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Monotonic extensions on economic domains

Author

Listed:
  • William Thomson

    (Department of Economics, University of Rochester, Rochester, NY 14627, USA)

Abstract

The property of "monotonicity" is necessary, and in many contexts, sufficient, for a solution to be Nash implementable (Maskin 1977). In this paper, we follow Sen (1995) and evaluate the extent to which a solution may fail monotonicity by identifying the minimal way in which it has to be enlarged so as to satisfy the property. We establish a general result relating the "minimal monotonic extensions" of the intersection and the union of a family of solutions to the minimal monotonic extensions of the members of the family. We then calculate the minimal monotonic extensions of several solutions in a variety of contexts, such as classical exchange economies, with either individual endowments or a social endowment, economies with public goods, and one-commodity economies in which preferences are single-peaked. For some of the examples, very little is needed to recover monotonicity, but for others, the required enlargement is quite considerable, to the point that the distributional objective embodied in the solution has to be given up altogether.

Suggested Citation

  • William Thomson, 1999. "Monotonic extensions on economic domains," Review of Economic Design, Springer;Society for Economic Design, vol. 4(1), pages 13-33.
    • Handle: RePEc:spr:reecde:v:4:y:1999:i:1:p:13-33
    Note: Received: 21 September 1996 / Accepted: 17 August 1998
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    Cited by:

    1. Klaus, Bettina & Dimitrov, Dinko & Haake, Claus-Jochen, 2006. "Bundling in exchange markets with indivisible goods," Economics Letters, Elsevier, vol. 93(1), pages 106-110, October.
    2. İpek Özkal-Sanver, 2013. "Minimal conversely consistent extension of the men-optimal solution," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 40(1), pages 89-99, January.
    3. Benoit, Jean Pierre & Ok, Efe A. & Sanver, M. Remzi, 2007. "On combining implementable social choice rules," Games and Economic Behavior, Elsevier, vol. 60(1), pages 20-30, July.
    4. Matthew O. Jackson, 2001. "A crash course in implementation theory," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(4), pages 655-708.
    5. M. Remzi Sanver, 2017. "Nash implementing social choice rules with restricted ranges," Review of Economic Design, Springer;Society for Economic Design, vol. 21(1), pages 65-72, March.
    6. Michele Lombardi & Naoki Yoshihara, 2017. "Natural implementation with semi-responsible agents in pure exchange economies," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(4), pages 1015-1036, November.
    7. Pablo Amorós, 2009. "Unequivocal majority and Maskin-monotonicity," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 33(4), pages 521-532, November.
    8. Claus-Jochen Haake & Bettina Klaus, 2009. "Monotonicity and Nash implementation in matching markets with contracts," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 41(3), pages 393-410, December.
    9. Olivier Bochet, 2007. "Implementation of the Walrasian correspondence: the boundary problem," International Journal of Game Theory, Springer;Game Theory Society, vol. 36(2), pages 301-316, October.
    10. Bochet, Olivier & Maniquet, François, 2010. "Virtual Nash implementation with admissible support," Journal of Mathematical Economics, Elsevier, vol. 46(1), pages 99-108, January.
    11. Takashi Hayashi & Toyotaka Sakai, 2009. "Nash implementation of competitive equilibria in the job-matching market," International Journal of Game Theory, Springer;Game Theory Society, vol. 38(4), pages 453-467, November.
    12. Thomson, William, 2005. "Divide-and-permute," Games and Economic Behavior, Elsevier, vol. 52(1), pages 186-200, July.
    13. Remzi Sanver, M., 2006. "Nash implementation of the majority rule," Economics Letters, Elsevier, vol. 91(3), pages 369-372, June.
    14. İpek Özkal-Sanver & M. Sanver, 2006. "Nash implementation via hyperfunctions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 26(3), pages 607-623, June.
    15. Maskin, Eric & Sjostrom, Tomas, 2002. "Implementation theory," Handbook of Social Choice and Welfare, in: K. J. Arrow & A. K. Sen & K. Suzumura (ed.), Handbook of Social Choice and Welfare, edition 1, volume 1, chapter 5, pages 237-288, Elsevier.
    16. Velez, Rodrigo A. & Thomson, William, 2012. "Let them cheat!," Games and Economic Behavior, Elsevier, vol. 75(2), pages 948-963.
    17. Eun Jeong Heo & Vikram Manjunath, 2017. "Implementation in stochastic dominance Nash equilibria," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 48(1), pages 5-30, January.
    18. Ehlers, Lars, 2000. "Indifference and the uniform rule," Economics Letters, Elsevier, vol. 67(3), pages 303-308, June.
    19. Dietzenbacher, Bas & Yanovskaya, Elena, 2021. "Self-antidual extensions and subsolutions," Mathematical Social Sciences, Elsevier, vol. 114(C), pages 105-109.
    20. Orhan Erdem & M. Sanver, 2005. "Minimal monotonic extensions of scoring rules," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 25(1), pages 31-42, October.
    21. Maskin, Eric & Sjostrom, Tomas, 2002. "Implementation theory," Handbook of Social Choice and Welfare,in: K. J. Arrow & A. K. Sen & K. Suzumura (ed.), Handbook of Social Choice and Welfare, edition 1, volume 1, chapter 5, pages 237-288 Elsevier.

    More about this item

    Keywords

    Maskin-monotonicity; Nash-implementation; minimal monotonic extension; fair allocation;
    All these keywords.

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • D50 - Microeconomics - - General Equilibrium and Disequilibrium - - - General
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
    • D78 - Microeconomics - - Analysis of Collective Decision-Making - - - Positive Analysis of Policy Formulation and Implementation

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