DAN S. FELSENTHAL, MOSHE´ MACHOVER and WILLIAM ZWICKER
THE BICAMERAL POSTULATES AND INDICES OF A PRIORI VOTING POWER
ABSTRACT. If K is an index of relative voting power for simple voting games, the bicameral postulate requires that the distribution of K -power within a voting assembly, as measured by the ratios of the powers of the voters, be independent of whether the assembly is viewed as a separate legislature or as one chamber of a bicameral system, provided that there are no voters common to both chambers. We argue that a reasonable index – if it is to be used as a tool for analysing abstract, ‘uninhabited’ decision rules – should satisfy this postulate. We show that, among known indices, only the Banzhaf measure does so. Moreover, the Shapley– Shubik, Deegan–Packel and Johnston indices sometimes witness a reversal under these circumstances, with voter x ‘less powerful’ than y when measured in the simple voting game G1 , but ‘more powerful’ than y when G1 is ‘bicamerally joined’ with a second chamber G2 . Thus these three indices violate a weaker, and correspondingly more compelling, form of the bicameral postulate. It is also shown that these indices are not always co-monotonic with the Banzhaf index and that as a result they infringe another intuitively plausible condition – the price monotonicity condition. We discuss implications of these findings, in light of recent work showing that only the Shapley–Shubik index, among known measures, satisfies another compelling principle known as the bloc postulate. We also propose a distinction between two separate aspects of voting power: power as share in a fixed purse (P -power) and power as influence (I -power). KEY WORDS: Banzhaf, Deegan–Packel, index of voting power, Johnston, paradoxes of voting power, Penrose, postulates for index of voting power, Shapley value, Shapley–Shubik, simple voting game, weighted voting game.
1. INTRODUCTION
Of the indices used for measuring the a priori relative voting power of a voter in a simple voting game (SVG), two have received greatest attention. The first was proposed by Lionel Penrose (1946) and re-invented independently by Banzhaf (1965) – after whom it is generally named in the literature – and then again by Coleman (1971). It was extended to cooperative games in general by Owen (1978a and 1978b). The second index, proposed by Shapley and Shubik (1954), is a special case of the Shapley value (1953) for a
Theory and Decision 44: 83–116, 1998. c 1998 Kluwer Academic Publishers. Printed in the Netherlands.
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cooperative game. We will refer to these indices using the abbreviations ‘Bz’ and S-S’, respectively. Two other indices were proposed respectively by Johnston (1978) and by Deegan and Packel (1978) (referred to in the sequel as ‘Js’ and ‘D-P’). Although the Shapley value has received considerable attention from game theorists (see Roth, 1988a, and numerous further references therein), the Bz index alone was endorsed by a US court of law and has thereby acquired legal status (see Lucas, 1982). While papers have been written on the relative merits of these indices (Straffin, 1982; Brams, Affuso and Kilgour, 1989, for example), it seems fair to say that no index has achieved general recognition as the one correct way to measure voting power. Authors have tended to use both the S-S and Bz indices. However, it has recently been shown (see Felsenthal and Machover, 1995) that the Bz index displays paradoxes that apparently cast serious doubt on its claim to serve as a reasonable measure of relative voting power. It transpires that in a weighted voting game (WVG) the value assigned by the relative Bz index to a voter a may actually increase purely as a result of a donating some weight to another voter b. Even worse, there are cases in which two voters a and b form a bloc a&b and the relative Bz index assigns to the bloc a&b in the resulting new SVG a smaller value than it assigns to a alone in the original SVG. Thus the relative power of a voter a, as measured by the Bz index, may diminish as a result of a ‘swallowing’ another voter b and taking over the latter’s voting rights. The same paradoxes – called the donation and bloc paradox, respectively – are also displayed by the D-P and Js indices. Indeed, the D-P index behaves even more badly inasmuch as in a WVG it may assign a smaller value to a voter a than to a voter b, although a has greater weight than b. The S-S index, on the other hand, satisfies the transfer postulate (see Felsenthal and Machover, 1995, pp. 218 and 227) which guarantees that it is free from all the paradoxes mentioned in the preceding paragraph. It therefore seemed as though the S-S index alone was well behaved, whereas its three rivals, of which the Bz index is by far the most important, must be discarded. In the present paper we introduce a new axiom – the bicameral postulate – and argue that it ought to be satisfied by any truly a priori
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measure of voting power. Roughly speaking, the axiom stipulates that the ratios of power assigned to the voters within a voting system do not change when that system becomes one chamber of a bicameral system, providing that no voter belongs to both chambers. We show that the Bz index satisfies the postulate, but that the other proposed indices do not. The counterexamples are numerous, and include some very simple, weighted systems. Furthermore, we demonstrate that when power is measured by the S-S, Js or D-P indices, it sometimes happens that voter x has ‘more power’ than voter y when x and y are considered as voters in the system H1 , yet y has ‘more power’ than x when they are viewed as voters in the first chamber of the bicameral system H1 ^ H2 . (In the examples provided, H2 has only one voter, so that forming the bicameral meet, H1 ^ H2 , is equivalent to adding a single, new veto-wielding player to H1 .) Thus, these three indices violate a weaker, and correspondingly more compelling, version of the bicameral postulate. (However, in the case of the S-S and Js indices this more extreme failure cannot happen when H1 is weighted.) We also propose a distinction between two separate aspects of voting power: power as a voter’s expected share in a fixed purse to be distributed among the voters (P-power), and power as a voter’s a priori ability to influence decisions arrived at by voting (I-power). The remainder of the paper is organized as follows: Section 2 lays down preliminary definitions and terminology; the main argument for adopting the bicameral postulate as an axiom is in Section 3 and the key examples, explanations, and proof for the four indices are in Section 4; in Section 5 we discuss the combined package of the bloc and bicameral postulates and show that the package addresses the two fundamental issues of power redistribution. Precise statements and proofs of propositions from this section, including connections with the composition axiom of Dubey (1975), are in the Appendix. In Section 6 we propose the distinction between P-power and I-power and consider some of its implications; we also show that the S-S, Js and D-P indices are not always co-monotonic with the Bz index and consequently these three indices violate a condition related to the bicameral postulate – the price monotonicity condition (PMC). Our conclusions and suggestions for further research are presented in Section 7.
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2. DEFINITIONS AND TERMINOLOGY
For the basic definitions and terminology relating to the notions of simple voting game (abbreviated SVG) and weighted voting game (abbreviated WVG), we refer the reader to Shapley (1962). In particular, we represent an SVG as an ordered pair G = (N; W ), where N is a finite set and W is a collection of subsets of N , satisfying certain simple conditions (see Shapley, 1962, pp. 60–61). We refer to N as the assembly, to its members as voters, to its subsets as coalitions and to the members of W as winning coalitions of G. Shapley’s definition admits improper SVGs – in which two winning coalitions may be disjoint – but all our specific examples of SVGs (in Section 4) will be proper. If G = (N; W ) and a 2 S 2 W but S fag 62 W , we say that in the SVG G voter a is critical in the coalition S . We put n = jN j. The Bz score of voter x in an SVG G, denoted by ‘x (G)’, is the number of coalitions in which x is critical. The Bz index of voter x in G, denoted by ‘Bzx (G)’, is obtained from the Bz score by normalization: Bzx (G) = x = fy (G) : y 2 N g, so that fBzx (G) : x 2 N g = 1. The absolute Bz measure of voter x in G, which we shall denote by ‘ x (G)’, is defined by: x(G) = x(G)=2n 1. The S-S score of voter x in G is the sum (jS j 1)!(n jS j)!, where the summation is over all coalitions S in which x is critical. We denote this score by ‘x (G)’. As is well known and easy to prove, fx (G) : x 2 N g = n!. The S-S index of voter x in the SVG G, which we denote by ‘S-Sx (G)’, is obtained from the S-S score by normalization: S-Sx (G) = x (G)=n!; so that fS-Sx (G) : x 2 N g = 1. For additional definitions and terminology relating to SVGs and WVGs, see Felsenthal and Machover (1995). Note that in the literature the S-S index is often denoted by ‘'’, the Bz index by ‘ ’ and the absolute Bz measure or ‘index’ by ‘ 0 ’. However, we wish to reserve the term ‘index’ for measures that are normalized so that they add up to 1. To obtain the Js score of a voter x in an SVG G, denoted ‘x (G)’, let c(S ) be the number of voters in coalition S which are critical in S ; then x(G) = 1=c(S ), where the summation is again over all
P
P
P
P
P
P
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coalitions S in which x is critical. The Js index of x in G, denoted ‘Jsx(G)’, is obtained from the Js score by normalization. The D-P score of voter x in an SVG G, denoted by ‘x (G)’, is defined in the same way as x (G), except that only minimal winning coalitions (MWCs) are taken into account. (Note that if S is an MWC then c(S ) = jS j.) The D-P index of x in G, denoted ‘D-Px (G)’, is obtained from the D-P score by normalization. DEFINITION 2.1. More generally, a power score is any function assigning a non-negative real number a (G) to every voter a of every SVG G satisfying the following two conditions: (i) At least one voter a from each SVG G is assigned a value a (G) > 0. (ii) Whenever f is an isomorphism from an SVG G to an SVG G0 and a is a voter of G, then a (G) = f (a) (G0 ). ( is invariant under isomorphism.)
