Voting Power and Procedures

This volume collects the invited essays presented in honour of Dan Felsenthal and Moshé Machover. Most of the papers were delivered at the Voting Power in Practice Symposium, Voting Power in Social/Political Institutions: Typology, Measurement, Applications held at the London School of Economics, 20–22 March 2011. The symposium had been planned both to mark the end of 8 years of Leverhulme Trust funding of the LSE’s Voting Power & Procedures (VPP) research programme and to celebrate the immense contribution to the field of voting theory by Felsenthal and Machover’s (F&M) critically acclaimed monograph The Measurement of Voting Power (MVP) published a decade earlier.

The Dan Felsenthal and Moshé Machover research partnership (F&M hereafter) has been one of the most important influences on the modern development of voting theory. The focus of this well-deserved Festschrift to honour their work has been on voting power and its measurement, the subject of their landmark volume, The measurement of voting power: theory and practice, problems and paradoxes (MVP). The focus is entirely apt. The book is a remarkable work that played a major role—possibly the major role—in resuscitating the voting power field that until its appearance had critically stalled.

Felsenthal and Machover have made substantial contributions to the measurement of voting power. It is worth bearing in mind, however, that the notion of political power is actually a quite general one of which voting power is one instantiation. In this brief paper, we consider political power in the general sense and propose a definition. We will show that when applied specifically to voting power, our definition is a generalization of Banzhaf’s definition.

The main purpose of the present paper is to disentangle the mix-up of the notions of success and satisfaction which is prevailing in the voting power literature. We demonstrate that both notions are conceptually distinct, and discuss their relationship and measurement. We show that satisfaction contains success as one component, and that both coincide under the canonical set-up of a simultaneous decision-making mechanism as it is predominant in the voting power literature. However, we provide two examples of sequential decision-making mechanisms in order to illustrate the difference between success and satisfaction. In the context of the discussion of both notions we also address their relationship to different types of luck.

Voting power science is a field of co-operative game theory concerned with calculating the influence a voter can exert on the outcome of a voting game. The techniques used to calculate voting power have names like the Shapley-Shubik index, and the Banzhaf measure. They are invaluable when used to design democratically fair voting games.

In this paper we examine these different techniques, with the specific aim of trying to understand what they are measuring. Many commentators have argued that the techniques are similar, albeit with different probability models. But by focusing upon the less well know differences that exist in the underlying measures themselves, it soon becomes apparent that the dissimilarities between the techniques extend far beyond their methods of counting voting coalitions.

Voting power is commonly measured using a probability. But what kind of probability is this? Is it a degree of belief or an objective chance or some other sort of probability? The aim of this paper is to answer this question. The answer depends on the use to which a measure of voting power is put. Some objectivist interpretations of probabilities are appropriate when we employ such a measure for descriptive purposes. By contrast, when voting power is used to normatively assess voting rules, the probabilities are best understood as classical probabilities, which count possibilities. This is so because, from a normative stance, voting power is most plausibly taken to concern rights and thus possibilities. The classical interpretation also underwrites the use of the Bernoulli model upon which the Penrose/Banzhaf measure is based.

Felsenthal and Machover (1998, The Measurement of voting power – theory and practice, problems and paradoxes. Cheltenham: Edward Elgar) celebrated monograph on The Measurement of Voting Power set off a renewed impetus on the analysis of weighted voting systems. Their presentation strikes a balance between the game-theoretic and the probabilistic approaches to the subject. The present paper holds that the probabilistic view may be profitably extended even further, in providing helpful language as well as motivating new results.

The problem of designing an optimal weighted voting system for the two-tier voting, applicable in the case of the Council of Ministers of the European Union (EU), is investigated. Various arguments in favor of the square root voting system, where the voting weights of member states are proportional to the square root of their population are discussed and a link between this solution and the random walk in the one-dimensional lattice is established. It is known that the voting power of every member state is approximately equal to its voting weight, if the threshold q for the qualified majority in the voting body is optimally chosen. We analyze the square root voting system for a generic “union” of M states and derive in this case an explicit approximate formula for the level of the optimal threshold: \(q \simeq 1/2 + 1/\sqrt{\pi M}\). The prefactor \(1/\sqrt{\pi }\) appears here as a result of averaging over the ensemble of “unions” with random populations.

We consider two-tier voting systems and try to determine optimal weights for a fair representation in such systems. A prominent example of such a voting system is the Council of Ministers of the European Union. Under the assumption of independence of the voters, the square root law gives a fair distribution of power (based on the Penrose–Banzhaf power index) and a fair distribution of weights (based on the concept of the majority deficit), both given in the book by Felsenthal and Machover.

In this paper, special emphasis is given to the case of correlated voters. The cooperative behaviour of the voters is modeled by suitable adoptions of spin systems known from statistical physics. Under certain assumptions we are able to compute the optimal weights as well as the average deviation of the council’s vote from the public vote which we call the democracy deficit.

Representatives from differently sized constituencies take political decisions by a weighted voting rule and adopt the ideal point of the weighted median amongst them. Preferences of each representative are supposed to coincide with the constituency’s median voter. The paper investigates how each constituency’s population size should be mapped to a voting weight for its delegate when the objective is to maximize the total expected utility generated by the collective decisions. Depending on the considered utility functions, this is equivalent to approximating the sample mean or median voter of the population by a weighted median of sub-sample medians. Monte Carlo simulations indicate that utilitarian welfare is maximized by a square root rule if the ideal points of voters are all independent and identically distributed. However, if citizens are risk-neutral and their preferences are sufficiently positively correlated within constituencies, i.e., if heterogeneity between constituencies dominates heterogeneity within, then a linear rule performs better.

