Power indices expressed in terms of minimal winning coalitions

A voting situation is given by a set of voters and the rules of legislation that determine minimal requirements for a group of voters to pass a motion. A priori measures of voting power, such as the Shapley–Shubik index and the Banzhaf value, show the influence of the individual players in a voting situation and are calculated by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the legislative rules. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions and derive explicit formulae for the Shapley–Shubik and Banzhaf values. This new approach logically appealing as it writes measures as functions of the rules of the legislation. For certain classes of games that arise naturally in applications the logical shortcut drastically simplifies the numerical calculations to obtain the indices. The technique generalises directly to all semivalues.     

Voting power when using preference ballots

In this paper we provide a generalized power index which gives a measurement of voting power in multi-candidate elections with weighted voting using preference ballots. We use the power index to compare the power of various players between an election using plurality and one using the Borda method. The power index is based upon the Banzhaf power index.     

Simple plurality versus plurality runoff with privately informed voters

This paper compares two voting methods commonly used in presidential elections: simple plurality voting and plurality runoff. In a situation in which a group of voters have common interests but do not agree on which candidate to support due to private information, information aggregation requires them to split their support between their favorite candidates. However, if a group of voters split their support, they increase the probability that the winner of the election is not one of their favorite candidates. In a model with three candidates, due to this tension between information aggregation and the need for coordination, plurality runoff leads to higher expected utility for the majority than simple plurality voting if the information held by voters about the candidates is not very accurate. Received: 12 September 2000/Accepted: 8 November 2001     

A probabilistic synopsis of binary decision rules

Decision rules for Yes–No voting systems are placed in a probabilistic framework. Selfdual and permutationally invariant distributions are introduced. Under such distributions, the mean success margin of the majority rule and of the unanimity rule are shown to bound the mean success margin of all other decision rules. For bloc decision rules in the Penrose/Banzhaf model, a product formula for the voters’ influence probabilities is derived. Other indices and the Shapley/Shubik model are also discussed.     

Power indices and minimal winning coalitions

The Penrose–Banzhaf index and the Shapley–Shubik index are the best-known and the most used tools to measure political power of voters in simple voting games. Most methods to calculate these power indices are based on counting winning coalitions, in particular those coalitions a voter is decisive for. We present a new combinatorial formula how to calculate both indices solely using the set of minimal winning coalitions.     

Jury theorems with multiple alternatives

I consider a game in which imperfectly informed jurors vote to select one of several possible choices when there is a natural ordering of the possibilities. Each juror votes for the largest alternative the juror would like to implement, and the alternative that is selected is the largest alternative supported by a given number of jurors. For non-unanimous voting rules, the probability of a mistaken judgment goes to zero as the number of jurors goes to infinity. I also give necessary and sufficient conditions to obtain asymptotic efficiency under unanimous voting rules, and show that unanimous rules may lead to a bias in which moderate outcomes are never chosen.     

The probability of majority rule instability in the 2D euclidean model with an even number of voters

The classic instability theorems of Euclidean voting theory definitively treat all cases except that of an even number of voters in two dimensions. For that case, all that has been known is that the set of stable configurations is neither measure 0 nor measure 1. We prove that instability occurs with probability converging rapidly to 1 as the population increases.     

The inverse Banzhaf problem

Let ${\mathcal{F}}Let F{\mathcal{F}} be a family of subsets of the ground set [n] = {1, 2, . . . , n}. For each i ? [n]{i \in [n]} we let p(F,i){p(\mathcal{F},i)} be the number of pairs of subsets that differ in the element i and exactly one of them is in F{\mathcal{F}}. We interpret p(F,i){p(\mathcal{F},i)} as the influence of that element. The normalized Banzhaf vector of F{\mathcal{F}}, denoted B(F){B(\mathcal{F})}, is the vector (B(F,1),...,B(F,n)){(B(\mathcal{F},1),\dots,B(\mathcal{F},n))}, where B(F,i)=\fracp(F,i)p(F){B(\mathcal{F},i)=\frac{p(\mathcal{F},i)}{p(\mathcal{F})}} and p(F){p(\mathcal{F})} is the sum of all p(F,i){p(\mathcal{F},i)}. The Banzhaf vector has been studied in the context of measuring voting power in voting games as well as in Boolean circuit theory. In this paper we investigate which non-negative vectors of sum 1 can be closely approximated by Banzhaf vectors of simple voting games. In particular, we show that if a vector has most of its weight concentrated in k < n coordinates, then it must be essentially the Banzhaf vector of some simple voting game with n − k dummy voters.     

