xs(1-x)m-s-1 gi(S, x)]dx
xs(1-x)m-s-1gi(S, x)dx
(11)
which can be found by searching over all subsets of M-{i}, integrating out x by numerical quadrature at each subset then summing. The Banzhaf index β'i is obtained from (10) on setting x=0.5 instead of integrating it out, then summing to give β'i = ƒi(0.5).
11
The summation in expression (10) above is over all subsets of M-{i}, but it is clear that operationally we can search over all subsets of M since any set which includes i has a zero probability of a swing for i. Writing gi(S,x) = 0 for all S where i ∈ S, and as expression (9) where i∉ S then we can rewrite (10) and (11) as: ƒi(x) =
∑
xs(1-x)m-s-1 gi(S, x),
(12)
S⊆ M
φi =
∑ ∫
S⊆ M
1 0
xs(1-x)m-s-1gi(S, x)dx,
(13)
and the Banzhaf index β i' =
∑
0.5m-1gi(S,0.5) = ƒi(0.5).
(14)
S⊆ M
It is therefore possible to compute the indices for both major and minor players in a single search over the subsets of M. (2) Minor Players Now the computation of the indices for the smaller players, i ∈ N-M, is described. The subset S can now be considered to be any subset of M. Since we are now treating the votes of all m major players as random (not just m-1 of them), the probability of the subset S is: Pr(S) = p(s, m, x) = xs(1-x)m-s. The behavior of the minor players other than i is described by a random variable yi(x) =
∑
j∈N− M −{i}
vj which has an approximate normal distribution with moments: µi(x) = xw(N-M-{i})
and
σi(x)2 = x(1-x)h(N-M-{i}).
Hence we can evaluate the conditional swing probability gi(S, x) which now can be written as gi(S, x) = Pr[q - w(S) - wi ≤ yi(x) < q - w(S)], approximately by the normal probability in expression (9) after making the required notational substitutions.
12
Writing ƒi(x) =
∑
p(s, m, x) gi(S, x),
(15)
S⊆ M
the Shapley-Shubik index is found again by quadrature, then summing, φi =
∑ ∫
S⊆ M
1 0
p(s, m, x) gi(S, x) dx
(16)
and the Banzhaf index by setting x=0.5, then summing, β i' =
∑
0.5m-gi(S,0.5) = ƒi(0.5).
(17)
S⊆ M
within the same subset search as before, S ⊆ M. These algorithms have proved to be efficient and accurate in large games (in Leech (1998) has n>180 and in Leech (2000b) n>300) with a moderate choice of m. The obvious rule for choosing the value of m is that it should be large enough to ensure accuracy without being too large to permit all subsets of M to be enumerated in a reasonable computing time. Some examples of the application of the approach are given in the next section. V. Some Examples This section presents the results of using the algorithms in practical computations. Two examples are presented: the United States Electoral College States game as reported in Lambert (1988), and an artificial example with weights chosen to be very unequal. The application to the Electoral College States game is given in order to establish the accuracy of the algorithms against previously published results: this example is useful for this because we are in the fortunate position of having exact Shapley-Shubik indices. For this case however the Owen algorithms are reasonably accurate without the modifications proposed here. I have therefore presented a second example, in which the weights have been chosen to be much more unequal, in order to illustrate how accuracy is much improved by the use of the algorithms.
