A General Model for Multi-Parameter Weighted Voting Games

Bhattacherjee, Sanjay and Chakravarty, Satya R. and Sarkar, Palash (2022): A General Model for Multi-Parameter Weighted Voting Games.

Abstract

This article introduces a general model for voting games with multiple weight vectors. Each weight vector characterises a parameter representing a distinct aspect of the voting mechanism. Our main innovation is to model the winning condition by an arbitrary dichotomous function which determines whether a coalition is winning based on the weights that the coalition has for the different parameters. Previously studied multi-parameter games are obtained as particular cases of the general model. We identify a new and interesting class of games, that we call hyperplane voting games, which are compactly expressible in the new model, but not necessarily so in the previous models. For the general model of voting games that we introduce, we describe dynamic programming algorithms for determining various quantities required for computing different voting power indices. Specialising to the known classes of multi-parameter games, our algorithms provide unified and simpler alternatives to previously proposed algorithms.

• Item Type: MPRA Paper
• Language: English
• Keywords: weighted majority voting game, multi-parameter games, Boolean formula, voting power, dynamic programming
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C71 - Cooperative Games

References

Algaba, E. J. M., Bilbao, J.M., Garcı́a, F. and López, J. J.: Computing power indices in weighted multiple majority games. Mathematical Social Sciences 46, 63–80 (2003).

Algaba, E., Bilbao, J.M. and Fernández, J.: The distribution of power in the European constitution. European Journal of Operational Research 176 (3), 1752–1766 (2007).

Alonso-Meijide, J. M., Bilbao, J., Casas-Méndez, B. and Fernández, J.: Weighted multiple majority games with unions: generating functions and applications to the European Union. European Journal of Operational Research 198 (2), 530–544 (2009).

Banzhaf, J. F.: Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review 19, 317-343 (1965).

Bhattacherjee, S. and Sarkar, P.: Correlation and inequality in weighted majority voting games, in Dasgupta, I, and Mitra, M.(eds.) Deprivation, Inequality and Polarization: Essays in Honor of S.R. Chakravarty, pp. 161-191. Springer, New York (2019).

Bolus, S.: Power Indices of simple games and vector-weighted majority games by means of binary decision diagrams. European Journal of Operational Research 210 (2), 258–272 (2011).

Chakravarty, S.R., Mitra, M. and Sarkar, P.: A Course on Cooperative Game Theory. Cambridge University Press, Cambridge (2015).

Coleman, J.S.: Control of collectives and the power of a collectivity to act, in: Lieberman, B. (eds.) Social Choice, pp. 269-298. Gordon and Breach, New York (1971).

Deı̆neko, V. G. and Woeginger, G. J.: On the dimension of simple monotonic games. European Journal of Operational Research 170, 315-318 (2006).

Einy E. and Peleg B.: Linear measures of inequality for cooperative games. Journal of Economic Theory 53, 328-344 (1991).

Elkind, E., Goldberg, L. A., Goldberg, P. W. and Wooldridge, M. J.: On the computational complexity of weighted voting games. Annals of Mathematics and Artificial Intelligence 56, 109-131 (2009).

Faliszewski, P., Elkind E. and Wooldridge M.: Boolean Combinations of Weighted Voting Games. In Proceedings of 8th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), editors Sichmann, S. and Castelfranchi, 185–192 (2009).

Felsenthal, D.S. and Machover, M.: The Measurement of Voting Power. Edward Elgar (1998).

Holler, M.J.: Forming coalitions and measuring voting power. Political Studies 30, 262-271 (1982).

Johnston, R. J.: On the measurement of power: some reactions to Laver. Environment and Planning A 10, 907-914 (1978).

Kurz, S.: Computing the Power Distribution in the IMF. Arxiv, https://doi.org/10.48550/arXiv. 1603.01443, (2016).

Kurz, S. and Napel, S.: Dimension of the Lisbon voting rules in the EU Council: a challenge and new world record. Optimization Letters 10, 1245-1256 (2016).

Laruelle A. and Valenciano F.: Inequality in voting power. Social Choice and Welfare 22, 413-431 (2004).

Laruelle, A. and Valenciano, F.: Voting and Collective Decision-Making: Bargaining and Power. Cambridge University Press, Cambridge (2008).

Leech, D.: Voting power in the governance of the International Monetary Fund. Annals of Operations Research 109, 375-397 (2002a).

Leech, D.: Designing the voting system for the council of the European Union. Public Choice 113, 437-464 (2002b).

Maschler, M. and Owen, G.: The consistent Shapley value for hyperplane games. International Journal of Game Theory 18, 389–407 (1989).

Matsui, T. and Matsui, Y.: A survey of algorithms for calculating power indices for weighted majority games. Journal of the Operations Research Society of Japan 43, 71-86 (2000).

Penrose, L.S.: The elementary statistics of majority voting. Journal of the Royal Statistical Society 109(1), 53–57 (1946).

Rae, D.: Decision rules and individual values in constitutional choice. American Political Science Review 63, 40-56 (1969).

Shapley, L .S. and Shubik, M, J.: A method for evaluating the distribution of power in a committee system. American Political Science Review 48, 787-792 (1954).

Strand, J.R., Rapkin, D.P, and Wood, D.B.: The political economy of a priori voting power in the World Bank and IMF. Mimeo (1997).

Taylor, A. D. and W. S. Zwicker: Weighted voting, multicameral representation, and power. Games and Economic Behavior 5, 170-181 (1993).

Taylor, A. D. and W. S. Zwicker: Simple games and magic squares. Journal of Combinatorial Theory, Series A, 71, 67-88 (1995).

Taylor, A. D. and W. S. Zwicker: Simple Games: Desirability Relations, Trading, and Pseudoweightings. Princeton University Press, New Jersey, USA (1999).

Uno, T.: Efficient Computation of Power Indices for Weighted Majority Games. In: Algorithms and Computation, editors Chao, K.-M., Sheng Hsu, T. and Lee, D.-T., Springer, 679–689, (2012).

Weber, M.: Two-tier voting: measuring inequality and specifying the inverse power problem. Mathematical Social Sciences 79, 40–45 (2016).

Wilms, I.: Dynamic programming algorithms for computing power indices in weighted multi-tier games. Mathematical Social Sciences 108, 175–192 (2020).

Yu, C.: Hyperplane games, prize games and NTU values. Theory and Decision 93, 359–370 (2022).