Any such can be normalized by setting
X
Ka (G) = a (G)= fy (G) : y 2 N g; at which point the new function K satisfies normalization: X fK (G) : y 2 N g = 1: y We refer to such a K as a power index. In this paper we frame the theoretical discussion in terms of power indices, but much of it could easily be rephrased in terms of power scores. Note that Definition 2.1 implies that any power score or index must be symmetric: it assigns equal values to equivalent voters in any SVG. (Two voters a and b of G are equivalent if there is an automorphism of G that maps a to b.) We may sometimes omit explicit reference to G and write, for example, ‘Kx ’ instead of ‘Kx (G)’. DEFINITION 2.2. We say that a power index K has the dummy property if Kd (G) = 0 for every dummy voter d of an SVG G. The strong dummy property requires that the addition of a new dummy voter to an SVG never changes the power assigned to the other voters.
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It is easy to check that the S-S, Bz, D-P and Js indices have the strong dummy property. DEFINITION 2.3. Let G1 = (N1 ; W1 ) and G2 = (N2 ; W2 ) be any two SVGs whose assemblies, N1 and N2 , are disjoint. Let G1 N2 be the SVG whose assembly is N = N1 [ N2 and whose collection of winning coalitions, W1 N2 , is given by
X 2 W1 N2 , X = Y [ Z for some Y 2 W1 and some Z N2 : Thus, G1 N2 is obtained from G1 by adding in the voters from N2 as dummies. Similarly, G2 N1 is obtained from G2 by adding in the voters from N1 as dummies. We now define the bicameral meet of G1 and G2 to be the SVG
G1 ^ G2 =df (N; W1 N2 \ W2 N1 ): A typical MWC of G1 ^ G2 has the form X1 [ X2 , where X1 is an MWC of G1 and X2 is an MWC of G2 . Clearly, the bicameral meet is a form of product, closely akin to the product of measure spaces. We think of G1 ^ G2 as a bicameral legislature in which passage of a law requires approval in both ‘chambers’, G1 and G2 . DEFINITION 2.4. The bicameral postulate requires of a power index K that if a and b are any two non-dummy voters of an SVG G1 , and G2 is any SVG with an assembly disjoint from that of G1 , then
Ka (G1 )=Kb(G1) = Ka(G1 ^ G2 )=Kb(G1 ^ G2): The weak bicameral postulate requires of a power index K that if a, b, G1 and G2 are as above, then Ka (G1 ) < Kb(G1) , Ka(G1 ^ G2 ) < Kb(G1 ^ G2 ): This form of the postulate is clearly implied by the first version.
3. RATIONALE FOR THE BICAMERAL POSTULATE
If an SVG G1 is a component in a bicameral meet with G2 , part of the total power that G1 would have had if it were a stand-alone SVG
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must be transferred to the members of G2 , so there is a common effect on the power of G1 ’s voters. However, the assemblies of G1 and G2 are assumed to be disjoint, so that if a and b are voters in G1 , the interaction of a with the voters of G2 is no different from that of b. (To appreciate the role of disjointness, note, for example, that if in G1 voter a’s vote is only critical when voter b votes ‘no’, and if voter b belongs not only to G1 but also to G2 , where he has veto power, then a becomes a dummy in G1 ^ G2 .) Once the voters of G2 have been allocated their shares in the total power of G1 ^ G2 , the voters of G1 are, relative to each other, in the same positions in G1 ^ G2 as they were in the stand-alone G1 , except that now there is a smaller cake to divide among them. We see no mechanism for a differential effect that would skew the proportional distribution of the power of G1 that remains to be shared by its voters. Of course, the examples in the next section show that bicameral meet does have a differential effect on the particular measurement represented by the S-S index, and we also provide there an intuitively satisfying explanation of the source of this difference. No doubt, for anyone who finds the S-S method for measuring power to be so clearly correct that it indisputably measures what ‘a priori voting power’ means, this explanation may provide the missing mechanism referred to above. For the authors, something more compelling would be needed to dismiss the bicameral postulate. To express our argument in a general and abstract form, let us make explicit the tacit assumption that underlies the use of SVGs and power indices defined on SVGs to model real-life voting rules and voting power. Here is how Roth (1988b, p. 9) puts it: Analyzing voting rules that are modeled as [SVGs] abstracts from the particular personalities and political interests present in particular voting environments, but this abstraction is what makes the analysis focus on the rules themselves rather than on the other aspects of the political environment. This kind of analysis seems to be just what is needed to analyze the voting rules in a new constitution, for example, long before the specific issues to be voted on arise or the specific factions and personalities that will be involved can be identified. [Our emphasis]
It is especially noteworthy that this paragraph comes from a section of Roth’s text devoted precisely to explaining the S-S index. Here Roth is echoing a similar caveat sounded by Shapley and Shubik (1954, p. 791); see also Felsenthal and Machover (1995, p. 197– 198).
