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Path-Monotonicity and Incentive Compatibility

Author

Listed:
  • Berger André
  • Müller Rudolf
  • Naeemi Seyed Hossein

    (METEOR)

Abstract

We study the role of monotonicity in the characterization of incentive compatible allocation rules when types are multi-dimensional, the mechanism designer may use monetary transfers, and agents have quasi-linear preferences over outcomes and transfers. It is well-known that monotonicity of the allocation rule is necessary for incentive compatibility. Furthermore, if valuations for outcomes are either convex or differentiable functions in types, revenue equivalence literature tells that path-integrals of particular vector fields are path-independent. For the special case of linear valuations it is known that monotonicity plus path-independence is sufficient for implementation. We show by example that this is not true for convex or differentiable valuations, and introduce a stronger version of monotonicity, called path-monotonicity. We show that path-monotonicity and path-independence characterize implementable allocation rules if (1) valuations are convex and type spaces are convex; (2) valuations are differentiable and type spaces are path-connected. Next we analyze conditions under which monotonicity is equivalent to path-monotonicity. We show that an increasing difference property of valuations ensures this equivalence. Next, we show that for simply connected type spaces incentive compatibility of the allocation rule is equivalent to path-monotonicity plus incentive compatibility in some neighborhood of each type. This result is used to show that on simply connected type spaces incentive compatible allocation rules with a finite range are completely characterized by path--monotonicity, and thus by monotonicity in cases where path-monotonicity and monotonicity are equivalent. This generalizes a theorem by Saks and Yu to a wide range of settings.

Suggested Citation

  • Berger André & Müller Rudolf & Naeemi Seyed Hossein, 2010. "Path-Monotonicity and Incentive Compatibility," Research Memorandum 035, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
  • Handle: RePEc:unm:umamet:2010035
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    References listed on IDEAS

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    1. Paul Milgrom & Ilya Segal, 2002. "Envelope Theorems for Arbitrary Choice Sets," Econometrica, Econometric Society, vol. 70(2), pages 583-601, March.
    2. Birgit Heydenreich & Rudolf Müller & Marc Uetz & Rakesh V. Vohra, 2009. "Characterization of Revenue Equivalence," Econometrica, Econometric Society, vol. 77(1), pages 307-316, January.
    3. Jehiel, Philippe & Moldovanu, Benny, 2001. "Efficient Design with Interdependent Valuations," Econometrica, Econometric Society, vol. 69(5), pages 1237-1259, September.
    4. Muller, Rudolf & Perea, Andres & Wolf, Sascha, 2007. "Weak monotonicity and Bayes-Nash incentive compatibility," Games and Economic Behavior, Elsevier, vol. 61(2), pages 344-358, November.
    5. Kos, Nenad & Messner, Matthias, 2013. "Extremal incentive compatible transfers," Journal of Economic Theory, Elsevier, vol. 148(1), pages 134-164.
    6. Sushil Bikhchandani & Shurojit Chatterji & Ron Lavi & Ahuva Mu'alem & Noam Nisan & Arunava Sen, 2006. "Weak Monotonicity Characterizes Deterministic Dominant-Strategy Implementation," Econometrica, Econometric Society, vol. 74(4), pages 1109-1132, July.
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    8. Krishna, Vijay & Maenner, Eliot, 2001. "Convex Potentials with an Application to Mechanism Design," Econometrica, Econometric Society, vol. 69(4), pages 1113-1119, July.
    9. Jehiel, Philippe & Moldovanu, Benny & Stacchetti, Ennio, 1999. "Multidimensional Mechanism Design for Auctions with Externalities," Journal of Economic Theory, Elsevier, vol. 85(2), pages 258-293, April.
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    11. Olszewski, Wojciech & Chung, Kim-Sau, 2007. "A non-differentiable approach to revenue equivalence," Theoretical Economics, Econometric Society, vol. 2(4), December.
    12. Rochet, Jean-Charles, 1987. "A necessary and sufficient condition for rationalizability in a quasi-linear context," Journal of Mathematical Economics, Elsevier, vol. 16(2), pages 191-200, April.
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    Cited by:

    1. Mishra, Debasis & Sen, Arunava, 2012. "Robertsʼ Theorem with neutrality: A social welfare ordering approach," Games and Economic Behavior, Elsevier, vol. 75(1), pages 283-298.
    2. Mishra, Debasis & Roy, Souvik, 2013. "Implementation in multidimensional dichotomous domains," Theoretical Economics, Econometric Society, vol. 8(2), May.
    3. Mishra, Debasis & Pramanik, Anup & Roy, Souvik, 2014. "Multidimensional mechanism design in single peaked type spaces," Journal of Economic Theory, Elsevier, vol. 153(C), pages 103-116.

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