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Journal of Global Optimization

, Volume 56, Issue 2, pp 737–756 | Cite as

Modeling optimal social choice: matrix-vector representation of various solution concepts based on majority rule

  • Fuad Aleskerov
  • Andrey SubochevEmail author
Article

Abstract

Various Condorcet consistent social choice functions based on majority rule (tournament solutions) are considered in the general case, when ties are allowed: the core, the weak and strong top cycle sets, versions of the uncovered and minimal weakly stable sets, the uncaptured set, the untrapped set, classes of k-stable alternatives and k-stable sets. The main focus of the paper is to construct a unified matrix-vector representation of a tournament solution in order to get a convenient algorithm for its calculation. New versions of some solutions are also proposed.

Keywords

Solution concept Majority relation Tournament Matrix-vector representation Condorcet winner Core Top cycle Uncovered set Weakly stable set Externally stable set Uncaptured set Untrapped set k-stable alternative k-stable set 

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References

  1. 1.
    Fishburn P. (1977) Condorcet social choice functions. SIAM J. Appl. Math. 33: 469–489CrossRefGoogle Scholar
  2. 2.
    Miller N. (1980) A new solution set for tournaments and majority voting: further graph-theoretical approaches to the theory of voting. Am. J. Pol. Sci. 24: 68–96CrossRefGoogle Scholar
  3. 3.
    Richelson, J.T.: Majority Rule and Collective Choice. Mimeo (1981)Google Scholar
  4. 4.
    Bordes G. (1983) On the possibility of reasonable consistent majoritarian choice: some positive results. J. Econ. Theory 31: 122–132CrossRefGoogle Scholar
  5. 5.
    McKelvey R. (1986) Covering, dominance and institution-free properties of social choice. Am. J. Pol. Sci. 30: 283–314CrossRefGoogle Scholar
  6. 6.
    Duggan, J.: Uncovered sets. Mimeo (2006)Google Scholar
  7. 7.
    Duggan J. (2007) A systematic approach to the construction of non-empty choice sets. Soc. Choice Welf 28: 491–506CrossRefGoogle Scholar
  8. 8.
    Wuffl A., Feld S., Owen G. (1989) Finagle’s law and the Finagle’s point, a new solution concept for two-candidate competition in spatial voting games without a core. Am. J. Pol. Sci. 33(2): 348–375CrossRefGoogle Scholar
  9. 9.
    Aleskerov F., Kurbanov E. (1999) A degree of manipulability of known social choice procedures. In: Alkan A., Aliprantis Ch., Yannelis N. (eds) Current Trends in Economics: Theory and Applications. Springer, Berlin, pp 13–27CrossRefGoogle Scholar
  10. 10.
    Subochev, A.: Dominant, weakly stable, uncovered sets: properties and extensions. Working paper (preprint) WP7/2008/03. Moscow: State University, Higher School of Economics (2008)Google Scholar
  11. 11.
    Ward B. (1961) Majority rule and allocation. J. Confl. Resolut. 5: 379–389CrossRefGoogle Scholar
  12. 12.
    Schwartz T. (1970) On the possibility of rational policy evaluation. Theory Decis. 1: 89–106CrossRefGoogle Scholar
  13. 13.
    Schwartz T. (1972) Rationality and the myth of the maximum. Noûs. 6: 97–117CrossRefGoogle Scholar
  14. 14.
    Schwartz T. (1977) Collective choice, separation of issues and vote trading. Am. Pol. Sci. Rev. 71(3): 999–1010CrossRefGoogle Scholar
  15. 15.
    Good I. (1971) A note on Condorcet sets. Public Choice 10: 97–101CrossRefGoogle Scholar
  16. 16.
    Smith J. (1973) Aggregation of preferences with variable electorates. Econometrica 41(6): 1027–1041CrossRefGoogle Scholar
  17. 17.
    Aleskerov F., Subochev A. (2009) On stable solutions to the ordinal social choice problem. Doklady Math. 73(3): 437–439CrossRefGoogle Scholar
  18. 18.
    Subochev A. (2010) Dominating, weakly stable, uncovered sets: properties and extensions. Avtomatika i Telemekhanika (Automation & Remote Control) 1: 130–143Google Scholar
  19. 19.
    McGarvey D. (1953) A theorem on the construction of voting paradoxes. Econometrica 21: 608–610CrossRefGoogle Scholar
  20. 20.
    Laslier J.F. (1997) Tournament Solutions and Majority Voting. Springer, BerlinCrossRefGoogle Scholar
  21. 21.
    Gillies, D.B.: Solutions to general non-zero-sum games. In: Tucker, A.W., Luce, R.D. Contributions to the Theory of Games, vol. IV, Princeton University Press, Princeton (1959)Google Scholar
  22. 22.
    Banks J. (1985) Sophisticated voting outcomes and agenda control. Soc. Choice Welf 1: 295–306CrossRefGoogle Scholar
  23. 23.
    Miller N. (1977) Graph-theoretical approaches to the theory of voting. Am. J. Pol. Sci. 21: 769–803CrossRefGoogle Scholar
  24. 24.
    Deb R. (1977) On Schwartz’s rule. J. Econ. Theory 16: 103–110CrossRefGoogle Scholar
  25. 25.
    Roth A. (1976) Subsolutions and the supercore of cooperative games. Math. Oper. Res. 1(1): 43–49CrossRefGoogle Scholar
  26. 26.
    von Neumann J., Morgenstern O. (1944) Theory of Games and Economic Behavior. Princeton University Press, PrincetonGoogle Scholar
  27. 27.
    Laffond G., Lainé J. (1994) Weak covering relations. Theory Decis. 37: 245–265CrossRefGoogle Scholar
  28. 28.
    Levchenkov, V.: Cyclic Tournaments: A Matching Solution. Mimeo (1995)Google Scholar
  29. 29.
    Zhu X. et al (2010) New dominating sets in social networks. J. Glob. Optim. 48(4): 633–642CrossRefGoogle Scholar
  30. 30.
    Thai M., Pardalos P.M. (2011) Handbook of Optimization in Complex Networks: Communication and Social Networks. Springer, BerlinGoogle Scholar
  31. 31.
    Bomze I.M., Budinich M., Pardalos P.M., Pelillo M. (1999) The maximum clique problem. In: Du D.-Z., Pardalos P.M. (eds) Handbook of Combinatorial Optimization, Supplement, vol. A. Kluwer, Dordrecht, pp 1–74CrossRefGoogle Scholar
  32. 32.
    Xanthopoulos P., Arulselvan A., Boginski V., Pardalos P.M. (2009) A retrospective review of social networks. In: Memon N., Alhajj R. (eds) Proceedings of International Conference on Advances in Social Network Analysis and Mining. IEEE Computer Society, Washington, DC, pp 300–305Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  1. 1.DeCAn Lab and Department of Mathematics for EconomicsNational Research University Higher School of EconomicsMoscowRussia
  2. 2.Institute of Control SciencesRussian Academy of SciencesMoscowRussia

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