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Another way of putting it is that an SVG is an abstract shell, uninhabited by real agents, with real likes and dislikes, attractions and repulsions. And the ‘bills’ on which voting is supposed to take place are – as far as the theory of SVGs is concerned – generic issues without any predetermined relation to the interests of this or that voter. It is precisely because of this that we insist on saying that what we are studying are indices of a priori voting power. A truly a priori index must not presuppose any specific information as to the nature of the issues voted upon, the interests of the voters or the mutual relations between voters. Now, from the point of view of the voters of an SVG G1 , the formation of a bicameral meet imposes a selection on the class of bills in relation to which power within G1 is distributed. When G1 is considered as a separate legislature, all bills brought before it for a vote are relevant for judging the distribution of power among G1 ’s voters; but in the bicameral meet G1 ^ G2 only those bills which are approved by G2 count for the distribution of power within G1 . Indeed, if it transpires that a given bill has been (or will be) voted down by G2 , then it is a waste of time for G1 to consider it. Therefore, if the ratios of the voting powers of G1 ’s voters are altered by the formation of a bicameral meet G1 ^ G2 , this can only mean that the selection imposed by G2 is systematically biased compared to the unrestricted situation that prevails when G1 is a stand-alone legislature, and this bias favors some voters of G1 and disfavors others. Such bias is to be expected in real-life situations, where voters and groups of voters have, say, particular political inclinations; but it must surely be ruled out in the theory of SVGs, because it evidently violates the a prioricity caveat quoted above. The bicameral postulate may be viewed as imposing a requirement of independence – ‘independence’ not in the constitutional or juridicial sense of the term, but in a sense akin to that which is formalized in the theory of probability. We end this section with a ‘story’. Imagine that Country No. 1 has a unicameral legislature containing a single representative from each of the country’s provinces. As the provinces differ in population, this legislature employs a sophisticated decision rule – an SVG G1 for which the distribution of voting power, as measured by a long-accepted index K , accords well with population. For example,
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Province b has approximately thrice the population of Province a, and the ratio Kb (G1 )=Ka (G1 ) is about 3. Country No. 1 decides to form a political union with Country No. 2, in which legislation is determined by the SVG G2 . The countries decide to adopt in future only such new laws as are approved of by both legislatures. However, despite the fact that Country No. 1 now employs the same legislature as earlier to decide, via the same decision rule, what one may argue is the same question (‘Does Country No. 1 approve of the proposed new law?’), Province a objects, because it transpires that Kb (G1 ^ G2 )=Ka (G1 ^ G2 ) is approximately 5 instead of 3, indicating that the residents of Province a are now under-represented relative to those of Province b. Moreover, Province a also objects to the proposal that the new union begin with a base of legislation containing exactly those laws already common to both countries, because this would retroactively cheat Province a out of the influence due it. The complaints by residents of Province a would be slight, however, in comparison with those that would arise from b, were it to be discovered that the relative powers of these provinces had instead been thoroughly reversed via the bicameral meet, with Kb(G1 ^ G2) < Ka(G1 ^ G2 ). In fact, we believe that such a situation would be taken as evidence for an anomaly in the measuring instrument, K , rather than as an accurate measurement of a real effect. Doubt would, indeed, be cast on the validity of earlier votes – but only because they would be seen as having been based on a system designed with a flawed tool. 4. EXAMPLES, EXPLANATION AND PROOF
In the following examples we report the results of calculations, the details of which we omit. Readers interested in obtaining an outline of the detailed calculations can receive it on request from any of the authors. EXAMPLE 4.1. Let H1 be the SVG with assembly fa; b; c; d; e; f; gg and MWCs: fa; b; cg; fa; b; d; e; f g; fa; b; d; e; gg; fa; b; d; f; g; g fa; c; d; e; f g; fa; c; d; e; gg; fa; c; d; f; gg;
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fb; c; d; e; f g; fb; c; d; e; gg; fb; c; d; f; gg: H2 has assembly fj g with MWC fj g. Here forming H1 ^ H2 is
equivalent to adding j as a new blocker (vetoer) to H1 . In H1 the S-S scores of a and d are 1128 and 792, respectively, so S-Sa (H1 )/SSd (H1 ) = 1128 = 1:424: In H1 ^ H2 the respective S-S scores of a 792 and d are 5760 and 4320, so S-Sa (H1 ^ H2 )/S-Sd (H1 ^ H2 ) = 5760 = 4320 1:333: This example demonstrates that the Js and D-P indices also fail to satisfy the bicameral postulate. The Js scores of voters a and d in H1 are 88 and 42 , respectively; and their Js scores in H1 ^ H2 are 15 15 18 9 and 4 , respectively. Hence, Jsa (H1 )/Jsd (H1 ) = 88 = 2:095; but 4 42 Jsa (H1 ^ H2 )/Jsd (H1 ^ H2 ) = 189 = 2. Similarly, the D-P scores of voters a and d in H1 are 23 and 27 , respectively, and their D-P scores 15 15 15 18 in H1 ^ H2 are 12 and 12 , respectively. Hence, D-Pa (H1 )/D-Pd (H2 ) = 23 = 0:852; but D-Pa (H1 ^ H2 )/D-Pd (H1 ^ H2 ) = 15 = 0:833. 27 18 Note that H1 ; H2 , and H1 ^ H2 can all be represented as WVGs. In these representations, the quota is given first, separated by a semicolon and followed by the weights of the voters in alphabetical order: H1 : [12; 4; 4; 4; 2; 1; 1; 1]; H2 : [6; 6]; H1 ^ H2 : [18; 4; 4; 4; 2; 1; 1; 1; 6]: Thus S-S, Js and D-P violate the bicameral postulate even in the domain of WVGs. The following two considerably simpler examples illustrate simultaneous violation of the Bicameral postulate by two of the three indices S-S, Js, or D-P. In H1 : [3; 2; 1; 1];
H2 : [5; 5];
H1 ^ H2 : [8; 2; 1; 1; 5]
the bicameral postulate is violated by the S-S and Js indices (but not by the D-P index); whereas in H1 : [6; 5; 4; 1; 1]; H2 : [6; 6]; H1 ^ H2 : [12; 5; 4; 1; 1; 6] the bicameral postulate is violated by the Js and D-P indices (but not by the S-S index). We leave verification of these examples to
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the reader. But we have yet to find an instance of an SVG with two non-equivalent voters who are not dummies, for which the Js ratios fail to shift when a bicameral meet is formed. Examples showing that the S-S and Js indices fail the weak bicameral postulate seem to be significantly more complex: EXAMPLE 4.2. H1 has assembly fa; b; c; d; e; f; g; h; ig and MWCs:
fa; c; ig; fa; d; e; f; ig; fa; d; e; g; ig; fb; c; d; ig; fb; c; e; ig; fb; c; f; ig; fb; d; f; ig fb; e; f; g; h; ig:
H2 is as in Examples 4.1. In H1 the S-S scores of a and b are a (H1 ) = 39 744 and b (H1 ) = 38 488; whereas in the bicameral meet H1 ^ H2 the order of their S-S scores is reversed: a (H1 ^ H2 ) = 237 600 and b(H1 ^ H2 ) = 239 040. EXAMPLE 4.3. H1 has assembly and MWCs:
fa; b; c; d; e; f; g; h; i; j; k; l; mg
fa; c; d; e; f; g; k; l; mg; fa; c; d; e; f; h; k; l; mg; fa; c; d; e; f; i; k; l; mg; fa; c; d; e; f; j; k; l; mg; fa; c; d; e; g; h; k; l; mg; fa; c; d; e; g; i; k; l; mg; fa; c; d; e; g; j; k; l; mg; fa; c; d; e; h; i; k; l; mg; fa; c; d; e; h; j; k; l; mg; fa; c; d; e; i; j; k; l; mg; fa; c; d; f; g; i; k; l; mg; fa; c; d; f; h; i; k; l; mg; fa; c; d; g; h; i; j; k; l; mg; fa; d; e; f; g; h; i; j; k; l; mg; fb; c; d; j; k; l; mg; H2 has assembly fng with MWC fng. In H1 the Js scores of a and b are:
45 a(H1) = 88573 = 6:391; b (H1 ) = = 6:429: 13860 7
But in H1 ^ H2 the order of the scores is reversed: a(H1 ^ H2 ) = 39031 = 5:632; 6930 = 5:625: b(H1 ^ H2 ) = 45 8 In the SVGs H1 of Examples 4.2 and 4.3, there are pairs of voters that are not comparable by the dominance or desirability relation. (For
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a definition of this relation, see Felsenthal and Machover, 1995, p. 212.) Hence, in particular, these SVGs are not weighted. In fact, it is not difficult to show that for the S-S and Js indices this is a necessary feature of such an example. However, Example 4.4 demonstrates that the D-P index violates the weak bicameral postulate even in a WVG. EXAMPLE 4.4. H1 has assembly MWCs:
fa; b; c; d; e; f; g; h; i; j g
and
fa; c; d; e; f; g; j g; fa; c; d; e; f; h; j g; fa; c; d; e; f; i; j g; fb; j g: H2 has assembly fkg with MWC fk g. Both H1 and H1 ^ H2 can be
represented as WVGs:
H1 : [51; 4; 21; 4; 4; 4; 4; 1; 1; 1; 30]; H1 ^ H2 : [81; 4; 21; 4; 4; 4; 4; 1; 1; 1; 30; 30]: In this example, the five voters a; c; d; e and f are equivalent. In H1 , their D-P score is 37 , whereas that of b is 12 ; thus according to the D-P index b is ‘stronger’ than each of those five voters. However, in H1 ^ H2 the D-P score of the five voters is 38 , whereas that of b is 13 ; so now b is ‘weaker’ than them – in clear violation of the weak bicameral postulate. Note that this example also demonstrates that the D-P index fails to respect the dominance relation: although the weight of b is considerably larger than that of each of the five equivalent voters, the D-P score of b in H1 ^ H2 is smaller than theirs. To explain the reason why the bicameral postulates are violated by these indices, let us first look closely at how the formation of a bicameral meet G ^ H affects the S-S score of a voter a of G. For simplicity, let us assume that H has a sole voter v , so that the formation of the meet is tantamount to adding v to G as a new vetoer. Let S be a coalition of G in which a is critical, and let jS j = k . Then the contribution of S to the S-S score of a in G is (k 1)!(n k)!. This number is relatively large when k has extreme values (near 1 or near n) and is relatively small when k is near n=2. In fact, as n increases, the ratio between the highest and lowest values of (k 1)!(n k )! grows at a roughly exponential rate with n, so
