Moshé Machover

Postulates and paradoxes of relative voting power — A critical re-appraisal

This paper re-evaluates the problem of measuring thea priori relative voting power of a voter in an assembly. We propose several new intuitively compelling postualtes that any reasonable index of voting power ought to satisfy. At the same time we argue that most of the paradoxes of voting power discussed in the literature are paradoxical only in a weak sense, if at all. This leaves three crippling paradoxes — the well-known paradox ofweighted voting, and two new ones presented here: thebloc anddonation paradoxes. We evaluate the four main relative power indices discussed in the literature with respect to these three severe paradoxes. The Shapley-Shubik index is seen to be immune to all three paradoxes, while the Deegan-Packel index is vulnerable to all three. The Banzhaf and the Johnston indices are demonstrably immune to the paradox of weighted voting. However, they are shown to suffer from both the bloc and the donation paradoxes. We argue that this seriously undermines these indices in a hitherto unsuspected way. Several other theoretical issues relating to voting power are discussed.

Analysis of QM rules in the draft constitution for Europe proposed by the European Convention, 2003

Abstract.We analyse and evaluate the qualified majority (QM) decision rules for the Council of Ministers of the EU that are included in the Draft Constitution for Europe proposed by the European Convention [5]. We use a method similar to the one we used in [9] for the QM prescriptions made in the Treaty of Nice.

Voting power measurement: a story of misreinvention

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.

L.S. Penrose's limit theorem: proof of some special cases

L.S. Penrose was the first to propose a measure of voting power (which later came to be known as ‘the [absolute] Banzhaf (Bz) index’). His limit theorem—which is implicit in his booklet (1952) and for which he gave no rigorous proof—says that in simple weighted voting games (WVGs), if the number of voters increases indefinitely while the quota is pegged at half the total weight, then—under certain conditions—the ratio between the voting powers (as measured by him) of any two voters converges to the ratio between their weights. We conjecture that the theorem holds, under rather general conditions, for large classes of variously defined WVGs, other values of the quota, and other measures of voting power. We provide proofs for some special cases.

The Bicameral Postulates and Indices of a Priori Voting Power

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).

The weighted voting rule in the EU's Council of Ministers, 1958–1995: Intentions and outcomes

We consider the qualified majority voting rule in the Council of Ministers (CM) of the European Union (EU). Using the Banzhaf index--which, we argue, is the most appropriate for measuring the a priori voting power of members in the CM--we analyse the distribution of voting power of CM members in the various phases of expansion of the EU. We compare this distribution to the distribution of voting weights in the CM and the distribution of population in the EU. Finally, we assess the responsiveness (or volatility) of the voting rule in question.

L S Penrose's limit theorem: Tests by simulation

LS Penrose’s limit theorem (PLT) – which is implicit in Penrose [5, p. 72] and for which he gave no rigorous proof – says that, in simple weighted voting games, if the number of voters increases indefinitely while existing voters retain their weights and the relative quota is pegged, then – under certain conditions – the ratio between the voting powers of any two voters converges to the ratio between their weights. Lindner and Machover [3] prove some special cases of PLT; and conjecture that the theorem holds, under rather general conditions, for large classes of weighted voting games, various values of the quota, and with respect to several measures of voting power. We use simulation to test this conjecture. It is corroborated w.r.t. the Penrose–Banzhaf index for a quota of 50% but not for other values; w.r.t. the Shapley–Shubik index the conjecture is corroborated for all values of the quota (short of 100%).

Models and Reality: The Curious Case of the Absent Abstention

We discuss two inter-related puzzling features of the literature on a priori voting power. First, the mathematical model used in virtually all this literature does not recognize abstention as an option distinct from both a ‘yes’ and a ‘no’ vote. Second, real-life decision rules of voting bodies—in particular the US legislature and the UN Security Council—are misrepresented as though they did not allow abstention as a tertium quid. We suggest that these misrepresentations may be examples of what philosophers of science call ’theory-laden observation’.

Annexations and alliances: When are blocs advantageous a priori?

Abstract For any simple voting game (SVG), we consider the question posed in the title from two different points of view as to what voting power means. We also distinguish between blocs imposed by annexation and those formed voluntarily, and present some general theoretical results concerning these notions. We illustrate our theoretical findings with examples using both toy SVGs and the Qualified Majority Voting rule of the Council of Ministers of the European Community (CMEC). We show that when voting power is understood as influence (I-power), forming a voluntary bloc may be advantageous even if its voting power is smaller than the sum of the original powers of its members; and it may be disadvantageous even if its voting power is greater than that sum.

Who ought to be elected and who is actually elected? An empirical investigation of 92 elections under three procedures

Abstract Three voting procedures—Bordas Count (BR), Plurality Voting (PV), and the Single Transferable Vote (STV)—are compared with one another in relation to the outcomes of 92 real elections conducted in Britain by various trade unions, professional associations, and non-profit organizations. Given the social preference ordering (SPO) among the n competing candidates, of whom m candidates must be elected ( n > m ⩾ 1), each of these procedures is evaluated by means of two new indices that measure the discrepancy between the m candidates elected and the prescription of the majoritarian principle. For two of these procedures (PV and STV), we also investigate the relative frequency with which they display the recently discovered Discontinuity Paradox, namely, cases where a candidate who would be elected when m candidates must be elected would not also be elected, ceteris paribus , if m + 1 candidates were to be elected. Our analysis supports the following three conclusions: 1. (i) When m = 1, STV (or rather its Alternative Vote variant) conforms better with the SPO than do the other two procedures, particularly PV. 2. (ii) When m > 1, BR conforms with the SPO significantly better than do the other two procedures, particularly STV. 3. (iii) PV is likely to display the Discontinuity Paradox more often than STV. Since BR is not susceptible to the Discontinuity Paradox, we conclude that, in terms of avoiding this paradox and conforming with the SPO, BR performs better than both PV and STV.