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The outcome of competitive equilibrium rules in buyer–seller markets when the agents play strategically

Abstract

We analyze the two-stage games induced by competitive equilibrium rules for the buyer–seller market of Shapley and Shubik (Int J Game Theory 1:111–130, 1972). In these procedures, first sellers and then buyers report their valuation and the outcome is determined by a competitive equilibrium outcome for the market reported by the agents. We provide results concerning buyers and sellers’ equilibrium strategies. In particular, our results point out that, by playing first, sellers are able to instigate an outcome that corresponds to the sellers’ optimal competitive equilibrium allocation for the true market.

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Notes

  1. Kelso and Crawford (1982) extend the analysis to many-to-one matching models. Sotomayor (2007) introduces the concept of a competitive equilibrium payoff for the multiple-partners assignment game and extends the previous results for this environment.

  2. Demange and Gale (1985) extend the theorem to a model where the utilities are continuous in money, but are not necessarily linear. Pérez-Castrillo and Sotomayor (2013) prove that buyers (respectively, sellers) do not have an incentive to misreport their valuation if the buyer-optimal (respectively, seller-optimal) competitive equilibrium is used by the designer in a one-to-many (respectively, many-to-one) buyer–seller market.

  3. Papers analyzing the consequences of manipulation in marriage and the college admission models, that is, in models where there are no prices, include Gale and Sotomayor (1985a, (1985b), Roth (1985), Roth and Sotomayor (1990), Sotomayor (2008), Kojima and Pathak (2009), Ma (2010), Sotomayor (2012), and Jaramillo et al. (2013).

  4. Tie breaking rules are common in mechanism design. See, for instance, Pérez-Castrillo and Sotomayor (2002) for the assignment game. Some papers use alternatives to tie breaking rules to ensure existence of equilibria. For instance, for the combinatorial assignment problem where monetary transfers are not allowed, Budish (2011) proposes the use of “approximate competitive equilibrium” notions.

  5. See Roth and Sotomayor (1990) for an overview of this model.

  6. We use the notation \(\sum _{j}\) for the sum over all \(b_{j}\) in \(B,\,\sum _{k}\) for the sum over all \(s_{k}\) in S and \(\sum _{j,k}\) for the sum over all \(b_{j}\) in B and \(s_{k}\) in S.

  7. This result was also proved in Sotomayor (2000) by using combinatorial arguments.

  8. See Demange (1982) and Leonard (1983).

  9. The definition of \(\sigma (r^{\prime } )\) implies that the buyers can signalize any optimal matching x for \(M(a^{\prime } (r^{\prime } ),\,r^{\prime } )\) by choosing \(\sigma (r^{\prime } ) =\,x\). More generally, they can signalize any subset S of optimal matchings for \(M(a^{\prime } (r^{\prime } ),\,r^{\prime } )\) by selecting \(\sigma _{jk}(r^{\prime } ) = 1\) if there is some matching x in S such that \(x_{jk}\,= 1\) and \(\sigma _{jk}(r^{\prime } ) = 0\) otherwise.

  10. We can also consider that each matching in this set has the same probability of being selected.

  11. We write \(a_{-j}^{\prime }\) to denote the decision profile for the buyers other than \(b_{j}\).

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Acknowledgments

We thank three reviewers and an Associate Editor for very helpful comments. Marilda Sotomayor is a research fellow at CNPq-Brazil. David Pérez-Castrillo is a fellow of MOVE and CESIfo. He acknowledges financial support from the Ministerio de Ciencia y Tecnología (ECO2015-63679-P), Generalitat de Catalunya (2014SGR-142), the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0563) and ICREA Academia.

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Pérez-Castrillo, D., Sotomayor, M. The outcome of competitive equilibrium rules in buyer–seller markets when the agents play strategically. Econ Theory 64, 99–119 (2017). https://doi.org/10.1007/s00199-016-0997-9

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Keywords

  • Assignment game
  • Competitive price
  • Optimal matching
  • Competitive rule

JEL Classification

  • C78
  • D78