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the difference between the contributions of very small or very large coalitions and middle-sized ones can be very spectacular indeed. Thus voters can achieve approximately equal S-S scores by being critical in just a few coalitions of extreme size (very small or very large), or by being critical in many middle-sized coalitions. However, when a new vetoer v is added in, this has quite different effects on the contributions of coalitions of different sizes. In the meet, S is replaced by S [ fv g, whose contribution to the S-S score of a is k!(n k )!. This means that the contribution of S to the score of a is increased k -fold. Thus, the large contributions of large coalitions (those with k near n) are boosted by a large factor; the large contributions of small coalitions are boosted by a small factor; and the small contributions of middle-sized coalitions are boosted by a middle-sized factor. So the effect of the advent of v on the S-S score of voter a depends on the precise mix of sizes of coalitions from which the score of a in G was derived. It is therefore only to be expected that the S-S scores of non-equivalent voters of G – scores derived perhaps in very different ways (from coalitions of different sizes) – may be affected in different ways. Similar analysis applies also to the Js and D-P indices, except that here the differential effect of v on the contribution of S to the score of a depends on S in other ways. In particular, a large minimal coalition S , containing many critical members, tends to make a relatively small contribution to the Js and D-P scores of its members, whereas its contribution to their S-S score is relatively large. These considerations have guided us in constructing the examples presented above. A somewhat similar way of reasoning was also suggested by Roger Myerson (personal communication), and appears to be related to Thomas Quint’s (1994) decomposition of powerlessness into two components. The Bz index alone is unaffected in any way, because the contribution of S to the Bz score of a is just 1, irrespective of the size of S or of the number of other critical members of S . THEOREM 4.5. The Bz index satisfies the bicameral postulate. Proof. Notice that if S is a coalition of the SVG G1 ^ G2 , then the voter a of G1 is critical in S if and only if S = S1 [ S2 , where
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a is critical in the coalition S1 of G1 and S2 is a winning coalition of G2 . Thus the Bz score of a in G1 ^ G2 is equal to the Bz score of a in G1 multiplied by the number of winning coalitions in G2 . As this holds as well for any other voter b of G2 , the ratio Bza /Bzb is unchanged by bicameral meet.
5. A PACKAGE OF DESIRABLE PROPERTIES FOR POWER INDICES
Consider the following Package A of properties: (1) The bloc postulate; (2) The bicameral postulate; (3) The strong dummy property (with, perhaps, a few similarly unobjectionable properties added on). At present, we know of no Package A index (one having all three properties), nor can we rule out such indices in principle. An intuitively plausible notion of power which led to such an index might be quite interesting. If, however, an ‘impossibility theorem’ were to be proved, should the event properly be seen as significant and disappointing? We would say ‘Yes’, but not everyone agrees. Some argue either that one ought to be satisfied with an assortment of indices possessing complementary virtues – much as the mean and median represent distinct ways of determining the ‘center’ of a statistical distribution – or that one must be so satisfied, because impossibilities are common in the mathematical social sciences. We take up these matters in turn. Certainly, both the mean and the median, being compressions of higher-dimensional information into a one-dimensional measure, are bound to be imperfect, and this situation is not viewed as being at all ‘paradoxical’. However, the analogy with voting power is very incomplete. Note that there is a good and rather secure intuition for what the mean and median say, and fail to say, about a distribution. One inference may be clearly supported by citing the mean as evidence, while another may call for the median, or for both. But on the question of what it is that the S-S index and Bz index say about voting power, our hands-on feel is, as yet, much less sharp. For
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example, suppose (as in the story at the end of Section 3) our goal is to design a voting system in which Province b has thrice the influence of Province a, and that in some candidate system, G, b has too much influence according to the S-S index but has too little according to the Bz index. What are we to make of such a situation? Indeed, this type of design problem is central to the study of voting power and appears to call for a single index that is simultaneously fair in all ways that are appropriate to the intended purpose of the system being designed. Actually, this line of reasoning opens the door to an entirely different argument in favor of a plurality of indices. Perhaps our original, vague idea of unspecified ‘voting power’ conceals more than one precise idea, because there is more than one type of voting. The history of science knows many instances of intuitive notions that, when subjected to rigorous explication and analysis, yielded two or more precise notions that had previously been conflated with each other. (For such an instance in pure mathematics, concerning one of its most fundamental notions, that of set, see Kreisel (1967, pp. 143–145).) If so, there may develop a consensus that each of several power indices measures something real and useful (but only if each index satisfies all desiderata for some type of voting). In the next section, we propose just such a split, arguing that a voter’s ability to influence the outcome of a vote (‘I-power’) is intuitively distinct from his expected share in a fixed purse to be allocated among the voters (‘P-power’). Multiple indices, perhaps ones different from all those known at present, thus represent a possible future for power indices – but not the only possible future. For the time being we must also be willing to contemplate the possibility that there is no precise idea concealed within the original, broad notion of voting power. If so, then at least one impossibility theorem is waiting to be discovered. However, if such impossibilities turn out to be ‘cheap’, in that there are twenty other packages, each as compelling and short a list of apparently mild properties as is Package A, then this outcome would be seen more as a predictable reflection of the general state of affairs within the field than as an important comment on voting power per se. How compelling, then, is Package A – either in terms
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of the individual properties that make it up, or as a package qua package – and how many competing packages are there? Regarding the individual components of Package A, we have argued that the bicameral postulate is intuitively compelling for any a priori measure of voting power. The bloc postulate is advocated in Felsenthal and Machover (1995). (However, in Section 6 we qualify that advocacy in terms of the distinction between ‘P-power’ and ‘I-power’.) We are unaware of competing packages that both go beyond the unobjectionable properties common to virtually all known indices in order to distinguish among known indices, and that are also intuitively compelling. In particular, Dubey’s composition axiom is quite important because of what it says, structurally, about both the S-S and Bz indices and because of its usefulness in characterization theorems. DEFINITION 5.1. If G1 = (N; W1 ) and G2 = (N; W2 ) are SVGs with a common assembly, N , then the union and intersection of G1 and G2 are defined as follows: G1 [ G2 =df (N; W1 [ W2 ) and G1 \ G2 =df (N; W1 \ W2 ):
Dubey’s axiom requires, of a power score K , that
Ka (G1 ) + Ka(G2) = Ka (G1 [ G2) + Ka (G1 \ G2) holds whenever G1 = (N; W1 ) and G2 = (N; W2 ) are SVGs with a common assembly N , and a 2 N . Dubey (1975) introduced his axiom as a replacement for Shapley’s (1953) additivity axiom, which is satisfied by the Shapley value for general cooperative games, but has no meaning in the more restricted class of SVGs. Dubey’s axiom is satisfied by the S-S score, the S-S index, and – as shown by Dubey and Shapley (1979) – by the Bz score (but not by the Bz index). However, to our knowledge no one has offered a convincing argument in favor of adopting it as a principle that ought to be satisfied by any reasonable voting-power index (nor was that an intended purpose). In fact, there is an important sense in which the bicameral postulate and Dubey’s axiom arise from alternative, yet related, visions of
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how the universe of all SVGs arises. The comparison, which follows, supports the claim that Package A is natural as a package, and also reveals that Dubey’s axiom implies a weak form of the postulate. Suppose that we begin with the class E consisting of those WVGs for which each player has weight 1 or weight 0, and then provide a short list of operations which, when repeatedly applied to members of the class E , suffice to generate all SVGs. Two examples (see Taylor and Zwicker, 1993) of such generating sets of operations are:
formation of voting blocs (within a single game), taking the intersection of two games having a common assembly;
and
formation of voting blocs (within a single game), taking the union of two games having a common assembly.