13
Example 1: The US Electoral College States Game, 1980 Census Lambert (1988) computes the Shapley-Shubik indices for the 51-player states game in the US Presidential Electoral College by the method of generating functions due to Mann and Shapley (1962). This is therefore a suitable example to use to establish the utility of the algorithms. The example is used to check that the algorithm for the Shapley-Shubik index works and to investigate the relationship between its accuracy and the parameter m. I have computed the power indices for the 51 states for all values of m from 0 to 20. The results are shown in Table 1 for the ShapleyShubik index for m =0, 5, 10, 15 and 20. The algorithm is very accurate, agreeing with Lambert's to four decimal places for m≥15; such slight differences as there are seem to be due to greater rounding error in the earlier results9 . Figure 1 shows the effect of changing m on the accuracy of the indices for certain states: those numbered i=1 (California), 2 (New York), 3 (Texas), 18 (one of the four states with 10 votes each) and 51 (one of seven states with three votes). Figure 1(a) shows the effect of changing m on the relative error in the computed Shapley-Shubik indices, taking Lambert's figures as accurate. Figures 1(b) and 1(c) show the comparable graphs for the Non-Normalized Banzhaf index and the Normalized Banzhaf index respectively; since there are no accurate figures for this index, the values obtained when m=20 are taken as definitive. The approximation errors in Owens's algorithm are small in this example (the case m=0), being always well under 0.4%. Increasing m reduces the errors Computing time taken by the algorithms increases significantly with m and there is a tradeoff between it and accuracy. Computing time approximately doubles each time m is increased by 1 and therefore this places a severe limitation on the feasible size of m. The program used for this example computed both the indices simultaneously for one search over subsets and took 21 seconds for m=11, 343 seconds for m=15 and 10,259 seconds when m=20, on unix system. It is therefore feasible to choose a relatively large value of m (say m=15) to gain maximum accuracy. However this is not really a large game and computing times increase with n as well as m (though 14
linearly rather than exponentially) so there is not usually so much leeway in the choice of m. However the next example suggests that this need not cause problems because where Owen's method is inaccurate is where there are a few players with very large voting weight and the effect of this can be reduced by choosing m greater than their number. Table 1: US Electoral College Game: Shapley-Shubik Index Shapley-Shubik Indices for States Votes States m=0 m=5 m=10 m=15 m=20 Exact 47 1 0.0930 0.0928 0.0928 0.0928 0.0928 0.0928 36 1 0.0695 0.0694 0.0693 0.0693 0.0693 0.0693 29 1 0.0552 0.0551 0.0550 0.0550 0.0550 0.0550 25 1 0.0472 0.0471 0.0471 0.0470 0.0470 0.0470 24 1 0.0452 0.0451 0.0451 0.0451 0.0451 0.0451 23 1 0.0433 0.0431 0.0431 0.0431 0.0431 0.0431 21 1 0.0393 0.0392 0.0392 0.0392 0.0392 0.0392 20 1 0.0374 0.0373 0.0373 0.0373 0.0373 0.0373 16 1 0.0297 0.0296 0.0296 0.0296 0.0296 0.0296 13 2 0.0240 0.0239 0.0239 0.0239 0.0239 0.0239 12 3 0.0221 0.0220 0.0220 0.0220 0.0220 0.0220 11 3 0.0202 0.0202 0.0202 0.0202 0.0202 0.0202 10 4 0.0183 0.0183 0.0183 0.0183 0.0183 0.0183 9 2 0.0165 0.0164 0.0164 0.0164 0.0164 0.0164 8 5 0.0146 0.0146 0.0146 0.0146 0.0146 0.0146 7 4 0.0128 0.0127 0.0127 0.0127 0.0127 0.0127 6 2 0.0109 0.0109 0.0109 0.0109 0.0109 0.0109 5 3 0.0091 0.0091 0.0091 0.0091 0.0091 0.0091 4 7 0.0073 0.0072 0.0072 0.0072 0.0072 0.0072 3 7 0.0054 0.0054 0.0054 0.0054 0.0054 0.0054 Total 51 1.0032 1.0006 1.0002 1.0001 1.0001 1.0000 Exact indices taken from Lambert (1988). Lambert used single precision arithmetic and this probably accounts for the slight differences with my results in the fifth decimal place. The quota used was q=269.5 following Owen (1972) because the method relies on a normal approximation and quadrature. However exactly the same results are obtained using q=270.
The general conclusion from this example is that the algorithms perform well and that there is very little to be gained in improved accuracy by increasing m much beyond 5 when the relative accuracy for all the indices is around one tenth of one percent.
15
Figure 1: US Presidential Electoral College Relative Approximation Errors for Selected States Figure 1(a): Shapley-Shubik Index 1
0.9
0.8
0.7
0.6
i=1 i=2 i=3 i=18 i=51
0.5
0.4
0.3
0.2
0.1
0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
m
Figure 1(b): Non-Normalised Banzhaf Index 3
2.5
2
i=1 i=2 i=3 i=18 i=51
1.5
1
0.5
0 0
1
2
3
4
5
6
7
8
9
10
m
16
12
13
14
15
16
17
18
19
20
Fiigure 1(c):Normalised Banzhaf Index 1
0.5
0
i=1 i=2 i=3 i=18 i=51
-0.5
-1
-1.5 0
1
2
3
4
5
6
7
8
9
10
12
13
14
15
16
17
18
19
20
m
Example 2: An Artificial Example The second example is a game with artificially generated weights constructed to capture the characteristic pattern often encountered in real voting bodies with a large number of members, in which there are a few members with large weights and many with small weights10 The presence of these large players, and the inequality between players, means that Owen's algorithms are likely to be inaccurate. Very good results are obtained using the new algorithms, however. The game was chosen with n=100, q=55% and the weights have been generated at random (before being expressed as percentages) from a suitable distribution: w1 , w2 , . . . , w100 are a sample from the Lognormal Distribution Λ(4.3, 3). The resulting distribution of votes is relatively concentrated: one player has 29%, two further players have over 10% and a further 2 more than 5% of the total votes. The smallest player has 0.00001 percent of the votes. Although this concentration 17
makes it necessary to choose a value of m greater than zero to get good accuracy, for the same reason it is not necessary to use a large value of m. The results in Table 2 have been obtained using m=12. Table 2 shows the results for the players with the largest 10 weights and those for i =30, 60, 100, taken as representative. It shows the two normalized power indices, which sum to 1 over the players. The relative computation errors for these players are shown in Figure 2. Owen's MLE approximation methods are very inaccurate (the case m=0), the index for i=2 being overestimated by 15% and that for i=1 by 5%. However for the large players the errors in all indices are negligible for m≥5. For the small players the errors in the Shapley-Shubik indices and the Nonnormalized Banzhaf indices become negligible for m≥5 and those in the Normalized Banzhaf indices for m ≥ 7 in the case of the smallest player, i=100.
