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Voting power measurement: a story of misreinvention

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Abstract

In this account of the history of voting-power measurement, we confine ourselves to the concept of a priori voting power. We show how the concept was re-invented several times and how the circumstances in which it was reinvented led to conceptual confusion as to the true meaning of what is being measured. In particular, power-as-influence was conflated with value in the sense of transferable utility cooperative game theory (power as share in constant total payoff). Influence was treated, improperly, as though it were transferable utility, and hence an additive and distributive quantity. We provide examples of the resulting misunderstanding and mis-directed criticism.

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Notes

  1. References [32] and [41], which were published later, were available to us in 2002 in mimeograph form.

  2. [50 p. 294]. Although Martin's intuition that a voter's a priori chance of winning in a vote is not generally proportional to the voter's weight was correct, it should be pointed out that in the particular case which was his concern he had in fact no grounds for complaint. It so happens that if the delegates of each of the 13 states in the US House of Representatives would have indeed always voted as a bloc, then in the resulting weighted voting game, [33; 10, 8, 8, 6, 6, 5, 5, 5, 4, 3, 3, 1, 1], the Penrose–Banzhaf voting power of each of these state blocs would have been almost exactly proportional to its weight.

  3. [50 p. 294].

  4. Lionel Sharples Penrose (1898–1972) studied mathematics at St John's College, Cambridge; awarded degree of Moral Sciences Tripos (Cambridge, 1921); did a medical course at St Thomas's Hospital, London, and was awarded the degree of MD, (1930). His main research was in genetics as well as psychiatry. Together with his son, the famous mathematician Roger Penrose, he invented the Penrose triangle and the Penrose stairs, impossible objects popularized by the artist MC Escher.

  5. Initially he assumes one vote per voter; but then goes on to admit bloc votes of several voters acting together. So the decision rules he considers are, in effect, weighted voting rules with quota set at half, or very slightly over half, of the total weight.

  6. [45, p. 53].

  7. In what follows, whenever we speak of probability simpliciter, we mean a priori probability in this sense.

  8. [45, p. 57]. Although he does not say so explicitly, it is clear from the context that he is assuming representatives always vote according to the majority opinion in their respective constituencies.

  9. ‘In discussing the question of voting I have introduced the assumption of indifference which does not correspond closely with the facts as usually understood. However, since the future decisions of an elected representative cannot be easily predicted, voting is much more at random than it appears to be to the voter. Thus, the general theorems which can be established by using the assumption may be more often valid than might at first be supposed.’

  10. [46 p. 7]. The assumption that the other voters are ‘a large crowd’ [of individuals with one vote each] is not needed for the definition itself but for the approximation formula for P, based on the Central Limit Theorem of probability theory, which Penrose states immediately following this definition.

  11. Penrose does not offer any explanation for the shift in his definition. In fact, his 1946 paper is not mentioned in the 1952 booklet, although it is listed in its Bibliography.

  12. The proof outlined by Penrose is faulty: the conditions he assumes are not sufficient. On the other hand, the result does hold for a large variety of cases, many of which were not envisaged by him. This topic is currently being intensively researched; see [41].

  13. The exception that proves the rule is Morriss [43, p. 160], which gives him full credit, but does not belong to the mainstream. Morriss (private communication) recalls that he found out about Penrose's work from Fielding and Liebeck [30, p. 249], which also does not belong to the mainstream, where Penrose is given credit for the square-root rule, but not for his measure of voting power. This attribution of the square-root rule is cited also in Grofman and Scarrow [33, p. 171] (included in [14]), which does belong to the mainstream.

  14. [53, p. 59].

  15. The family of all such games was described precisely and classified by Shapley [53].

  16. [54, p. 789].

  17. As evidence for their claim S&S refer in a footnote to the uniqueness proof in [52]. But this is—at best—misleading: the proof in [52] applies to the class of all cooperative games, but not to that of simple games, because it is not closed under the algebraic operation of game addition. In 1975 Dubey [20] gave a correct unique characterization of the S-S index; but one of the conditions postulated by him—Dubey's axiom—is by no means compelling for a measure of voting power.

  18. This is clearly the case in the UN Security Council, which is one of the two examples cited by Shapley [53, p. 59] where allegedly ‘the acquisition of power is the payoff’.

