Voting power in a system of compound majority rule

In 1973, the United States Supreme Court ruled that a water district's voting scheme that apportioned votes on the basis of the assessed valuation of acreage in the district was constitutional. Among the justifications for the constitutionality of this scheme was the concurrent requirement that legislation be approved by a majority of voters as well as by a majority of weighted votes. However, analysis of this voting scheme in game-theoretic terms indicates that this justification is only partial: when two sets of winning coalitions must form simultaneously in order to pass legislation, the voting power of each voter in the combined system equals the mean of the voting power afforded each voter in each simple system. The results can be generalized to three or more concurrent requirements.     

Minimal-axiom characterizations of the Coleman and Banzhaf indices of voting power

Coleman [Coleman, J.S., 1971. Control of collectives and the power of a collectivity to act. In: Lieberman, B. (Ed.), Social Choice. Gordon and Breach, New York, pp. 269–298] suggested two indices of voting power, power to prevent an action and power to initiate an action. This paper develops two-axiom (and hence, minimal) characterizations of both the indices. We also show that similar characterizations can be done for other indices.     

Spatial power indices with applications on real voting data from the Chamber of Deputies of the Czech Parliament

Participants in parliamentary voting—usually political parties—are evaluated with a value that assigns them predicted power with respect to their strength by so called power indices. However, in real world, political parties’ representatives act not strictly as predicted in theory. One possibility how the representatives differ from theory is the way they form coalitions; the coalitions can be announced (for example official governmental coalition parties in multiparty parliaments) or hidden. To incorporate the coalition forming influence, Bilal et al. (http://aei.pitt.edu/2052/1/001591_1.pdf, 2001) proposed to consider additional weights to possible coalitions into power indices. This article applies the concept of additional weights to calculate an ex post power distribution using Shapley–Shubik power index together with Banzhaf power index on real voting data, namely the data from the Chamber of Deputies of the Czech Parliament with the emphasis on the State Budget voting issues during 2006–2010 parliamentary period.     

The exact lower bound for the Coleman index of the power of a collectivity for a special class of simple majority games

Coleman [Coleman, J.S., 1971. Control of collectivities and the power of a collectivity to act. In: Leberman, B. (Ed.), Social Choice. Gordon and Breach, New York] proposed a measure of the power that a decision-making body has to pass any bill that comes before it. He termed it “the power of a collectivity to act”. In this paper we provide some numerical bounds on the values that this measure can take when the collectivity takes decisions under simple majority voting rule.     

Heuristic and exact solutions to the inverse power index problem for small voting bodies

Power indices are mappings that quantify the influence of the members of a voting body on collective decisions a priori. Their nonlinearity and discontinuity makes it difficult to compute inverse images, i.e., to determine a voting system which induces a power distribution as close as possible to a desired one. The paper considers approximations to this inverse problem for the Penrose-Banzhaf index by hill-climbing algorithms and exact solutions which are obtained by enumeration and integer linear programming techniques. They are compared to the results of three simple solution heuristics. The heuristics perform well in absolute terms but can be improved upon very considerably in relative terms. The findings complement known asymptotic results for large voting bodies and may improve termination criteria for local search algorithms.     

The Banzhaf power index for ternary bicooperative games

In this paper we analyze ternary bicooperative games, which are a refinement of the concept of a ternary voting game introduced by Felsenthal and Machover. Furthermore, majority voting rules based on the difference of votes are simple bicooperative games. First, we define the concepts of the defender and detractor swings for a player. Next, we introduce the Banzhaf power index and the normalized Banzhaf power index. The main result of the paper is an axiomatization of the Banzhaf power index for the class of ternary bicooperative games. Moreover, we study ternary bicooperative games with two lists of weights and compute the Banzhaf power index using generating functions.     

On Some Properties of the Hoede-Bakker Index

In this article, we analyze the decisional power index, the so called Hoede-Bakker index. This index takes the preferences of the players into account, as well as the social structure in which players may influence each other. We investigate the properties of the Hoede-Bakker index. We adapt some postulates for power indices and voting power paradoxes for the Hoede-Bakker set up. We investigate whether this decisional power index displays the adapted voting power paradoxes and whether it satisfies the adapted postulates for power indices.     

Voting power in the European Union enlargement

The Shapley–Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. The Banzhaf power index depends on the number of ways in which each voter can effect a swing. If there are n players in a voting situation, then the function which measures the worst case running time for computing these indices is in O(n2ⁿ). We present a combinatorial method based in generating functions to compute these power indices efficiently in weighted double or triple majority games and we study the time complexity of the algorithms. Moreover, we calculate these power indices for the countries in the Council of Ministers of the European Union under the new decision rules prescribed by the Treaty of Nice.     

Minimal-axiom characterizations of the Coleman and Banzhaf indices of voting power

Coleman [Coleman, J.S., 1971. Control of collectives and the power of a collectivity to act. In: Lieberman, B. (Ed.), Social Choice. Gordon and Breach, New York, pp. 269–298] suggested two indices of voting power, power to prevent an action and power to initiate an action. This paper develops two-axiom (and hence, minimal) characterizations of both the indices. We also show that similar characterizations can be done for other indices.     

The minimum sum representation as an index of voting power

We propose a new power index based on the minimum sum representation (MSR) of a weighted voting game. The MSR offers a redesign of a voting game, such that voting power as measured by the MSR index becomes proportional to voting weight. The MSR index is a coherent measure of power that is ordinally equivalent to the Banzhaf, Shapley–Shubik and Johnston indices. We provide a characterization for a bicameral meet as a weighted game or a complete game, and show that the MSR index is immune to the bicameral meet paradox. We discuss the computation of the MSR index using a linear integer program and the inverse MSR problem of designing a weighted voting game with a given distribution of power.     

