The Basic Concepts On Distribution of Decision Power Between The Players and Manipulation in Weighted Voting Games

  • Zdravko Dimitrov Slavov Varna Free University
Keywords: annex, split, merge, manipulation, decision power, weighted voting game


It is known that voting is a widely used method in social choice theory. In the present paper we consider some concepts of distribution of voting powers between the player and the process of manipulation in weighted voting games. The aim is to show some basic problems in social choice theory by studying the decision powers of players and the three processes of manipulation in weighted voting games: by merging of two players into a single player, by players splitting into a number of smaller units, and by annexation of a part or all of the voting weights of another player.


Download data is not yet available.


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

Barua R., S. Chakravarty, S. Roy, On the Coleman Indices of Voting Power, European Journal of Operational Research vol. 172 (2006), 273-289.

Barua R., S. Chakravatry, P. Sarkar, Minimal-axiom Characterizations of the Coleman and Banzhaf Indices of Voting Power, Mathematical Social Sciences vol. 58 (2009), 367-375.

Burgin M., L. Shapley, Enhanced Banzhaf Power Index and its Mathematical Properties, WP-797, Department of Mathematics, UCLA, 2001.

Coleman J., "Control of collectives and the power of a collectivity to act" in Social Choice, ed. by B. Lieberman, Gordon and Breach, New York, 1971, 269-298.

Conitzer V., T. Sandholm, J. Lang, When are elections with few candidates hard to manipulate? Journal of ACM vol. 54 no. 4, Article 14, 2007.

Dubey P., L. Shapley, Mathematical Properties of the Banzhaf Power Index, Mathematics of Operations Research vol. 4 no. 2 (1979), 99-151.

Gamson W., A Theory of Coalition Formation, American Sociological Review vol. 26 no. 3 (1961), 373-382.

Gibbard A., Manipulation of Voting Schemes, Econometrica vol. 41 no. 4 (1973), 587-601.

Gonzalez-Diaz J., I. Garcia-Jurado, M. Fiestras-Janeiro, An Introductory Course on Mathematical Game Theory, American Mathematical Society, 2010.

Houy N., W. Zwicker, The Geometry of Voting Power: Weighted Voting and Hyper-Ellipsoids, Games and Economic Behavior vol. 84 (2014), 7-16.

Knudsen P., L. Osterdal, Merging and Splitting in Cooperative Games: Some (I’m)possibility Results, International Journal of Game Theory vol. 41 (2012), 763-774.

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

Lippman D., Math in Society, Creative Commons BY-SA, 2012.

Masser N., Decision-Making in Committees: Game-Theoretic Analysis, Springer, 2010.

Peleg B., "Game-Theoretic Analysis of Voting in Committee" in Handbook of Social Choice and Welfare vol. 1 chap. 8, ed. by K. Arrow, A. Sen and K. Suzumura, Elsevier, 2002, pp. 396-423.

Satterthwaite M., Strategy-proofness and Arrow’s Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions, Journal of Economic Theory vol. 10 no. 2 (1975), 187-217.

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

Slavov Z., C. Evans, On the Voting Paradox of Luxembourg and Decision Power Indices, Mathematics and Education in Mathematics vol. 43 (2014), 138-144.

Slavov Z., C. Evans, Voting Games with Positive Weights and Dummy Players: Facts and Theory, Applied Mathematical Sciences vol. 10 no. 53 (2016), 2637-2646. doi:10.12988/ams.2016.67209

Slavov Z., C. Evans, Manipulation by Merging in Weighted Voting Games, Mathematics and Education in Mathematics vol. 46 (2017), 201-208.

Slavov Z., C. Evans, Manipulation by Merging and Annexation in Weighted Voting Games, Serdica Journal of Computing vol. 11 no. 1 (2017), BAS, 59-72.

Straffin P., "Power and stability in Politics" in Handbook of Game Theory with Economics, ed. by R. Aumann and S. Hart, vol. II, chap. 32, Elsevier, 1994, 1127-1151.

Taylor A., A. Pacelli, Mathematics and Politics: Strategy, Voting, Power and Proof. Stringer, 2008.

Von Neumann J., O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1944.

How to Cite
Slavov, Z. (2018). The Basic Concepts On Distribution of Decision Power Between The Players and Manipulation in Weighted Voting Games. MathLAB Journal, 1(3), 268-285. Retrieved from
Research Articles