    This is quite apart from the fact that Shapley misrepresents the decision rule of the UNSC as a [binary] simple game. In reality, it cannot be so represented because abstention by a permanent member has a different effect from both a ‘yes’ and a ‘no’ vote. On this, see Subsection 10.1 below.

  19. This was to remain his chief concern throughout.

  20. [2, p. 331].

  21. [3, p. 1316].

  22. [3, p. 1316].

  23. [4, p. 307] Note the terms ‘chance’ and ‘likely’. On the following page he explains and justifies very lucidly the aprioristic nature of this assumption, and draws a distinction between a priori power ‘inherent in the rule’ and actual power.

  24. For example, in \({\user1{\mathcal{U}}} = {\left[ {3;1,1,1,0} \right]}\) the value of ψ for the first voter is \(\frac{1} {4}\) and that of β is \(\frac{1} {3}\). In \({\user1{\mathcal{V}}} = {\left[ {3;1,1,1,1} \right]}\) the respective values of ψ and β for the first voter are \(\frac{3} {8}\) and \(\frac{1} {4}\). Thus the first voter has in \({\user1{\mathcal{V}}}\) more absolute power but less relative power than in \({\user1{\mathcal{U}}}\) . In their invaluable 1979 study [21], Dubey and Shapley note that the common normalization of the Banzhaf score is ‘not as innocent as it seems’. Instead, they propose, and study in the rest of the paper, ‘another normalization [that] is in many respects more natural’, namely ψ, which they call ‘the swing probabilities’ (p. 102). This major understatement was unheeded by most users of the Bz index.

  25. [2, p. 319].

  26. [2, p. 331].

  27. See [2, p. 331 fn. 32]. However, he also argues against the S–S index on the grounds that ‘it... attaches an importance to the order in which legislators appear in each minimal voting coalition rather than simply to the number of minimal voting coalitions in which each appears’ (loc. cit.). This criticism of the S–S index, which has been made by several other authors, is in our opinion based on a misunderstanding; see [24, p. 200].

  28. [3, p. 1317]. The references listed in the footnotes to this claim leave no doubt that the latter measure is the S-S index.

  29. [4, p. 312, fn. 28], where he also reports that Riker, Shapley and Mann had predicted that ‘the two techniques yield substantially similar results for large numbers of voting units’. This prediction is correct under certain conditions; but it is known to be false for so-called oceanic weighted decision rules, in which there are a few ‘heavy’ voters and a large number of very nearly insignificant ones (see e.g. [57, Example 2, p. 1134]).

  30. Dubey and Shapley [21, p. 124f] prove Penrose's identity, and remark that ‘[i]t was not noticed for several years that this “Rae index” is nothing but the Banzhaf index in disguise.’ In fact, as we saw in Section 2, Penrose had been aware of this identity 33 years earlier, 23 years before the publication of Rae's paper.

  31. Reprinted in [17].

  32. [15, p. 271f].

  33. [15, p. 297].

  34. A=ω/2n where n is the number of voters and ω is the number of winning coalitions. The values of ψ x for all voters x are not sufficient to determine A. But from ψ a and A we can obtain γ a (=ψ a /2A) and γ a *(=ψ a /2(1−A)). Conversely, from γ a (or γ a *) and A we can retrieve ψ a . Also, ψ a can be obtained from γ a and γ a * jointly, as their harmonic mean: ψ a =[(γ a −1+γ a *-1)/2]−1.

  35. This was somewhat strange in view of Banzhaf's high public profile in matters concerning social choice. Even more strangely, Banzhaf's work is not mentioned in Coleman's 1973 contribution [16] to the subject.

  36. See [12].

  37. It is widely believed that the Shapley value, and hence the S–S index, also relies for its justification on a specific bargaining model, namely the model in which the grand coalition is formed in random order, and each player, on admission, demands and is promised the marginal amount that s/he contributes to the worth of the coalition. This belief is mistaken; see [24, Comments 6.2.8, 6.3.9].

  38. For a detailed discussion see our [24, Section 6.4 and Ch 7 passim].

  39. In his 1977 paper [36], Johnston does not refer to Banzhaf—of whom he may not have been aware at that time. Instead, at the end of this paper (p. 577) he appends an Acknowledgement: ‘The procedure outlined here was developed from one used by Coleman (1972) and Rae (1972).’ (These are references to [15] and [48] in the 1972 edition of [40]). Laver must have missed this Acknowledgement. In his 1978 response [37] to Laver, Johnston tries to put the record straight (p. 907): ‘At the outset, however, I should make it clear that the power index that I used was not devised by me.... All that I did was to modify slightly ... an index apparently developed independently by Coleman (1972) and by Rae (1972) and very akin to that used by Banzhaf (1968).’ (This is a reference to [4]) A few pages later, Johnston cites Banzhaf's 1965 paper [2], borrowing from it an argument against the S–S index, which Laver favoured.

