Dan S. Felsenthal

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.

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).

Two-person bargaining behavior in fixed discounting factors games with infinite horizon

Abstract We consider a noncooperative game in which two bargainers have to divide a given surplus (“pie”) by alternating in making and responding to proposals over the division of the surplus; time is discounted, the number of rounds is unlimited, and the discount factors are common knowledge. Two experiments using both equal and unequal discount factors are reported. Their major purpose is to test the subgame perfect equilibrium and compare it to two simpler models incorporating the norms of equality and equity. Analyses of final agreements, number of rounds to agreement, and counteroffers reject the subgame perfect equilibrium model and partially support the other two models.

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.

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.

Review of Paradoxes Afflicting Procedures for Electing a Single Candidate

Three factors motivated me to write this chapter: The recent passage (25 February 2010) by the British House of Commons of the Constitutional Reform and Governance Bill, clause #29 of which states that a referendum will be held by 31 October 2011 on changing the current single member plurality (aka first-past-the-post, briefly FPTP) electoral procedure for electing the British House of Commons to the (highly paradoxical) alternative vote (AV) procedure (aka Instant Runoff ).1 Similar calls for adopting the alternative vote procedure are voiced also in the US. My assessment that both the UK and the US will continue to elect their legislatures from single-member constituencies, but that there exist, from the point of view of social-choice theory, considerably more desirable voting procedures for electing a single candidate than the FPTP and AV procedures. A recent report by Hix et al. (2010) – commissioned by the British Academy and entitled Choosing an Electoral System – that makes no mention of standard social-choice criteria for assessing electoral procedures designed to elect one out of two or more candidates.

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.

New Chairman Paradoxes

Power is one of the most alluring but also one of the most intractable concepts in political science. It leads to a number of paradoxes (Brams, 1976, ch. 7), which continue to manifest themselves in real-life voting bodies like the European Community Council of Ministers (Brams and Affuso, 1985; Brams, 1985: 101–104). For example, one implication of some definitions of power is that the greater proportion of resources (such as votes) that an actor controls, the greater is his power. In this paper, we shall show that this implication in a certain context may be false.