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Strategy-proofness versus symmetry in economies with an indivisible good and money

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Abstract

We consider the problem of allocating a single indivisible good among \(n\) agents when monetary transfers are allowed. We study the possibility of constructing strategy-proof, symmetric, and budget balanced mechanisms. We show that there is no strategy-proof, symmetric, and budget balanced mechanism (under the weak domain condition that the set of agent’s possible valuations includes at least \(n+1\) common valuations). Moreover, this result implies that there is no strategy-proof, symmetric, and budget balanced mechanism (i) in the model where agents may have non-quasilinear preferences, and/or (ii) in the unit-demand model with \(n\) heterogeneous indivisible goods.

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Notes

  1. See Sprumont (1995) and Barberà (2001; 2012) for excellent surveys of the literature on strategy-proofness.

  2. Symmetry is also called equal treatment of equals.

  3. Ohseto (2000) and Schummer (2000) formally prove this statement in related models.

  4. This result also follows as a corollary of Holmström (1979). See Saitoh and Serizawa (2008), Sakai (2008; 2013a), Ashlagi and Serizawa (2012) and Sprumont (2013) for alternative characterizations of the Vickrey mechanism.

  5. The definitions of strategy-proofness and budget balance can be extended to larger domains straightforwardly. The definition of symmetry can be extended to larger domains in the weak sense that it requires the same condition on the set of all valuation profiles and it requires nothing on the outside of that set. See Hashimoto and Saitoh (2010, Definition 4) for this type of extension in the context of public good economies.

  6. Saitoh and Serizawa (2008, Theorems 2 and 3) also provide characterizations on some domains of non-quasilinear preferences that do not include any quasilinear preferences.

  7. Ignore this equation in the two-agent case since there is no non-consumer whose valuation is \(a\).

  8. See Fujinaka and Sakai (2007) for a general definition of Maskin monotonicity. In our model, a mechanism \(f\) is Maskin monotonic if for all \(v,v^{\prime }\in V\), if \(v_{i}^{\prime }\ge v_{i}\) for \(i\in C(v)\) and \(v_{j}^{\prime }\le v_{j}\) for all \(j\in NC(v)\), then \(f(v^{\prime })=f(v)\).

  9. A mechanism \(f\) is weakly symmetic if for all \(v\in V\), \([v_{i}=v_{j}\) for all \(i,j\in N]\Longrightarrow [U(f_{i}(v);v_{i})=U(f_{j}(v);v_{i})\) for all \(i,j\in N]\).

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Acknowledgments

We would like to thank Kazutoshi Ando, an associate editor, and two anonymous referees for helpful suggestions and comments. This research was partially supported by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science.

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Authors and Affiliations

  1. Tohoku Seikatsu Bunka College, Sendai, Miyagi, 981-8585, Japan

    Miki Kato

  2. Faculty of Economics, Tohoku University, Sendai, Miyagi, 980-8576, Japan

    Shinji Ohseto & Shohei Tamura

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  1. Miki Kato
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  2. Shinji Ohseto
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  3. Shohei Tamura
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Correspondence to Shinji Ohseto.

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Kato, M., Ohseto, S. & Tamura, S. Strategy-proofness versus symmetry in economies with an indivisible good and money. Int J Game Theory 44, 195–207 (2015). https://doi.org/10.1007/s00182-014-0425-y

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