As we show in the Appendix, Theorem A.2, a third is formation of voting blocs (within a single game), taking the bicameral meet of two games having disjoint assemblies.
Because relative voting power is noncontroversial for games in E (each of the m players with weight 1 has relative voting power 1=m), one natural approach to settling the issue of voting power would be to choose a particular generating set of operations and then offer intuitively defensible, quantitative rules specifying exactly how power is redistributed each time an operation is applied. Such rules would then inductively determine the exact voting power distribution in all SVGs. While no such wholesale analysis exists, some important work on power can be seen as heading down this road. In particular, the bloc postulate restricts (but does not completely specify) the way voting power shifts when a voting bloc is formed, and Dubey’s composition axiom restricts voting power redistribution under intersection and union (but because the axiom mentions both operations, it does not completely specify the redistribution). We have already argued (in Section 3) that the existence of members common to two assemblies creates potentially complex interactions in the bicameral system; this is why it is difficult to invoke reliable intuition as to how power should redistribute under the operation of (non-disjoint) intersection or union – and therefore difficult to
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argue for Dubey’s axiom on the grounds of intuition. The bicameral postulate, however, is based on the substitute operation of (disjoint) bicameral meet, which is more transparent with respect to power redistribution. Furthermore, bicameral meet is a cousin to intersection, and the dual operation of bicameral join (see Appendix) is similarly related to union, so that Dubey’s axiom actually implies a weak form of the bicameral postulate: for any power index satisfying Dubey’s axiom any internal shift (shift in proportions within one of the assemblies) of power under bicameral meet must be balanced by an exactly compensating shift under bicameral join (see Corollary A.8 for the exact statement). In particular, S-S yields two non-zero shifts which compensate, while Bz yields zero shifts for both operations. This result outlines more clearly the task at hand for anyone seeking to justify the behavior of S-S with respect to the bicameral operations. It also raises some questions. Does the bicameral postulate in turn imply some weak form of Dubey’s axiom, and, if so, does this open the door to constructing an intuitive defense of that axiom?
6. P-POWER AND I-POWER; THE PRICE MONOTONICITY CONDITION
As mentioned in the Introduction, the S-S index is a special case of the Shapley value for cooperative (coalitional) games. Although Shapley (1953) did not put it quite like this, the standard – and indeed natural– way of interpreting the Shapley value is as a prediction of the expected payoff allocation to players out of a total ‘purse’ (see, for example, Myerson, 1991, p. 436). In a general coalitional game – given by means of a real-valued characteristic function v , defined for all subsets of the set N of players – the total purse is equal to v(N ) units of transferable utility. The S-S index is obtained by considering an SVG (N; W ) as a coalition game, with v (S ) = 1 if S 2 W and v (S ) = 0 otherwise. In particular, the total purse v (N ) is always equal to one unit of utility. Thus the intuitive scenario behind the S-S index seems to be the following. Once a winning coalition is formed – presumably after some process of bargaining – it gains control over the purse, and divides it among its members as it sees fit. This constitutes one
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play of the game. A voter about to participate in this game is faced with uncertainty: will I be included in the winning coalition? and if so, what will be my share of the purse? S-Sx is then offered as an estimate of voter x’s expected payoff in this situation of uncertainty. A similar scenario – division of a fixed amount of ‘spoils’ among the members of a winning coalition – is also offered, quite explicitly, by Deegan and Packel (1978) as the rationale for their index; indeed, the title of their (1982) paper declares this in slogan form. However, they insist that only MWCs are in fact formed, each with equal probability, and that once such a coalition is formed it divides the ‘spoils’ equally among its members. For brevity, we shall refer to this notion of voting power – a voter’s expected or estimated share in a fixed purse, to be divided among the members of a winning coalition that wins a vote – as Ppower. It is epitomized by Shapley’s (1962, p. 59) dictum that SVGs can be used as formal representations of voting bodies ‘in which the acquisition of power is the payoff’. The Bz index is justified by its proponents – see Penrose (1946, p. 53) Banzhaf (1965, p. 331) and Coleman (1971, passim) – in terms of a rather different notion of voting power: their idea is that a voter’s power is proportional to his or her ability to influence the outcome of a vote. We shall refer to this notion of voting power as I-power. Under this notion, a voter (or indeed any other person) may obtain a certain payoff, depending on whether a given bill is passed or not; but this payoff is not conceived of as a voter’s portion in some constant sum to be shared among all voters who vote for the bill. Our distinction between P-power and I-power is, in essence, the same as that made by Coleman (1971); see also Morriss (1987, pp. 157–166). Banzhaf was concerned with measuring the ratio of the I-powers of two voters. But Penrose (1946) proposed a scale for measuring absolute I-power, whereby a voter’s absolute I-power in two different SVGs may be meaningfully compared. He used a probabilistic model, based implicitly on the following assumption about the behavior of voters in an SVG G. The same assumption is stated explicitly by Dubey and Shapley (1979, pp. 102–193).
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ASSUMPTION 6.1. Voters act independently of one another, each voting ‘yea’ or ‘nay’ with equal probability of 12 . This assumption will be crucial for what follows. Later on we shall discuss its methodological significance and justification; for the time being let us just accept it as a working hypothesis. Suppose that the assembly is called upon to vote on a bill. Consider a given voter, x, and let S be the coalition consisting of all those among the remaining n 1 voters who vote ‘yea’. S can be chosen in 2n 1 distinct ways, and by Assumption 6.1 they are all equiprobable. But our x is able to affect the outcome of the vote iff x is critical in the coalition S [fxg; and the number of such S is exactly x . Hence THEOREM 6.2. Under Assumption 6.1, the probability that voter x can affect the outcome of the assembly’s vote is x = x =2n 1 .
Straffin (1982, p. 267) suggests that x is a reasonable measure of voter x’s absolute voting power. This, as it happens, is a slightly modified version of Penrose’s measure of I-power: x =2n. We shall use the latter measure, which will allow us to quantify absolute Ipower in units of transferable utility; for, as observed by Morriss (1987, p. 226), Penrose’s measure equals the price at which a voter’s vote may be sold to an interested outsider. (According to normal practice in game theory, we ignore questions of legality or morality arising from such sale. Besides, in some voting situations – such as a shareholders’ meeting – these questions do not arise at all.) Let us suppose that an outsider (not one of the voters) stands to gain a payoff of one unit of transferable utility if the assembly were to pass a given bill. Then her expected payoff (in units of utility) is numerically equal to the probability that the bill will pass. Now suppose she considers buying voter x’s vote. If such a purchase is made, then from the buyer’s point of view voter x will indeed be known to vote ‘yea’ but Assumption 6.1 is still maintained for the other voters. Let us denote by ‘ x (G)’ the increment that would be added to the prospective buyer’s expected payoff if voter x were to vote ‘yea’ with certainty instead of probability 12 . This magnitude can be regarded as the price (Greek: ) of voter x’s vote, because the buyer should clearly be prepared to pay for it any sum smaller than x . Now, the only advantage that the buyer can secure by buying x’s vote lies in
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preventing the event that x frustrates the adoption of the bill. This event is the conjunction of two events: first, that x is able to affect the outcome and, second, that x votes ‘nay’. Under Assumption 6.1 these two events are independent. Hence by Theorem 6.2 we have: THEOREM 6.3. Under Assumption 6.1, the price x=2 = x=2n.
