Bhattacherjee, Sanjay and Sarkar, Palash (2017): Correlation and inequality in weighted majority voting games.
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Abstract
In a weighted majority voting game, the weights of the players are determined based on some socioeconomic parameter. A number of measures have been proposed to measure the voting powers of the different players. A basic question in this area is to what extent does the variation in the voting powers reflect the variation in the weights? The voting powers depend on the winning threshold. So, a second question is what is the appropriate value of the winning threshold? In this work, we propose two simple ideas to address these and related questions in a quantifiable manner. The first idea is to use Pearson's Correlation Coefficient between the weight vector and the power profile to measure the similarity between weight and power. The second idea is to use standard inequality measures to quantify the inequality in the weight vector as well as in the power profile. These two ideas answer the first question. Both the weightpower similarity and inequality scores of voting power profiles depend on the value of the winning threshold. For situations of practical interest, it turns out that it is possible to choose a value of the winning threshold which maximises the similarity score and also minimises the difference in the inequality scores of the weight vector and the power profile. This provides an answer to the second question. Using the above formalisation, we are able to quantitatively argue that it is sufficient to consider only the vector of swings for the players as the power measure. We apply our methodology to the voting games arising in the decision making processes of the International Monetory Fund (IMF) and the European Union (EU). In the case of IMF, we provide quantitative evidence that the actual winning threshold that is currently used is suboptimal and instead propose a winning threshold which has a firm analytical backing. On the other hand, in the case of EU, we provide quantitative evidence that the presently used threshold is very close to the optimal.
Item Type:  MPRA Paper 

Original Title:  Correlation and inequality in weighted majority voting games 
Language:  English 
Keywords:  Voting games, weighted majority, power measure, correlation, inequality, Gini index, coefficient of variation. 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C15  Statistical Simulation Methods: General D  Microeconomics > D7  Analysis of Collective DecisionMaking > D72  Political Processes: RentSeeking, Lobbying, Elections, Legislatures, and Voting Behavior D  Microeconomics > D7  Analysis of Collective DecisionMaking > D78  Positive Analysis of Policy Formulation and Implementation Y  Miscellaneous Categories > Y1  Data: Tables and Charts 
Item ID:  86363 
Depositing User:  Dr. Sanjay Bhattacherjee 
Date Deposited:  25 Apr 2018 06:49 
Last Modified:  28 Sep 2019 19:15 
References:  Banzhaf, J. F. (1965). Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review, 19:317–343. Barua, R., Chakravarty, S. R., and Sarkar, P. (2009). Minimalaxiom characterizations of the coleman and banzhaf indices of voting power. Mathematical Social Sciences, 58:367–375. Brink, R. and Laan, G. (1998). Axiomatizations of the normalized banzhaf value and the shapley value. Social Choice and Welfare, 15(4):567–582. Chakravarty, S. R. (2017). Analyzing Multidimensional WellBeing: A Quantitative Approach. John Wiley, New Jersey. in press. Chakravarty, S. R. and Lugo, M. A. (2016). Multidimensional indicators of inequality and poverty. In Adler, M. D. and Fleurbaey, M., editors, Oxford Handbook of WellBeing and Public Policy, pages 246–285. Oxford University Press, New York. Chakravarty, S. R., Mitra, M., and Sarkar, P. (2015). A Course on Cooperative Game Theory. Cambridge University Press. Coleman, J. S. (1971). Control of collectives and the power of a collectivity to act. In Lieberma, B., editor, Social Choice, pages 269–298. Gordon and Breach, New York. Cowell, F. A. (2016). Inequality and poverty measures. In Adler, M. D. and Fleurbaey, M., editors, Oxford Handbook of WellBeing and Public Policy, pages 82–125. Oxford University Press, New York. Deegan, J. and Packel, E. W. (1978). A new index of power for simple nperson games. International Journal of Game Theory, 7(2):113–123. Dubey, P. and Shapley, L. S. (1979). Mathematical properties of the banzhaf power index. Mathematics of Operations Research, 4(2):99–131. Einy, E. and Peleg, B. (1991). Linear measures of inequality for cooperative games. Journal of Economic Theory, 53(2):328–344. Felsenthal, D. S. and Machover, M. (1998). The Measurement of Voting Power. Edward Elgar, Chel tenham. Holler, M. J. (1982). Forming coalitions and measuring voting power. Political Studies, 30(2):262–271. Holler, M. J. and Packel, E. W. (1983). Power, luck and the right index. Journal of Economics, 43(1):21–29. Laruelle, A. and Valenciano, F. (2001). Shapleyshubik and banzhaf indices revisited. Mathematics of Operations Research, 26(1):89–104. Laruelle, A. and Valenciano, F. (2004). Inequality in voting power. Social Choice and Welfare, 22(2):413– 431. Laruelle, A. and Valenciano, F. (2011). Voting and collective decisionmaking. Cambridge University Press, Cambridge. Leech, D. (2002a). Designing the voting system for the council of the european union. Public Choice, 113:437–464. Leech, D. (2002b). Power in the governance of the international monetary fund. Annals of Operations Research, 109(14):375––397. Lehrer, E. (1998). An axiomatization of the banzhaf value. International Journal of Game Theory, 17(2):89–99. Matsui, T. and Matsui, Y. (2000). A survey of algorithms for calculating power indices of weighted voting games. Journal of Operations Research Society of Japan, 43:71–86. Shapley, L. S. (1953). A value for nperson games. In Kuhn, H. W. and Tucker, A. W., editors, Contributions to the Theory of Games II (Annals of Mathematics Studies), pages 307–317. Princeton University Press. Shapley, L. S. and Shubik, M. J. (1954). A method for evaluating the distribution of power in a committee system. American Political Science Review, 48:787–792. Shorrocks, A. F. (1980). The class of additively decomposable inequality measures. Econometrica, 48(3):613–625. Weber, M. (2016). Twotier voting: Measuring inequality and specifying the inverse power problem. Mathematical Social Sciences, 79:40–45. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/86363 
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Correlation and inequality in weighted majority voting games. (deposited 02 Jan 2018 12:55)
 Correlation and inequality in weighted majority voting games. (deposited 25 Apr 2018 06:49) [Currently Displayed]