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Computing Tournament Solutions using Relation Algebra and REL VIEW

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  • Rudolf Berghammer

    ()
    (Institut für Informatik - Universitat Kiel)

  • Agnieszka Rusinowska

    ()
    (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon-Sorbonne)

  • Harrie De Swart

    ()
    (Department of Philosophy - Erasmus University Rotterdam)

Abstract

We describe a simple computing technique for the tournament choice problem. It rests upon a relational modeling and uses the BDD-based computer system RelView for the evaluation of the relation-algebraic expressions that specify the solutions and for the visualization of the computed results. The Copeland set can immediately be identified using RelView's labeling feature. Relation-algebraic specifications of the Condorcet non-losers, the Schwartz set, the top cycle, the uncovered set, the minimal covering set, the Banks set, and the tournament equilibrium set are delivered. We present an example of a tournament on a small set of alternatives, for which the above choice sets are computed and visualized via RelView. The technique described in this paper is very flexible and especially appropriate for prototyping and experimentation, and as such very instructive for educational purposes. It can easily be applied to other problems of social choice and game theory.

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Paper provided by HAL in its series Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) with number halshs-00639942.

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Date of creation: Oct 2011
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Handle: RePEc:hal:cesptp:halshs-00639942

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Related research

Keywords: Tournament; relational algebra; RelView; Copeland set; Condorcet non-losers; Schwartz set; top cycle; uncovered set; minimal covering set; Banks set; tournament equilibrium set.;

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  1. Bolus, Stefan, 2011. "Power indices of simple games and vector-weighted majority games by means of binary decision diagrams," European Journal of Operational Research, Elsevier, vol. 210(2), pages 258-272, April.
  2. Hudry, Olivier, 2009. "A survey on the complexity of tournament solutions," Mathematical Social Sciences, Elsevier, vol. 57(3), pages 292-303, May.
  3. repec:hal:journl:halshs-00308741 is not listed on IDEAS
  4. Dutta, Bhaskar, 1988. "Covering sets and a new condorcet choice correspondence," Journal of Economic Theory, Elsevier, vol. 44(1), pages 63-80, February.
  5. Michel Grabisch & Agnieszka Rusinowska, 2010. "A model of influence in a social network," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00308741, HAL.
  6. John Duggan, 2011. "Uncovered Sets," Wallis Working Papers WP63, University of Rochester - Wallis Institute of Political Economy.
  7. Deb, Rajat, 1977. "On Schwartz's rule," Journal of Economic Theory, Elsevier, vol. 16(1), pages 103-110, October.
  8. Berghammer, Rudolf & Rusinowska, Agnieszka & de Swart, Harrie, 2010. "Applying relation algebra and RelView to measures in a social network," European Journal of Operational Research, Elsevier, vol. 202(1), pages 182-195, April.
  9. I. Good, 1971. "A note on condorcet sets," Public Choice, Springer, vol. 10(1), pages 97-101, March.
  10. Agnieszka Rusinowska & Harrie de Swart & Jan-Willem van der Rijt, 2005. "A new model of coalition formation," Social Choice and Welfare, Springer, vol. 24(1), pages 129-154, 09.
  11. Berghammer, Rudolf & Rusinowska, Agnieszka & de Swart, Harrie, 2009. "An interdisciplinary approach to coalition formation," European Journal of Operational Research, Elsevier, vol. 195(2), pages 487-496, June.
  12. Brandt, Felix & Fischer, Felix, 2008. "Computing the minimal covering set," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 254-268, September.
  13. Berghammer, Rudolf & Rusinowska, Agnieszka & de Swart, Harrie, 2007. "Applying relational algebra and RelView to coalition formation," European Journal of Operational Research, Elsevier, vol. 178(2), pages 530-542, April.
  14. Elizabeth Penn, 2006. "Alternate Definitions of the Uncovered Set and Their Implications," Social Choice and Welfare, Springer, vol. 27(1), pages 83-87, August.
  15. Smith, John H, 1973. "Aggregation of Preferences with Variable Electorate," Econometrica, Econometric Society, vol. 41(6), pages 1027-41, November.
  16. Nicolas Houy, 2009. "Still more on the Tournament Equilibrium Set," Social Choice and Welfare, Springer, vol. 32(1), pages 93-99, January.
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