x of voter x is
It seems to us that x is a valid measure of voter x’s absolute I-power. It tells us what x is able to obtain by selling his vote – his way of exercising influence – to an outsider. As in measuring P-power, the sum of one unit of utility is held fixed, irrespective of the particular SVG or particular voter being considered. But whereas in the case of P-power this fixed sum represents the total payoff to be divided among the voters, in the case of I-power it represents the total payoff of the outside buyer provided the bill is approved. Whereas the sum of the P-powers of all voters of an SVG must automatically be 1, this is by no means the case for I-power. Rather, the sum
(G) =
Xf (G) : x 2 N g x
seems to be an important characteristic of an SVG G = (N; W ). Dubey and Shapley (1979, p. 106) suggest that the total Bz score of all voters in G, that is to say the quantity (G) = fx (G) : x 2 N g, is a kind of ‘democratic participation index’, measuring the decision rule’s sensitivity to the ‘public will’. In our view, this description applies more properly to rather than to . For SVGs with a fixed number of n of voters, and do not differ significantly: merely by a constant scaling factor 2n . But if we wish to compare SVGs with different numbers of voters, then we must use quantities scaled upon a fixed unit of measurement; in the case of this is a unit of utility. Note also that if a dummy is added to G then the value of is doubled, whereas that of is unchanged. An index – a relative measure – of I-power is obtained from an absolute measure of I-power by normalization, so that the relative I-powers of all voters in an SVG add up to 1. Under Assumption 6.1, the index obtained (via Theorem 6.3) is precisely the Bz index. It may turn out that I-power and P-power are merely two aspects of a single notion that can be called ‘voting power’ simpliciter. However, if instead it turns out that we have here two distinct notions of
P
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voting power that cannot be synthesized into one, then at this point it could be argued that for a separate notion of I-power the primary concept is the absolute measure, whereas the index is merely a standardized derivative, useful for measuring the relative share of a voter in the total I-power. It could further be argued that the bloc postulate proposed by Felsenthal and Machover (1995, p. 209), which is undoubtedly compelling for a P-power index, cannot be insisted upon for an I-power index. If a voter a in an SVG G ‘swallows’ a non-dummy voter b, then it is intuitively obvious that in the resulting SVG, Gja&b, the expected share of the bloc a&b in the fixed purse must be greater than the expected share of a was in the original SVG G. Hence a valid index of P-power must satisfy the bloc postulate. It is equally clear that the absolute I-power of a&b in Gja&b ought to be greater than that of a in G. But since the sum of the absolute I-powers of all voters is not a fixed quantity, one cannot exclude the – admittedly paradoxical – possibility that the relative share of I-power possessed by a&b may be smaller than that possessed by a alone. As shown by Felsenthal and Machover (1995), the Bz, Js and D-P indices violate the bloc postulate. Hence they are in our view unacceptable as indices of P-power. However, by Theorem 11.1 of Felsenthal and Machover (1995, p. 226–227),
a&b (Gja&b) = a (G) + b (G):
So it may be argued that the Bz index’s violation of the bloc postulate is not sufficiently paradoxical to disqualify it from serving as a measure of relative I-power, as a separate notion. This matter will no doubt continue to be debated. For the Js and D-P indices we know of no result analogous to Theorem 6.2 or 6.3 which would allow us to define an ‘absolute’ Js or D-P measure in a natural way. Without such results, these indices cannot be accepted as valid indices of I-power. (The D-P index also fails to respect the dominance relation, which in our view disqualifies it as an index of any known aspect of voting power.) Let us now consider the significance of Assumption 6.1. We would like to argue that it is justified as embodying a priori ignorance regarding the view of an SVG as an ‘uninhabited shell’ in the sense explained in Section 3. Assumptions such as this were indeed made in classical probability theory in situations where nothing is known
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a priori about the voters’ inclinations. J. Bernoulli justified this sort of assumption by what is called the Principle of Insufficient Reason. Of course, different assumptions could be made, yielding different conclusions. For example, Straffin (1982, 1988) considers both a version of 6.1 and the following HOMOGENEITY ASSUMPTION. Whenever a vote is called, a number p is chosen at random, as a value of a random variable P having uniform distribution on the unit interval [0,1]; all voters then vote independently of one another, each voting ‘yea’ with probability p. Under this assumption, the probability that voter x is able to affect the outcome of the vote is equal to S-Sx (G). (The proof is due to Owen, 1972; see also Straffin, 1982 and 1988). Consequently, for the price x we would get S-Sx /2 instead of x /2. Note however that here the votes of two distinct voters x and y are independent only under the condition that p is specified; but if p is left unspecified their votes are very highly correlated. To be precise, it is easy to see that Probfx votes ‘yea’ jP = p & y votes ‘yea’g = Probfx votes ‘yea’ jP = pg = p; but Probfx votes ‘yea’ jy votes ‘yea’g =
2 3
while Probfx votes ‘yea’g =
1 : 2 Thus the homogeneity assumption amounts to positing a very strong measure of agreement among all voters. As Straffin (1988, p. 75) puts it: ‘We could think of voters as judging bills by some uniform standard, and the number p as the bill’s acceptability level by that standard.’ This of course flies in the face of the caveat quoted in Section 3, which warns us against assuming a priori that voters act in such a harmonious way. It must be stressed that, given the assumptions of our analysis, the choice between Assumption 6.1 and the homogeneity assumption should not be made according to their degree of verisimilitude,
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the extent to which they are truly descriptive of real-life voting situations. (In this respect both of them are unacceptable.) Rather, the question is which of them is a more reasonable expression of a priori ignorance. This can be made somewhat more precise by borrowing the notion of entropy from statistical mechanics. Consider an arbitrary SVG G with n voters. Then a vote may have any one of 2n distinguishable outcomes, according to which coalition of voters voted ‘yea’. This gives us a sample space with 2n points. The assumption that these 2n outcomes are indeed distinguishable corresponds to the Maxwell–Boltzman statistics of classical statistical mechanics. Under this assumption, the probability distribution yielding maximal entropy (hence embodying greatest ‘disorder’ and least information) is the one that assigns equal probabilities to all the 2n points. This is precisely Assumption 6.1. On the other hand, it is easy to see that if Ek is the event that exactly k voters vote ‘yea’, then the homogeneity assumption assigns to all Ek equal probability, 1=(n + 1) each. This yields the probability distribution with maximal entropy provided the space of outcomes is regarded as having not 2n but only n + 1 points, each corresponding to the number of ‘yea’ voters. This accords with the Bose-Einstein statistics, which in modern statistical mechanics is applied to photons and certain microparticles, precisely on the grounds that they must be regarded as indistinguishable clones! (On the Maxwell–Bolzmann and Bose–Einstein statistics, see Feller, 1957, p. 38f. As Feller notes, the term ‘statistics’ in this context is used in a sense ‘peculiar to physics’.) In our view Assumption 6.1 must be upheld as embodying absence of a priori information regarding voters’ motivations, intentions and interests. Of course, there may be situations when, on the basis of some partial knowledge, the homogeneity assumption may be preferable a posteriori. In such situations the S-S index may indeed serve as a measure of I-power; but as an a posteriori rather than a genuinely a priori measure. (See in this connection also Leech, 1990, p. 299, who states that the S-S index is appropriate only if there is reason to assume a high degree of uniformity among voters.) This ties in with what we have seen in preceding sections, because there is no warrant for insisting that an a posteriori measure satisfy the bicameral postulate.