Characterizations of two power indices for voting games with r alternatives

In the first three sections of this paper we present a set of axioms which provide a characterization of an extension of the Banzhaf index to voting games with r alternatives, such as the United Nations Security Council where a nation can vote “yes”, “no”, or “abstain”. The fourth section presents a set of axioms which characterizes a power index based on winning sets instead of pivot sets. Received: 4 April 2000/Accepted: 30 April 2001     

Welfarist evaluations of decision rules for boards of representatives

We consider a decision board with representatives who vote on proposals on behalf of their constituencies. We look for decision rules that realize utilitarian and (welfarist) egalitarian ideals. We set up a simple model and obtain roughly the following results. If the interests of people from the same constituency are uncorrelated, then a weighted rule with square root weights does best in terms of both ideals. If there are perfect correlations, then the utilitarian ideal requires proportional weights, whereas the egalitarian ideal requires equal weights. We investigate correlations that are in between these extremes and provide analytic arguments to connect our results to Barberà and Jackson (J Polit Econ 114(2):317–339, 2006) and to Banzhaf voting power.     

Impartiality,friendship-networks and voting behavior: Evidence from voting patterns in the Eurovision Song Contest

What is the extent to which a country's political institutions impact aggregate voting behavior in a comparative perspective? More specifically, are citizens in some countries more inclined vote on the basis of ‘quality’ or ‘merit’ over ‘friendship’ or ‘loyalty’, and if so, why? This paper seeks to address how the extent to which a country's political institutions are impartial (treats all citizens equally, free from corruption, strong rule of law) impact aggregate citizen behavior. When political institutions are more (less) impartial, success in society is more often on the basis of merit (patrimonial ties). This test cases is voting in the Eurovision Song Contest (ESC) from 1975 to 2012 among pairs and blocs of ‘voting friends’. The theory elucidates that given that certain pairs or blocs exhibit systematic voting bias for one another over time, the bias will be considerably less among impartial states than those with highly partial institutions. Using several measures of ‘friendship’, I find strong empirical evidence for this claim, even when controlling for myriad alternative factors and taking into account various voting regimes. The analysis gives us new insights on how political institutions condition aggregate citizen behavior in general and that although there is much bias in ESC voting, not all bias is equal among friend-countries.     

Axiomatizations of the normalized Banzhaf value and the Shapley value

A cooperative game with transferable utilities– or simply a TU-game – describes a situation in which players can obtain certain payoffs by cooperation. A solution concept for these games is a function which assigns to every such a game a distribution of payoffs over the players in the game. Famous solution concepts for TU-games are the Shapley value and the Banzhaf value. Both solution concepts have been axiomatized in various ways. An important difference between these two solution concepts is the fact that the Shapley value always distributes the payoff that can be obtained by the `grand coalition' consisting of all players cooperating together while the Banzhaf value does not satisfy this property, i.e., the Banzhaf value is not efficient. In this paper we consider the normalized Banzhaf value which distributes the payoff that can be obtained by the `grand coalition' proportional to the Banzhaf values of the players. This value does not satisfy certain axioms underlying the Banzhaf value. In this paper we introduce some new axioms that characterize the normalized Banzhaf value. We also provide an axiomatization of the Shapley value using similar axioms. Received: 10 April 1996 / Accepted: 2 June 1997     

LITTLE WHITE LIES AND SOCIAL SCIENCE MODELS CORRELATED RESPONSE ERRORS IN A PANEL STUDY OF VOTING

One interpretation for the common survey finding that the backgroundcharacteristics of vote overreporters resemble those of actualvoters is that misreporters usually vote. This hypothesis—thatmisreporters regularly voted in earlier elections—is testedwith data from the 1972–74–76 Michigan ElectionPanel. It receives no support: the 1972 and 1974 validated turnoutof the 1976 misreporters was very low. Moreover, misreportingwas a fairly stable respondent characteristic: misreportingabout an election in one interview was correlated with misreportingabout the remaining elections in each of the other two interviews.A comparison of regressions predicting turnout using the validatedreports versus the self-reports shows that the respondent errorscan distort conclusions about the correlates of voting. Forexample, controlling for three other variables, education wasrelated to self-reported voting but not to validated voting.Here, as well as in surveys of other socially desirable or undesirableissues, respondent self-reports may bias survey data in favorof commonsense models of the world.     

Voting rules as statistical estimators

We adopt an ‘epistemic’ interpretation of social decisions: there is an objectively correct choice, each voter receives a ‘noisy signal’ of the correct choice, and the social objective is to aggregate these signals to make the best possible guess about the correct choice. One epistemic method is to fix a probability model and compute the maximum likelihood estimator (MLE), maximum a posteriori (MAP) estimator or expected utility maximizer (EUM), given the data provided by the voters. We first show that an abstract voting rule can be interpreted as MLE or MAP if and only if it is a scoring rule. We then specialize to the case of distance-based voting rules, in particular, the use of the median rule in judgement aggregation. Finally, we show how several common ‘quasiutilitarian’ voting rules can be interpreted as EUM.     

