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Computing Tournament Solutions using Relation Algebra and REL VIEW

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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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  • Rudolf Berghammer & Agnieszka Rusinowska & Harrie de Swart, 2011. "Computing Tournament Solutions using Relation Algebra and REL VIEW," Documents de travail du Centre d'Economie de la Sorbonne 11067, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    • Handle: RePEc:mse:cesdoc:11067
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    1. 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.
    2. Michel Grabisch & Agnieszka Rusinowska, 2010. "A model of influence in a social network," Theory and Decision, Springer, vol. 69(1), pages 69-96, July.
    3. 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.
    4. Deb, Rajat, 1977. "On Schwartz's rule," Journal of Economic Theory, Elsevier, vol. 16(1), pages 103-110, October.
    5. Agnieszka Rusinowska & Harrie de Swart & Jan-Willem van der Rijt, 2005. "A new model of coalition formation," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 24(1), pages 129-154, September.
    6. 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.
    7. Hudry, Olivier, 2009. "A survey on the complexity of tournament solutions," Mathematical Social Sciences, Elsevier, vol. 57(3), pages 292-303, May.
    8. 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.
    9. Dutta, Bhaskar, 1988. "Covering sets and a new condorcet choice correspondence," Journal of Economic Theory, Elsevier, vol. 44(1), pages 63-80, February.
    10. Elizabeth Penn, 2006. "Alternate Definitions of the Uncovered Set and Their Implications," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 27(1), pages 83-87, August.
    11. John Duggan, 2011. "Uncovered Sets," Wallis Working Papers WP63, University of Rochester - Wallis Institute of Political Economy.
    12. 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.
    13. Nicolas Houy, 2009. "Still more on the Tournament Equilibrium Set," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 32(1), pages 93-99, January.
    14. Smith, John H, 1973. "Aggregation of Preferences with Variable Electorate," Econometrica, Econometric Society, vol. 41(6), pages 1027-1041, November.
    15. I. Good, 1971. "A note on condorcet sets," Public Choice, Springer, vol. 10(1), pages 97-101, March.
    16. Brandt, Felix & Fischer, Felix, 2008. "Computing the minimal covering set," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 254-268, September.
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    Cited by:

    1. Berghammer, Rudolf & Schnoor, Henning, 2015. "Control of Condorcet voting: Complexity and a Relation-Algebraic approach," European Journal of Operational Research, Elsevier, vol. 246(2), pages 505-516.
    2. Andrea C. Hupman & Jay Simon, 2023. "The Legacy of Peter Fishburn: Foundational Work and Lasting Impact," Decision Analysis, INFORMS, vol. 20(1), pages 1-15, March.

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    Keywords

    Tournament; relational algebra; REL VIEW; Copeland set; Condorcet non-losers; Schwartz set; top cycle; uncovered set; minimal covering set; Banks set; tournament equilibrium set;
    All these keywords.

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C88 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Other Computer Software

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