Computing power indices: Multilinear extensions and new characterizations

Johnston [Johnston, R.J., 1978. On the measurement of power: some reactions to Laver. Environment and Planning A 10, 907–914], Deegan and Packel [Deegan, J., Packel, E.W., 1979. A new index of power for simple n-person games. International Journal of Game Theory 7, 113–123], and Holler [Holler, M.J., 1982. Forming coalitions and measuring voting power. Political Studies 30, 262–271] proposed three power indices for simple games: Johnston index, Deegan–Packel index, and the Public Good Index. In this paper, methods to compute these indices by means of the multilinear extension of the game are presented. Furthermore, a new characterization of the Public Good Index is given. Our methods are applied to two real-world examples taken from the political field.     

Generating functions for computing power indices efficiently

TheShapley-Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. TheBanzhaf power index depends on the number of ways in which each voter can effect a swing. We introduce a combinatorial method based ingenerating functions for computing these power indices efficiently and we study thetime complexity of the algorithms. We also analyze the meet of two weighted voting games. Finally, we compute the voting power in the Council of Ministers of the European Union with the generating functions algorithms and we present its implementation in the system Mathematica. This work has been partially supported by the Spanish Ministery of Science and Technology under grant SEC2000-1243.     

The distribution of power in the European Constitution

The aim of this paper is to analyze the distribution of voting power in the Constitution for the enlarged European Union. By using generating functions, we calculate the Banzhaf power indices for the European countries in the Council of Ministers under the decision rules prescribed by the Treaty of Nice and the new rules proposed by the European Constitution Treaty. Moreover, we analyze the power of the European citizens under the egalitarian model proposed by Felsenthal and Machover [D.S. Felsenthal, M. Machover, The measurement of voting power: Theory and practice, problems and paradoxes, Edward Elgar, Cheltenham, 1998].     

联合国安理会改革的势力分析

建立了联合国安理会加权投票系统模型,给出了联合国安理会加权投票系统的Shapley势力指数和Banzhaf势力指数,计算了联合国安理会各种改革方案下各理事国的势力指数。最后对联合国安理会各种改革方案下各理事国的势力指数的变化情况进行了比较分析。     

A simple “market value” bargaining model for weighted voting games: characterization and limit theorems

We offer a bargaining model for weighted voting games that is a close relative of the nucleolus and the kernel. We look for a set of weights that preserves winning coalitions that has the property of minimizing the difference between the weight of the smallest and the weight of the largest Minimum Winning Coalition. We claim that such a set of weights provides an a priori measure of a weighted voter’s bribeworthiness or market value. After introducing our model, we provide a characterization result for this model and show its links to other bargaining model approaches in the literature. Then we offer some limit results showing that, with certain reasonable conditions on the distributions of weights, as the size of the voting body increases, the values of bribeworthiness we calculate will approach both the weights themselves and the Banzhaf scores for the weighted voting game. We also show that, even for relatively small groups using weighted voting, such as the membership of the European Council of Ministers (and its predecessors) 1958–2003, similarities among the usual a priori power scores, bribeworthiness/market value, and the weights themselves, will be quite strong.     

The Banzhaf power index for political games

The Banzhaf index of a voting game is a measure of a priori power of the voters. The model on which the index is based treats the voters symmetrically, i.e. the ideology, outlook, etc., of the voters influencing their voting behavior is ignored. Here we present a nonsymmetric generalization of the Banzhaf index in which the ideology of the voters affecting their voting behavior is taken into account. A model of ideologies and issues is presented. The conditions under which our model gives the Shapley-Shubik index (another index of a priori power of the voters) are given. Finally several examples are presented and some qualitative results are given for straight majority and pure bargaining games.     

Asymptotic probability of majority inversion under a general binomial voting model

We obtain limits on the probability of majority inversion when the number of voters tends to infinity, for a binomial voting model specific to each state with different population sizes, and for different voting quotas in the two stages of the voting procedure. The case of weighted votes at the second stage is also discussed. For an important special case where the limit cannot be determined, we provide an exact expression for the inversion probability, but only for unweighted votes.     

On the inverse power index problem

Weighted voting games are frequently used in decision making. Each voter has a weight and a proposal is accepted if the weight sum of the supporting voters exceeds a quota. One line of research is the efficient computation of so-called power indices measuring the influence of a voter. We treat the inverse problem: Given an influence vector and a power index, determine a weighted voting game such that the distribution of influence among the voters is as close as possible to the given target value. We present exact algorithms and computational results for the Shapley–Shubik and the (normalized) Banzhaf power index.     

约束式赋权与基于指标得分的权重调整方法研究

在评价中经常会出现评价者对某些关键评价指标有特定要求或评价结果的预期,数学分析表明由这些要求或预期提出的约束蕴含着该指标的临界权重,且可以通过最优化方法求解临界权重并据此进行约束式赋权.在评价具体对象时,评价指标得分的分布情况会影响评价结果,由此提出根据全部评价指标的标准偏差和最低得分指标的离差对权重进行调整的方法.     

Evaluating correct classification probability for weighted voting classifiers with plurality voting

Weighted voting classifiers (WVCs) consist of N units that each provide individual classification decisions. The entire system output is based on tallying the weighted votes for each decision and choosing the winning one (plurality voting) or one which has the total weight of supporting votes greater than some specified threshold (threshold voting). Each individual unit may abstain from voting. The entire system may also abstain from voting if no decision is ultimately winning. Existing methods of evaluating the correct classification probability (CCP) of WVCs can be applied to limited special cases of these systems (threshold voting) and impose some restrictions on their parameters. In this paper a method is suggested which allows the CCP of WVCs with both plurality and threshold voting to be exactly evaluated without imposing constraints on unit weights. The method is based on using the modified universal generating function technique.