Costly voting when both information and preferences differ: is turnout too high or too low?

We study a model of costly voting over two alternatives, where agents’ preferences are determined by both (i) a private preference in favour of one alternative e.g. candidates’ policies, and (ii) heterogeneous information in the form of noisy signals about a commonly valued state of the world e.g. candidate competence. We show that depending on the level of the personal bias (weight on private preference), voting is either according to private preferences or according to signals. When voting takes place according to private preferences, there is an unique equilibrium with inefficiently high turnout. In contrast, when voting takes place according to signals, turnout is locally too low. Multiple Pareto-ranked voting equilibria may exist and in particular, compulsory voting may Pareto dominate voluntary voting. Moreover, an increase in personal bias can cause turnout to rise or fall, and an increase in the accuracy of information may cause a switch to voting on the basis of signals and thus lower turnout, even though it increases welfare. This is a substantially revised version of Department of Economics University of Warwick Working Paper 670, “Information Aggregation, Costly Voting and Common Values”, January 2003. We would like to thank B. Dutta, M. Morelli, C. Perrroni, V. Bhaskar and seminar participants at Warwick, Nottingham and the ESRC Workshop in Game Theory for their comments. We would also like to thank the editor and an anonymous referee for their comments.     

Strategy versus sincerity in mean voting

This paper focuses on the mean vote procedure for choosing the quantity of a public good, a social choice rule that selects the mean of the quantities voted for. The available theoretical models of mean voting give rise to conflicting predictions about the extent of the strategic bias that might arise in the individual vote. An experiment has been run in order to assess whether the participants disclose their (induced) preferences in voting or whether they strategically manipulate their vote, and in the latter case, which variables impact upon the likelihood of strategic bias and to what extent.     

Policy convergence in a two-candidate probabilistic voting model

We propose a generalization of the probabilistic voting model in two-candidate elections. We allow the candidates have general von Neumann–Morgenstern utility functions defined over the voting outcomes. We show that the candidates will choose identical policy positions only if the electoral competition game is constant-sum, such as when both candidates are probability-of-win maximizers or vote share maximizers, or for a small set of functions that for each voter define the probability of voting for each candidate, given candidate policy positions. At the same time, a pure-strategy local Nash equilibrium (in which the candidates do not necessarily choose identical positions) exists for a large set of such functions. Hence, if the candidate payoffs are unrestricted, the “mean voter theorem” for probabilistic voting models is shown to hold only for a small set of probability of vote functions.     

Condorcet efficiencies under the maximal culture condition

The Condorcet winner in an election is a candidate that could defeat each other candidate in a series of pairwise majority rule elections. The Condorcet efficiency of a voting rule is the conditional probability that the voting rule will elect the Condorcet winner, given that such a winner exists. The study considers the Condorcet efficiency of basic voting rules under various assumptions about how voter preference rankings are obtained. Particular attention is given to situations in which the maximal culture condition is used as a basis for obtaining voter preferences. Received: 4 February 1998/Accepted: 13 April 1998     

Exploiting polyhedral symmetries in social choice

A large amount of literature in social choice theory deals with quantifying the probability of certain election outcomes. One way of computing the probability of a specific voting situation under the Impartial Anonymous Culture assumption is via counting integral points in polyhedra. Here, Ehrhart theory can help, but unfortunately the dimension and complexity of the involved polyhedra grows rapidly with the number of candidates. However, if we exploit available polyhedral symmetries, some computations become possible that previously were infeasible. We show this in three well known examples: Condorcet’s paradox, Condorcet efficiency of plurality voting and in Plurality voting vs Plurality Runoff.     

The Efron dice voting system

About 50?years ago, Efron noted some counterintuitive properties of the long-term behavior of contests involving dice. For instance, consider the 6-sided dice whose sides are labeled (4,4,4,4,0,0), (3,3,3,3,3,3), (6,6,2,2,2,2), and (5,5,5,1,1,1). Each die has a 2/3 probability of rolling a higher number than the next one in the list and the last has the same 2/3 probability of rolling a higher number than the first. The non-transitivity of games involving non-identical dice was popularized by Gardner (Sci Am, 223:110–114, 1970). Although Gardner and other authors have observed that non-transitive dice serve to illustrate the complexities of the theory of voting, it does not seem that much attention has been paid to the corresponding voting system. Our purpose in this article is to present this voting system and compare its properties with those of other voting systems. One of the most interesting properties is the fact that cancellation with respect to the Efron dice voting system can replace cancellation with respect to pairwise preferences in Young’s characterization of the social choice function associated with the Borda Count.