The President of the United States is elected, not by a direct national popular vote, but by a two-tier Electoral College system in which (in almost universal practice since the 1830s) separate state popular votes are aggregated by adding up state electoral votes awarded, on a winner-take-all basis, to the plurality winner in each state. Each state has electoral votes equal in number to its total representation in Congress and since 1964 the District of Columbia has three electoral votes. At the present time, there are 435 members of the House of Representatives and 100 Senators, so the total number of electoral votes is 538, with 270 required for election (with a 269–269 tie possible). The U.S. Electoral College is therefore a two-tier electoral system: individual voters cast votes in the lower-tier to choose between rival slates of ‘Presidential electors’ pledged to one or other Presidential candidate, and the winning elector slates then cast blocs of electoral votes for the candidate to whom they are pledged in the upper tier. The Electoral College therefore generates the kind of weighted voting system that invites analysis using one of the several measures of a priori voting power. With such a measure, we can determine whether and how much the power of voters may vary from state to state and how individual voting power may change under different variants of the Electoral College system.

The a priori voting power indices concentrate on actor resource distributions and decision rules to determine the potential influence over outcomes by various actors. That these indices sometimes seem to be at odds with the intuitive distribution of real power in voting bodies follows naturally from their a priori nature. Indices based on actor preferences address this by equating an actor’s voting power with the proximity of voting outcomes to his/her ideal point. It is, however, shown that in some cases the preference-based indices are just as questionable as the classic ones. The main aim of this paper is to delineate the proper scope of power indices. In the pursuit of this aim we try to show that the procedures resorted to in making collective decisions are as important—if not more so—as the actor resource distribution. We review some results on agenda-systems to drive home this point. The proper role of power indices then turns out to be in the study of actor influences over outcomes when the actors are on the same level of aggregation (individuals, groups, states) and “comparable” in the sense of having similar sets of strategies at their disposal and preferences are not taken into consideration, e.g. because a veil of ignorance applies.

In the paper we deal with the comparison of the Shapley–Shubik index and Banzhaf–Coleman index in games with a coalition structure. We analyze two possible approaches in both cases: we calculate voters’ power in a composite game or we apply the modification of original indices proposed by Owen for games with a priori unions. The behavior of both indices is compared basing on the voting game with 100 voters and different coalition structures. We analyze changes of power (measured by means of BC index and SS index) implied by changes of the size and composition of coalition structures as well as by different methodology of measuring the voters’ power (composite game versus game with a priori unions).

This paper sets out from a discussion of the well-known fact that the PGI violates the axiom of local monotonicity (LM). It argues that cases of nonmonotonicity indicate properties of the underlying decision situations which cannot be brought to light by the more popular power measures, i.e., the Banzhaf index and the Shapley–Shubik index, that satisfy LM. The discussion proposes that we can constrain the set of games such that LM also holds for the PGI. A discussion of causality follows. It suggests that the nonmonotonicity can be the result of framing the decision problem in a particular way and perhaps even ask the “wrong question.” Correspondingly, the PGI can be interpreted as an indicator. The probabilistic relationship of Banzhaf index and PGI identifies the factor which is responsible for the formal difference between the two measures and therefore for the violation of LM that characterizes the PGI, but not the Banzhaf.

Drawing on insights about the geometric structure of majority rule spatial voting games with Euclidean preferences derived from the Shapley–Owen value (Shapley and Owen, Int J Game Theory 18:339–356, 1989), we seek to explain why the outcomes of experimental committee majority rule spatial voting games are overwhelmingly located within the uncovered set (Bianco et al., J Polit 68:837–50, 2006; Polit Anal 16:115-37, 2008). We suggest that it is not membership in the uncovered set, per se, that leads to some alternatives being much more likely to become final outcomes of majority decision-making than others, but the fact that alternatives differ in the size of their winsets. We show how winset size for any alternative is a function of its squared distance from the point with minimal win set, and how this point, referred to by Shapley and Owen (Int J Game Theory 18:339–356, 1989) as the strong point, is determined as a weighted average of voter ideal points weighted by their Shapley–Owen values. We show that, in experimental voting games, alternatives with small winsets are more likely to be proposed, more likely to beat a status quo, and are more likely to be accepted as the final outcome than alternatives with larger winsets.

This paper discusses whether the equilibrium of a popular formal bargaining model due to (Baron & Ferejohn, 1989, American Political Science Review, 83, 1181–1206) satisfies the postulates that (Felsenthal & Machover, 1998, The measurement of voting power: theory and practice, problems and paradoxes. Cheltenham: Edward Elgar) consider essential in any power measure. It is well known that the equilibrium does not satisfy two adequacy postulates, namely ignoring dummies and vanishing only for dummies. This paper shows that it does not respect dominance either. It is also argued that the equilibrium displays one of the less intuitive instances of the paradox of new members.

We consider the case of three-candidate elections with large electorates under the assumption of the Dual Culture Condition. It is well known that a perfect relationship does not exist between the probability that Condorcet’s Paradox will be observed and a simple measure of the social homogeneity of voters’ preferences under these conditions. However, we show that this intuitively appealing relationship actually is found to hold on an expected value basis, except for an aberration in one very small region of possible voter situations, and this aberration is completely explainable. It is also found that the expected value of the Condorcet Efficiency of Borda Rule consistently increases as the simple measure of social homogeneity increases, except in the same small region of aberration for voting situations that is noted above.

Generically, the problem of apportionment may be defined as the problem of assigning a vector of integer numbers to each of a number of entitled entities that comes as close as possible to giving each entity its proportionate share of representation. Within that, there are a number of sub-problems. The correct solution (if a uniquely best solution exists) to one may not be the correct solution to another.