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The Relation between Monotonicity and Strategy-Proofness

  • Bettina Klaus
  • Olivier Bochet

The Muller-Satterthwaite Theorem (Muller and Satterthwaite, 1977) establishes the equivalence between Maskin monotonicity and strategy-proofness, two cornerstone conditions for the decentralization of social choice rules. We consider a general model that covers public goods economies as in Muller and Satterthwaite (1977) as well as private goods economies. For private goods economies we use a weaker condition than Maskin monotonicity that we call unilateral monotonicity. We introduce two easy-to-check domain conditions which separately guarantee that (i) unilateral/Maskin monotonicity implies strategy-proofness (Theorem 1) and (ii) strategy-proofness implies unilateral/Maskin monotonicity (Theorem 2). We introduce and discuss various classical single-peaked domains and show which of the domain conditions they satisfy (see Propositions 1 and 2 and an overview in Table 1). As a by-product of our analysis, we obtain some extensions of the Muller-Satterthwaite Theorem as summarized in Theorem 3. We also discuss some new "Muller-Satterthwaite domains" (e.g.,Proposition 3).

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Paper provided by Université de Lausanne, Faculté des HEC, DEEP in its series Cahiers de Recherches Economiques du Département d'Econométrie et d'Economie politique (DEEP) with number 10.01.

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Length: 21 pages
Date of creation: Jan 2010
Date of revision:
Publication status: Published in Social Choice and Welfare, 40(1), January 2013, pp. 41-63
Handle: RePEc:lau:crdeep:10.01
Contact details of provider: Postal: Université de Lausanne, Faculté des HEC, DEEP, Internef, CH-1015 Lausanne
Phone: ++41 21 692.33.20
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  1. Takamiya, Koji, 2007. "Domains of social choice functions on which coalition strategy-proofness and Maskin monotonicity are equivalent," Economics Letters, Elsevier, vol. 95(3), pages 348-354, June.
  2. Dolors Berga & Bernardo Moreno, 2009. "Strategic requirements with indifference: single-peaked versus single-plateaued preferences," Social Choice and Welfare, Springer, vol. 32(2), pages 275-298, February.
  3. Maskin, Eric, 1999. "Nash Equilibrium and Welfare Optimality," Review of Economic Studies, Wiley Blackwell, vol. 66(1), pages 23-38, January.
  4. Muller, Eitan & Satterthwaite, Mark A., 1977. "The equivalence of strong positive association and strategy-proofness," Journal of Economic Theory, Elsevier, vol. 14(2), pages 412-418, April.
  5. H. Moulin, 1980. "On strategy-proofness and single peakedness," Public Choice, Springer, vol. 35(4), pages 437-455, January.
  6. Le Breton, Michel & Zaporozhets, Vera, 2006. "On the Equivalence of Coalitional and Individual Strategy-Proofness Properties," IDEI Working Papers 408, Institut d'Économie Industrielle (IDEI), Toulouse.
  7. David Cantala, 2004. "Choosing the level of a public good when agents have an outside option," Social Choice and Welfare, Springer, vol. 22(3), pages 491-514, 06.
  8. Satterthwaite, Mark A & Sonnenschein, Hugo, 1981. "Strategy-Proof Allocation Mechanisms at Differentiable Points," Review of Economic Studies, Wiley Blackwell, vol. 48(4), pages 587-97, October.
  9. Takamiya, Koji, 2001. "Coalition strategy-proofness and monotonicity in Shapley-Scarf housing markets," Mathematical Social Sciences, Elsevier, vol. 41(2), pages 201-213, March.
  10. M. Fleurbaey & F. Maniquet, 1997. "Implementability and Horizontal Equity Imply No-Envy," Econometrica, Econometric Society, vol. 65(5), pages 1215-1220, September.
  11. Dasgupta, Partha S & Hammond, Peter J & Maskin, Eric S, 1979. "The Implementation of Social Choice Rules: Some General Results on Incentive Compatibility," Review of Economic Studies, Wiley Blackwell, vol. 46(2), pages 185-216, April.
  12. Border, Kim C & Jordan, J S, 1983. "Straightforward Elections, Unanimity and Phantom Voters," Review of Economic Studies, Wiley Blackwell, vol. 50(1), pages 153-70, January.
  13. Reny, Philip J., 2001. "Arrow's theorem and the Gibbard-Satterthwaite theorem: a unified approach," Economics Letters, Elsevier, vol. 70(1), pages 99-105, January.
  14. Mark A. Satterthwaite & Hugo Sonnenschein, 1979. "Strategy-Proof Allocation Mechanisms," Discussion Papers 395, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  15. Koji Takamiya, 2003. "On strategy-proofness and essentially single-valued cores: A converse result," Social Choice and Welfare, Springer, vol. 20(1), pages 77-83.
  16. Sprumont, Yves, 1991. "The Division Problem with Single-Peaked Preferences: A Characterization of the Uniform Allocation Rule," Econometrica, Econometric Society, vol. 59(2), pages 509-19, March.
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