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For the rest of this section we shall continue to assume that the price of voter x’s vote is correctly given by the equality x = x =2n . As we have observed, this may be taken as a reasonable measure of I-power, rather than P-power. However, it seems highly plausible to assume that the two aspects of voting power ought to be closely related. It is interesting to note that although Shapley and Shubik did not propose any specific method for determining a voter’s price, they nevertheless explicitly expected not only that there should be a positive connection between a voter’s power and his price, but that a voter’s power ought to be exactly proportional to his price (cf. Shapley and Shubik, 1954, pp. 790–791). In this connection it must be recalled that the rationale of the S-S index proposed by them, rooted as it is in the Shapley value, is that of P-power. We would not like to go as far as suggesting that P-power ought to be proportional to price. Indeed, this would eliminate the possibility that P-power and I-power, as two distinct notions, may not be coextensive. Instead, we propose a much weaker connection, which seems to us very plausible: surely a more powerful voter must be more expensive than a less powerful voter in the same SVG. Let K be any power score (in particular, K may be a power index). We shall say that K satisfies the price monotonicity condition (PMC) if for every G and any two voters x and y of G,
Kx(G) < Ky (G) , x(G) < y (G):
The foregoing discussion suggests that the PMC should be included together with the bicameral and bloc postulates (as well as other intuitively compelling postulates, such as those proposed in Felsenthal and Machover, 1995) among the conditions that a reasonable index of a priori P-power ought to satisfy. The Bz index trivially fulfills the PMC, but (as we have argued) fails certain other tests of adequacy as a measure of P-power. As for the S-S index, any situation in which it violates the weak bicameral postulate yields automatically an example where it also violates the PMC. This follows easily from the fact that the Bz index satisfies the (plain) bicameral postulate. Thus Example 4.2 shows that the S-S index violates the PMC. A somewhat simpler example, an SVG having eight voters, in which the S-S index violates the PMC (but not the weak bicameral postulate), is given by Straffin (1988, p. 73).
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If the PMC is accepted as intuitively compelling, then it provides a conclusive argument for the weak bicameral postulate, because the latter is a logical consequence of the former. However, regardless of whether or not the PMC is accepted as an intuitively compelling postulate, these examples demonstrate that the S-S and Bz indices may fail to be co-monotonic and hence produce sharply conflicting conclusions: according to the S-S index voter a is ‘stronger’ than another voter b, yet according to the Bz index b is ‘stronger’ than a. Obviously, both cannot be right. By the way, Examples 4.3 and 4.4 demonstrate that Js and D-P respectively may also fail to be co-monotonic with Bz.
7. CONCLUSIONS
In Section 3 we argued that the bicameral postulate is intuitively compelling. Anyone rejecting this conclusion ought to show good reason why the distribution of relative voting power within a decisionmaking body G can be expected to shift when G forms a bicameral meet with another such body whose assembly is disjoint from that of G. In real life such meets are quite common: the resolutions of many decision-making bodies are not final but subject to assent by another body (which in some cases consists of one veto-wielding individual). A clear example of this is the European Community Council of Ministers (ECCM): the Treaty of Maastricht (1992) turned it into a chamber of a bicameral meet, by allowing its decisions on a range of important issues to be vetoed by an absolute majority of the members of the European Parliament. Arguably, another example is the US federal legislative system (consisting of the two Houses of Congress and the President), whose legislation can be annulled by the Supreme Court. Yet, as far as we know, all studies of the distribution of voting power within such real-life bodies G, which form part of a bicameral meet G ^ H, have assumed tacitly that this issue can be addressed as though G were a stand-alone body, and have not allowed for a possible effect of the meet; for example, there was no reconsideration of the ECCM distribution in 1992, based on an anticipated effect due to the added veto power. Nor are we aware that this tacit assumption has ever been criticized. Thus it seems that
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all those investigators have assumed that such shifting effect ought not to exist. That intuition is justified; their only mistake has been the tacit assumption that the specific instruments, the indices used to measure power, would in fact be indifferent to such effects. Of course, one might argue that the literature ignored the effect only because no one had noticed it. If this view prevails, it will be interesting to see how it will be put into practice in future studies of voting power. Since legislation is hardly ever completely final but may be overturned at some future stage, it is not clear in which SVG power is to be measured – where and when does the game end? The S-S index can be characterized as the unique index satisfying a certain set of conditions, proposed by Dubey (1975), based on Shapley (1953). One of these conditions, Dubey’s composition axiom, is not generally regarded as intuitively compelling (see, for example, Straffin 1982, p. 296, where he concurs with Roth, 1977). Indeed, many writers have gone on using the Bz index alongside the S-S index, regarding neither as uniquely valid. Felsenthal and Machover (1995) have conducted an evaluation of the voting-power indices proposed to date in light of several intuitively compelling postulates, and found that the S-S index satisfied all of them. However, seeing that the satisfaction of any set of intuitively compelling postulates by a voting-power index can only be regarded as a necessary but not a sufficient condition for the reasonableness of this index, they speculated that ‘the S-S index [may yet be] found to suffer from a hitherto unsuspected unacceptable anomaly’ (cf. Felsenthal and Machover, 1995, p. 225). This has now occurred: we view the bicameral postulates as intuitively very compelling, and hence regard their violation by the S-S index as a very severe pathology of this index. What one makes of the violation of the PMC by the S-S index depends on whether one accepts Assumption 6.1 as the most natural ‘null hypothesis’ in a state of ignorance of voters’ intentions. If so, then violation of the PMC provides a further argument against the S-S index. Some of the arguments we have presented apply not only to the S-S index but also to Shapley’s (1953) value for n-person games, from which the S-S index is derived. It therefore seems as though the Shapley value, too, is undermined to some extent. As is well known,
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the Shapley value is in any case somewhat vulnerable because it is conceptualized as a player’s expected payoff without, however, specifying in a convincingly realistic and fully general way a probability space in which this payoff is defined as a random variable. (Thus, the space of all permutations of the set of players, in which it is possible to define a random variable whose expectation is the Shapley value, is not remotely credible as a bargaining model for the formation of coalitions. On the other hand, the bargaining model proposed by Gul, 1989, which is arguably more realistic, only yields the Shapley value for a rather restricted class of n-person games, which excludes all non-trivial SVGs.) In general, such an approach may lead to incoherent results. (Cf. the observation by Luce and Raiffa, 1957, pp. 251–252, that the Shapley value does not have an independent, clear-cut definition, say in terms of a definite bargaining procedure.) Now, because the S-S index violates the bicameral postulates, this may perhaps be regarded as an instance of such incoherence. It seems, then, that none of the four extant indices can serve as a truly a priori measure of what we have called P-power, whereas arguably only the Bz index can serve as a truly a priori measure of what we have called I-power. Its violation of the bloc postulate is no doubt paradoxical, but perhaps excusable for an index of I-power, as argued in Section 6. The consequences for future research are therefore clear. First, it is worth exploring the question of whether there can exist any a priori relative voting-power index that is not ad hoc and that satisfies the bloc and bicameral postulates, in addition to having the strong dummy property and respecting the dominance relation. Second, it would be interesting to see whether there are other sets of principles that are significant additions or alternatives – compelling conditions that are more-or-less unrelated to the bloc and bicameral postulates. The results of such investigations will ultimately determine whether the future holds one voting power index, or several. If there are several, they may all be viewed as flawed, or each may be seen as appropriate for a different purpose. In any case, it remains unclear whether the Bz or S-S index will survive. On the other hand, it seems to us that both the Js and D-P indices should be disqualified as indices of any type of a priori voting power because the former
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violates both the bloc and bicameral postulates, whereas the latter also does not respect the dominance relation, even in WVGs. APPENDIX
We present here the definitions, exact theorem statements, and proofs pertinent to the material from the latter half of Section 5.
DEFINITION A.1. For any SVG G = (N; W ), a coalition X N is a blocking coalition if N X 62 W (the members of X can successfully block passage of a law if they collectively choose to do so); and is a minimal blocking coalition (abbreviated MBC) if it is a blocking coalition that does not include any other blocking coalitions. The game dual to G, denoted Gd , is defined as Gd = (N; W d );
2 W d , X is a blocking coalition of G. For any finite set N , and Y N , put Y = fZ N : Z \ Y 6= ;g. Let G (N; Y ) be the SVG (N; Y ). The class of all SVGs of the form G (N; Y ) will be denoted by ‘UMB’. Note that G (N; Y ) is completely characterized as the game on N where X
whose unique minimal blocking coalition is Y (hence the acronym ‘UMB’) and that it is a WVG: every voter in Y gets weight 1, everyone else gets weight 0, and the quota is 1. In particular, G (N; Y ) is in the class E mentioned in Section 5. Thus UMB E .
