Tournament solutions based on cooperative game theory (original) (raw)

Abstract

This paper considers the ranking problem of candidates for a certain position based on ballot papers filled by voters. We suggest a ranking procedure of alternatives using cooperative game theory methods. For this, it is necessary to construct a characteristic function via the filled ballot paper profile of voters. The Shapley value serves as the ranking method. The winner is the candidate having the maximum Shapley value. And finally, we explore the properties of the designed ranking procedure.

Key takeaways

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  1. The paper proposes a ranking procedure using cooperative game theory, specifically the Shapley value.
  2. It defines a characteristic function based on voter preferences from ballot papers for ranking candidates.
  3. The method accounts for candidate correlations within coalitions, improving traditional voting methods.
  4. The research demonstrates properties like unanimity and the Condorcet property in ranking procedures.
  5. A case study with 45 voters and 5 candidates illustrates the application and effectiveness of the proposed method.

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References (9)

  1. Brams SJ, Fishburn PC (1978) Approval Voting. The American Political Science Review, vol. 72, no. 3. Pp. 831-847.
  2. Brams SJ, Fishburn PC (2005) Going from Theory to Practice: The Mixed Success of Approval Voting. Social Choice and Welfare, vol. 25, no. 2- 3. Pp. 457-474.
  3. Hillinger C (2005) The Case for Utilitarian Voting, Homo Oeconomicus, vol. 23. Pp. 295-321.
  4. Gaertner W, Xu Y (2012) A General Scoring Rule, Mathematical Social Sciences, vol. 63, no. 3. Pp. 193-196.
  5. Smith WD (2000) Range Voting, Technical Report 56, NEC Research, Princeton, NJ, USA.
  6. Balinski M, Laraki R (2007) A Theory of Measuring, Electing, and Ranking, Proceedings of the National Academy of Sciences of the USA, vol. 104, no. 21. Pp. 8720-8725.
  7. Klamler C (2006) On the Closeness Aspect of Three Voting Rules: Borda-Copeland-Maximin, Group Decision and Negotiation, vol. 14, issue 3. Pp. 233-240.
  8. Brams SJ, Kilgour DM, Sanver MR (2007) A Minimax Procedure for Electing Committees, Public Choice, vol. 132, no. 3-4. Pp. 401-420.
  9. Kilgour DM (2010) Approval Balloting for Multi-Winner Elections, Handbook on Approval Voting, Springer. Pp. 105-124.

FAQs

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What advantages do cooperative game theory methods have in candidate ranking?add

The paper demonstrates that using cooperative game theory methods allows for a nuanced ranking process, integrating candidates' coalition capabilities through Shapley values. This method compares favorably against traditional ranking techniques like Borda and Copeland methods, shown in examples from 2010 FIA Formula One Championship.

How does the new ranking procedure leverage tournament matrices?add

The research utilizes tournament matrices to compute characteristic functions for coalitions of candidates, yielding collective ranking outcomes. This approach accounts for win-loss records among candidates, enhancing sensitivity to voter preferences compared to simpler voting methods.

What is the importance of the Condorcet and strong Condorcet properties?add

Unanimity and the Condorcet properties ensure that preferred candidates consistently rank higher in collective choices, while the strong Condorcet property accentuates the robustness of rankings under voter preference changes. The study confirms these properties are satisfied by the proposed Shapley value-based ranking method.

How does the Shapley value contribute to evaluating candidate power?add

The Shapley value quantitatively reflects each candidate's influence within various coalition formations, thus providing an impartial measure of candidate strength. Experimental results indicate that candidates with higher Shapley values tend to be those favored in winning scenarios, illustrating its practical relevance.

What distinguishes this paper's ranking method from traditional voting rules?add

This paper's ranking method incorporates cooperative game theory to analyze relationships between candidates in coalitions, contrasting with traditional rules like majority voting that may overlook nuanced voter preferences. Specifically, it converges on voter sentiments while maintaining computational efficiency, as evidenced in detailed examples.