Advertisement

Theory and Decision

, Volume 81, Issue 2, pp 213–253 | Cite as

Counting votes in coupled decisions

An efficient method for counting votes in coupled decisions with multiple inequality restrictions
  • Andreas Wendemuth
  • Italo Simonelli
Article
  • 111 Downloads

Abstract

We consider scenarios with distributed decision processes, e.g., coupled majorities and personal union in parliament chambers, supranational decisions and supervisory boards. When computing the adoption rate for reaching a decision in these scenarios, multiple linear inequality restrictions in combinatorial countings are present. These rates cannot be computed in closed form. We introduce a general method for incorporating multiple inequality conditions in multiple majority decisions, which significantly reduces the number of involved summations and removes restrictions on the summation indices. Exact solutions are provided through (a) integral representations which can be evaluated numerically, and (b) unrestricted, contracted sums over discrete events. Further, we provide methods to reduce the number of necessary summations by splitting or recurring the original problem to easier sub-problems. For five dedicated scenarios, full results are given which indeed require a single unrestricted summation only.

Keywords

Coupled decisions Double counting methods Multiple restricted majorities Personal union Voting game  Social choice 

Notes

Acknowledgments

One of us (A. Wendemuth) acknowledges continued support by the Transregional Collaborative Research Centre SFB/TRR 62 “Companion-Technology for Cognitive Technical Systems” (www.sfb-trr-62.de) funded by the German Research Foundation (DFG).

References

  1. Abramowitz, M., & Stegun, C. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; “Beta Function” §6.2. New York: Dover.Google Scholar
  2. Aigner, M., Ziegler, A., & Guenter, M. (1998). Proofs from THE BOOK. New York: Springer.CrossRefGoogle Scholar
  3. Australian Constitution Act—Sect 128 (1977). http://www.austlii.edu.au/au/legis/cth/consol_act/coaca430/s128.html.
  4. Banzhaf, J. F, III. (1965). Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review, 19(2), 317343.Google Scholar
  5. Brams, S. J. (2013). Game Theory and Politics. Mineola: Courier.Google Scholar
  6. Brandenburger, A. (2007). Cooperative Game Theory. New York University. http://www.uib.cat/depart/deeweb/pdi/lbm/arxius_decisions_and_games/cooperative_game_theory-brandenburger.
  7. Canada Constitution Act—Part V, 38 (1982). www.legislation.gov.uk/ukpga/1982/11.
  8. Coleman, S. (1971). Control of collectives and the power of a collectivity to act. In Bernhardt Lieberman (Ed.), Social Choice (Vol. 135, pp. 192–225). New York: Gordon and Breach.Google Scholar
  9. Deegan, J., & Packel, E. W. (1978). A new index of power for simple n-person games. International Journal of Game Theory, 7(2), 113123.CrossRefGoogle Scholar
  10. German Condominium Act (2014). (Bundesrepublik Deutschland—Wohnungseigentumsgesetz) §22 Abs. 2 https://dejure.org/gesetze/WEG.
  11. Gould, H. W. (2010). Formula (5.1). In J. Quaintance (Ed.), Manuscript Notebooks of H. W. Gould (1945-1990), vol 6. (pp. 20) http://www.math.wvu.edu/~gould/Vol.6.PDF.
  12. Graham, R., Knuth, D., & Patashnik, O. (1989). Concrete Mathematics. Reading: Addison-Wesley.Google Scholar
  13. Hodge, J. K., & Klima, R. E. (2005). The Mathematics of Voting and Elections: A Hands-on Approach. Mathematical World, vol. 22. Providence: American Mathematical Society.CrossRefGoogle Scholar
  14. Holler, M. J., & Packel, E. W. (1983). Power, luck and the right index. Journal of Economics, 43(1), 2129.CrossRefGoogle Scholar
  15. Houy, N. (2009). A characterization of majority voting rules with quorums. Theory and Decision, 67(3), 295–301.CrossRefGoogle Scholar
  16. Lex Abs (1965). Aktiengesetz vom 6. September 1965. In Bundesgesetzblatt (Deutschland) I, vol 48 (p. 1089). Deutschland: Bundesministerium der Justiz.Google Scholar
  17. Lisbon Treaty (2007). Treaty of Lisbon amending the Treaty on European Union and the Treaty establishing the European Community, signed at Lisbon, 13 December 2007. In Official Journal of the European Union, vol. 50 (p. C306/1). Publications Office of the European Union.Google Scholar
  18. Penrose, L. S. (1946). The elementary statistics of majority voting. Journal of the Royal Statistical Society, 109(1), 5357.CrossRefGoogle Scholar
  19. Petkovs̆ek, M., Wilf, H., & Zeilberger, D. (1996). A=B. Wellesley: Peters.Google Scholar
  20. Petkovs̆ek, M., & Wilf, H. (1997). When can the sum \((1/p)\)th of the binomial coefficients have closed form? The electronic journal of combinatorics, 42, 1–7.Google Scholar
  21. Shapley, L. S., & Shubik, M. (1954). A method for evaluating the distribution of power in a committee system. American Political Science Review, 48(3), 787–792.CrossRefGoogle Scholar
  22. Swiss Constitution (1999). Federal Constitution of the Swiss Confederation - Art. 142, http://www.admin.ch/ch/d/sr/c101.html.
  23. Wendemuth, A., & Simonelli, I. (2013). Majority decisions in overlapping committees and asymptotic size of dichotomies. SIAM Journal on Discrete Mathematics, 26(4), 1537–1549.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Cognitive Systems Group, Faculty of Electrical Engineering and Information TechnologyOtto-von-Guericke University MagdeburgMagdeburgGermany
  2. 2.Department of Mathematics and Computer ScienceMcDaniel CollegeWestminsterUSA

Personalised recommendations