  40. [39, p. 902].

  41. As we saw, in the case of the D–P index the argument for this was that the spoils won by a coalition must be divided among its members.

  42. By the way, his definition of this index [5, p. 186] is imprecise, and applies in fact to Holler's PGI (see above, end of Section [7]). He also claims (ibid.) that ‘it is generally the case that a change in voting rules that increases an actor's power on the Shapley–Shubik index also increases it on the Banzhaf index, and vice versa.’ This is false, as can be seen from [24, Examples 7.8.5, 7.9.16] and [51].

  43. [5, p. 191].

  44. It is from this book that we found out about Penrose's work in 1995; we wish we had come across it earlier.

  45. Morriss is however unfair to Coleman, whose measures he dismisses as ‘identical to Banzhaf's’ (p. 167). In our view, he is also too scathing about the Bz index, which he rejects as utterly useless and advocates ‘banishing’ it (p. 166). In our opinion, the Bz index is useful for certain purposes, provided it is handled with care and understanding.

  46. These can be found, with proofs, in [24, Cor. 3.4.10, Comment 6.2.24].

  47. Barry [5, p. 350] also mentions abstention as a significant factor.

  48. This is done, for example, by Penrose in his 1952 booklet [46] and by Banzhaf in his 1968 paper [4].

  49. Arguably, if j=2 then the inputs can automatically be regarded as naturally ordered; this is also true when j=3, provided one of the three possible inputs is abstention, or indifference between the other two. The same holds for the outputs where k=2; and also for k=3, provided one of the three outputs is a tie between the other two.

  50. In his 1965 paper, Banzhaf raises and dismisses the issue of abstention in a brief, and in our opinion inadequate, remark in a footnote [2, fn. 34]; but this is at least better than ignoring it, as others do.

  51. These are weighted rules whose possible inputs are x and y, which may (but need not) be ‘yes’ and ‘no’, and abstention; the possible outputs are x, y and a tie, which occurs if and only if the total weight of the x votes equals that of the y votes.

  52. For example, let \({\user1{\mathcal{U}}} = {\left[ {11;6,5,1,1,1,1,1} \right]}\). Now suppose the first voter (with weight 6) annexes the voting rights of one of the voters with weight 1. The result is \({\user1{\mathcal{V}}} = {\left[ {11;7,5,1,1,1,1} \right]}\) Now, the value of β for the first voter is \(\frac{{{\text{11}}}} {{{\text{23}}}}\) in \({\user1{\mathcal{U}}}\) and \(\frac{{17}} {{36}}\) in \({\user1{\mathcal{V}}}\). Since \(\frac{{17}} {{36}} <\frac{{11}} {{23}}\), it seems that the first voter has lost power as a result of the annexation. (This is [24, Example 7.8.14] and is somewhat simpler than the original example given in [22]).

  53. The I-power/P-power distinction is also further discussed and explained in subsequent papers, for example [25] and [42].

  54. See text to footnote [24].

  55. In the example of footnote [52], the value of ψ for the first voter is \(\frac{{33}} {{64}}\) in \({\user1{\mathcal{U}}}\) and increases to \(\frac{{17}} {{32}}\) in \({\user1{\mathcal{V}}}\).

  56. We too were guilty of (partly) reinventing the wheel. After that paper was published we discovered that Coleman [16] had addressed a similar problem almost 30 years earlier. However, his analysis was less comprehensive than ours.

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  1. University of Haifa, Haifa, Israel

    Dan S. Felsenthal

  2. King's College, London, UK

    Moshé Machover

  3. CPNSS, London School of Economics and Political Science, London, UK

    Dan S. Felsenthal & Moshé Machover

  4. 5 Milman Road, Queen's Park, London, NW6 6EN, UK

    Moshé Machover

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Felsenthal, D.S., Machover, M. Voting power measurement: a story of misreinvention. Soc Choice Welfare 25, 485–506 (2005). https://doi.org/10.1007/s00355-005-0015-9

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