Table 2: An Artificial Example: Power Indices for Selected Players i
Voting Weights
Normalized Banzhaf Index
Shapley-Shubik Index
1 2 3 4 5 6 7 8 9 10 ... 30 ... 60 ... 100 Sum
0.2915 0.1960 0.1170 0.0970 0.0640 0.0248 0.0222 0.0200 0.0179 0.0168 ... 0.000855 ... 0.0000636 ... 0.0000001 1.0000
0.3522 0.1711 0.1147 0.0930 0.0708 0.0205 0.0185 0.0168 0.0151 0.0142 ... 0.00073 ... 0.000054 ... 0.00000009 1.0000
0.3506 0.1850 0.1142 0.0898 0.0566 0.0217 0.0194 0.0175 0.0156 0.0146 ... 0.00073 ... 0.000054 ... 0.00000009 1.0000
q= 0.55
18
Figure 2: Artificial Example:Relative Approximation Errors
Figure 2(a): Shapley-Shubik Index 20
i=1 i=2 i=3 i=6 i=30 i=60 i=100
0
1
2
3
4
5
6
7
8
9
10
11
12
m
Figure 2(b): Non-Normalized Banzhaf Index 30 25 20 15
i=1 i=2 i=3 i=6 i=30 i=60 i=100
10 5 0 -5 -10 -15
0
1
2
3
4
5
6
m
19
7
8
9
10
11
12
Figure 2(c): Normalized Banzhaf Index
20
15
10 i=1 i=2 i=3 i=6 i=30 i=60 i=100
5
0
-5
-10
-15
0
1
2
3
4
5
6
7
8
9
10
11
12
m
VI. Conclusion This paper has described new algorithms for computing power indices for voting games, enabling the easier analysis of power in the kind of large weighted voting bodies which often occur in reality. They are capable of achieving a high degree of accuracy without excessive cost in terms of computing time in real applications and can be applied to voting bodies of any size. The algorithms are a hybrid between the direct application of the definitions of the indices, whose feasibility is severely limited by computing time, and approximation methods due to Owen, which are inaccurate when voting weights are very unequal. By treating a small number of members with large weight differently from the minor members, the methods achieve a significant reduction in approximation error. I suggest that advances in computing of this type might facilitate the wider use of empirical analyses of power in weighted voting bodies and legislatures which can not only enhance our understanding of the disposition of power in actual voting systems but also contribute to advances in the understanding of the relative properties and utility of the different indices, which is still an open 20
question11 . If we are able to make progress on this question, then there is the possibility of using power indices in the design of constitutions for international organizations and others which use weighted voting.
