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A Mathematical Model for Optimal Decisions in a Representative Democracy

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  • Malik Magdon-Ismail
  • Lirong Xia

Abstract

Direct democracy is a special case of an ensemble of classifiers, where every person (classifier) votes on every issue. This fails when the average voter competence (classifier accuracy) falls below 50%, which can happen in noisy settings where voters have only limited information, or when there are multiple topics and the average voter competence may not be high enough for some topics. Representative democracy, where voters choose representatives to vote, can be an elixir in both these situations. Representative democracy is a specific way to improve the ensemble of classifiers. We introduce a mathematical model for studying representative democracy, in particular understanding the parameters of a representative democracy that gives maximum decision making capability. Our main result states that under general and natural conditions, 1. Representative democracy can make the correct decisions simultaneously for multiple noisy issues. 2. When the cost of voting is fixed, the optimal representative democracy requires that representatives are elected from constant sized groups: the number of representatives should be linear in the number of voters. 3. When the cost and benefit of voting are both polynomial, the optimal group size is close to linear in the number of voters. This work sets the mathematical foundation for studying the quality-quantity tradeoff in a representative democracy-type ensemble (fewer highly qualified representatives versus more less qualified representatives).

Suggested Citation

  • Malik Magdon-Ismail & Lirong Xia, 2018. "A Mathematical Model for Optimal Decisions in a Representative Democracy," Papers 1807.06157, arXiv.org.
  • Handle: RePEc:arx:papers:1807.06157
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    References listed on IDEAS

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    1. Marcus Pivato, 2013. "Voting rules as statistical estimators," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 40(2), pages 581-630, February.
    2. Kanazawa, Satoshi, 1998. "A brief note on a further refinement of the Condorcet Jury Theorem for heterogeneous groups," Mathematical Social Sciences, Elsevier, vol. 35(1), pages 69-73, January.
    3. Nitzan, Shmuel & Paroush, Jacob, 1980. "Investment in Human Capital and Social Self Protection under Uncertainty," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 21(3), pages 547-557, October.
    4. Kaushik Mukhopadhaya, 2003. "Jury Size and the Free Rider Problem," Journal of Law, Economics, and Organization, Oxford University Press, vol. 19(1), pages 24-44, April.
    5. Emmanuelle Auriol & Robert Gary-Bobo, 2012. "On the optimal number of representatives," Public Choice, Springer, vol. 153(3), pages 419-445, December.
    6. Daniel Berend & Luba Sapir, 2007. "Monotonicity in Condorcet’s Jury Theorem with dependent voters," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 28(3), pages 507-528, April.
    7. Karotkin, Drora & Paroush, Jacob, 1995. "Incentive schemes for investment in human capital by members of a team of decision makers," Labour Economics, Elsevier, vol. 2(1), pages 41-51, March.
    8. Timothy Besley & Stephen Coate, 1997. "An Economic Model of Representative Democracy," The Quarterly Journal of Economics, Oxford University Press, vol. 112(1), pages 85-114.
    9. Scott Feld & Bernard Grofman, 1984. "The accuracy of group majority decisions in groups with added members," Public Choice, Springer, vol. 42(3), pages 273-285, January.
    10. Sapir, Luba, 2005. "Generalized means of jurors' competencies and marginal changes of jury's size," Mathematical Social Sciences, Elsevier, vol. 50(1), pages 83-101, July.
    11. Ruth Ben-Yashar & Mor Zahavi, 2011. "The Condorcet jury theorem and extension of the franchise with rationally ignorant voters," Public Choice, Springer, vol. 148(3), pages 435-443, September.
    12. Ruth Ben‐Yashar & Jacob Paroush, 2003. "Investment in Human Capital in Team Members Who Are Involved in Collective Decision Making," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 5(3), pages 527-539, July.
    13. Gradstein, Mark & Nitzan, Shmuel, 1987. "Organizational decision-making quality and the severity of the free-riding problem," Economics Letters, Elsevier, vol. 23(4), pages 335-339.
    14. Mark Fey, 2003. "A note on the Condorcet Jury Theorem with supermajority voting rules," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 20(1), pages 27-32.
    15. Daniel Berend & Jacob Paroush, 1998. "When is Condorcet's Jury Theorem valid?," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 15(4), pages 481-488.
    16. Daniel Berend & Luba Sapir, 2005. "Monotonicity in Condorcet Jury Theorem," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 24(1), pages 83-92, August.
    17. Ruth Ben-Yashar & Jacob Paroush, 2000. "A nonasymptotic Condorcet jury theorem," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 17(2), pages 189-199.
    18. Young, H. P., 1988. "Condorcet's Theory of Voting," American Political Science Review, Cambridge University Press, vol. 82(4), pages 1231-1244, December.
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