THEOREM A.2. The operations of voting-bloc formation, and of taking the bicameral meet of two games (having a disjoint assembly), together generate the class of all SVGs (up to isomorphism) from the class E . Proof. It suffices to show that all SVGs are generated via these operations from the subclass UMB. First we show that an arbitrary SVG G = (N; W ) is an intersection of games in UMB. Indeed, let (Y1 ; Y2 ; . . . ; Yk ) enumerate the MBCs of G. Then
X 2 W , N X is not a blocking coalition , N X does not include any MBC , X 2 Y1 \ Y2 \ \ Yk :
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Now consider two SVGs G1 = (N; W1 ) and G2 = (N; W2 ) with a common assembly N = f1; 2; . . . ; ng. Let N = f 1; 2; . . . ; ng and let G2 = (N ; W2 ) be an isomorphic copy of G2 via the isomorphism that takes each i to i. Let G = G1 ^ G2 . One at a time, for i = 1; 2; . . . ; n, form the bloc of voter i and her mirror image i, obtaining the sequence of SVGs Gj1& 1, (Gj1& 1)j2& 2, and so on. The last in this sequence is clearly isomorphic to G1 \ G2 . This establishes our theorem. DEFINITION A.3. A power index K is dual if Kx (G) = for every voter x of every SVG G.
Kx(Gd)
Both S-S and Bz are known to be dual. DEFINITION A.4. Using the notation of Definition 2.3, we define the parallel join of two SVGs G1 and G2 : G1 _ G2 =df (N; W1 N2 [ W2 N2 ); here, passage of a law requires approval in either chamber, G1 or G2 . A power index K is said to satisfy the parallel postulate if
Ka (G1 )=Kb(G1) = Ka(G1 _ G2 )=Kb(G1 _ G2) whenever a and b are non-dummy voters in an SVG G1 and G2 is
another SVG whose assembly is disjoint from that of G1 .
THEOREM A.5. Let K be any dual power index. Then K satisfies the bicameral postulate if and only if K satisfies the parallel postulate. Proof. The key idea here is an analogy between the set-theoretic operations \, [ and c (intersection, union, complementation) and the game-theoretic operations ^, _ and d (bicameral meet, parallel join, dualization). The analog of one of De Morgan’s laws, (X1 \ X2)c = X1c [ X2c , is (G1 ^ G2 )d = G1 d _ G2 d ;
the validity of which is easy to check. Now, let K be any index satisfying the parallel postulate, a and b be any non-dummy voters of some SVG G1 , and let G2 be any
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second SVG. If H and J are any two SVGs we shall write ‘H to abbreviate the statement
J’
Ka (H)=Kb(H) = Ka (J)=Kb(J) for all a and b
who are non-dummy voters in both H and J:
To see that K must satisfy the bicameral postulate, note that G1
G1 d G1d _ G2d = (G1 ^ G2)d G1 ^ G2 :
A similar argument derives the parallel postulate from the bicameral postulate, for dual indices. COROLLARY A.6. The Bz index satisfies the parallel postulate and the S-S index does not. THEOREM A.7. If K is a power score that satisfies Dubey’s composition axiom and has the strong dummy property, then
Ka (G1 ) = Ka(G1 ^ G2 ) + Ka(G1 _ G2 ) for any voter a in the SVG G1 and any SVG G2 with a disjoint
assembly. Proof. Take two SVGs G1 = (N1 ; W1 ) and G2 = (N2 ; W2 ) with disjoint assemblies, and a 2 N1 . Then if we apply Dubey’s axiom to G1 N2 and G2 N1 , we obtain
Ka (G1 N2) + Ka (G2 N1) = Ka ((G1 N2 ) [ (G2 N1 )) + Ka ((G1 N2 ) \ (G2 N1 )): The right-hand side is recognizable as Ka (G1 ^ G2 ) + Ka (G1 _ G2 ) and the left-hand side is equal to Ka (G1 ) + 0, if the index K has the
strong dummy property. COROLLARY A.8. If K , G1 , G2 , dummy voter in G1 , then
a are as above and b is a non-
Ka (G1 )=Kb(G1) = (Ka (G1 ^ G2) + Ka (G1 _ G2))= (Kb (G1 ^ G2 ) + Kb (G1 _ G2 )):
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In particular, Corollary A.8 holds when K is the S-S index. The interpretation seems clear: if according to the S-S index the ratio of the power of a to the power of b decreases (or increases) when the bicameral meet with G2 is formed, then this ratio must increase (or decrease, respectively) when the parallel join with G2 is formed, and by an amount that compensates exactly (in the sense of Corollary A.8). ACKNOWLEDGMENTS
The authors would like to thank Robert Aumann, Ken Binmore, Steven Brams, V.S. Levchenkov, William Lucas, Roger Myerson, Guillermo Owen, Amnon Rapoport, Thomas Quint, Alan Taylor, and two anonymous referees for their helpful comments. W.Z. was supported by NSF grant DMS-9101830. Earlier versions of this paper were presented at the International Workshop on Game Theory, Chicago IL, August 8–12, 1994; and at the annual meeting of the Public Choice Society, Long Beach CA, March 24–26, 1995. REFERENCES Banzhaf, J.F.: 1965, ‘Weighted Voting Doesn’t Work: A Mathematical Analysis’, Rutgers Law Review 19, 317–343. Brams, S.J., Affuso, P.J. and Kilgour, D.M.: 1989, ‘Presidential Power: A GameTheoretic Analysis’, in: Brace, P., Harrington, C.B. and King, G. (Eds.), The Presidency in American Politics, New York: New York University Press, pp. 55–74. Coleman, J.S.: 1971, ‘Control of Collectivities and the Power of a Collectivity to Act’, in: Lieberman, B. (Ed.), Social Choice, New York: Gordon and Breach, pp. 269–300. Deegan, J. and Packel, E.W.: 1978, ‘A New Index of Power for Simple n-Person Games’, International Journal of Game Theory 7, 113–123. Deegan, J. and Packel, E.W.: 1982, ‘To the (Minimal Winning) Victors Go the (Equally Divided) Spoils: A New Index of Power for Simple n-Person Games’, in: Brams, S.J., Lucas, W.F. and Straffin, P.D. (Eds.), Political and Related Models (Vol. 2 in series Models in Applied Mathematics, Ed. W.F. Lucas), New York: Springer, pp. 239–255. Dubey, P.: 1975, ‘On the Uniqueness of the Shapley Value’, International Journal of Game Theory 4, 131–140. Dubey, P. and Shapley, L.S.: 1979, ‘Mathematical Properties of the Banzhaf Power Index’, Mathematics of Operations Research 4, 99–131. Feller, W.: 1957, An Introduction to Probability Theory and its Applications, Vol. I, 2nd. Edn., New York: Wiley.
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Straffin, P.D.: 1988, ‘The Shapley–Shubik and Banzhaf Power Indices as Probabilities’, in: Roth, A.E. (Ed.), The Shapley Value, Cambridge: Cambridge University Press, pp. 71–81. Taylor, A. and Zwicker, W.: 1993, ‘Weighted Voting, Multicameral Representation, and Power’, Games and Economic Behavior 5, 170–181.
DAN S. FELSENTHAL
MOSHE´ MACHOVER
Dept of Political Science University of Haifa Haifa 31905 Israel
Dept. of Philosophy King’s College London Strand London WC2R 2LS U.K.
E-mail:
[email protected]
E-mail:
[email protected] WILLIAM ZWICKER
Dept. of Mathematics Union College Schnectady, NY 12038 U.S.A. E-mail:
[email protected]
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