References Banzhaf, John F (1965), “Weighted Voting Doesn’t Work: A Mathematical Analysis”, Rutgers Law Review, 19, 317-343 Coleman, J.S. (1971), "Control of Collectivities and the Power of a Collectivity to Act," in B.Lieberman (ed), Social Choice, New York, Gordon and Breach, reprinted in J.S. Coleman, 1986, Individual Interests and Collective Action, Cambridge University Press. Dubey, P. and L.S. Shapley (1979), "Mathematical Properties of the Banzhaf Value," Mathematics of Operational Research, 4, 99-131. Felsenthal, D. S. and M. Machover (1995), ""Postulates and Paradoxes of Relative Voting Power: a Critical Re-Appraisal," Theory and Decision, 38,195-229. Felsenthal, D. S. and M. Machover (1998) ,The Measurement of Voting Power, Edward Elgar. Felsenthal, D. S., M. Machover and W.Zwicker (1998), "The Bicameral Postulates and Indices of a Priori Voting Power," Theory and Decision, 44, 83-116. La Porta, R., Florencio Lopez-de-Silanes, Andrei Shleifer and R.W. Vishny, (1999) "Corporate Ownership around the World," Journal of Finance, vol. 32, 3(July): 1131-50. Lambert, J.P. (1988), "Voting Games, Power Indices and Presidential Elections," UMAP Journal, 9, 216-277. Leech, D. (1988) "The Relationship between Shareholding Concentration and Shareholder Voting Power in British Companies: a Study of the Application of Power Indices for Simple Games," Management Science, 34, 509-527. Leech, D. (1998), Power Relations in the International Monetary Fund: A Study of the Political Economy of A Priori Voting Power Using the Theory of Simple Games, Center for the Study of Globalization and Regionalization Working Paper No. 06/98, University of Warwick, revised 2000, presented to the meetings of the Internatonal Studies Society, Los Angeles, March 2000. Leech, D. (2000a), "An Empirical Comparison of the Performance of Classical Power Indices," Warwick Economic Research Papers, Number 563; presented at Games2000, Bilbao, Spain, the first World Congress of the Game Theory Society, July. Leech, D. (2000b), "Shareholder Power and Corporate Governance," Warwick Economic Research Papers, Number 563. Lucas, W F (1983), “Measuring Power in Weighted Voting Systems,” in S. Brams, W. Lucas and P. Straffin (eds.), Political and Related Models, Springer. 21
Mann, I. and L.S Shapley (1960), Values of Large Games IV: Evaluating the Electoral College by Montecarlo Techniques, RM-2651, The Rand Corporation, Santa Monica. Mann, I. and L.S. Shapley (1962), Values of Large Games VI: Evaluating the Electoral College Exactly, RM-3158, The Rand Corporation , Santa Monica. Owen, Guillermo (1972), “Multilinear Extensions of Games,” Management Science, vol. 18, no. 5, Part 2, P-64 to P-79. Owen, Guillermo (1975), "Multilinear Extensions and the Banzhaf Value," Naval Research Logistics Quarterly, 22, pp.741-50. Owen, Guillermo (1995), Game Theory,(3rd Edition), Academic Press. Penrose, L.S. (1946), "The Elementary Statistics of Majority Voting," Journal of the Royal Statistical Society, 109, 53-57. Shapley, L. S. and M. Shubik (1954), “A Method for Evaluating the Distribution of Power in a Committee System,” American Political Science Review, 48, pp. 787-92. Straffin, Philip D (1994), "Power and Stability in Politics," chapter 32 of Aumann, Robert J and Sergiu Hart (eds.), Handbook of Game Theory, Volume 2, North-Holland. Garrett G. and G. Tsebelis (1999), "Why Resist the Temptation to Apply Power Indices to the European Union," in Journal of Theoretical Politics (Special Issue on Power Indices and the European Union, 11, 3, 291-308. Widgren, Mika (1994), "Voting Power in the EC Decision Making and the Consequences of Two Different Enlargements," European Economic Review, 38, 1153-1170.
1
For example La Porta,, Lopez de Silanes, Shleifer and Vishny (1999).
2
There has recently been an increase in interest in the approach with studies of the European Union council and other international organizations being published but this has produced a backlash from some authors rejecting it out of hand. See the vituperative exchange between Garrett and Tsebelis (1999) and the other authors in the recent special issue of the Journal of Theoretical Politics on the use of power indices for the European Union. Garrett and Tsebelis are attacking a straw man by suggesting that because power indices are unable to predict the outcome of any particular vote because they ignore the preferences of the players, they are therefore useless. But power indices are not intended to do that, but to measure the influence of members in a general sense without regard to particular issues. They are therefore useful in the design of constitutions where it is necessary to conceptualise voting in a priori terms allowing for all possible constellations of preferences. The point is made elegantly and concisely in his important paper on the measurement of voting power by Coleman (1971). There would be no disagreement if all would accept power indices on their own terms rather than criticise them for what they do not purport to do. Theoretical discussions of the relative merits of the indices can be found in Felsenthal et. al. (1998), Holler (1998), Felsenthal and Machover (1995, 1998). 3
Actually originally proposed by Penrose (1946). See Felsenthal and Machover (1998)for the history of the measurement of voting power. 4
Studied by Leech (1998).
22
5
The term used by Shapley and Shubik was pivot.
6
This method has the advantage that it can be applied not only to evaluating power indices for simple games but more generally it can be easily adapted to find Shapley values (and other value concepts for cooperative games which assign a characteristic function to each coalition of players). 7
Lucas (1983) discusses the applicability of the method of generating functions. See Widgren (1994) for a case where the use of the exact MLE approach did not prove computationally feasible for a game with 19 members and the MLE approximation method had to be used instead. 8
For example Leech (1988).
9
Lambert used single-precision.
10
For example the International Monetary Fund. In 1946 its members included the United States with 33% and the United Kingdon with 15% of the votes respectively. 11
See Leech (2000a).
23
