Fixed Price Approximability of the Optimal Gain from Trade

  • Riccardo Colini-Baldeschi
  • Paul Goldberg
  • Bart de Keijzer
  • Stefano Leonardi
  • Stefano Turchetta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10660)

Abstract

Bilateral trade is a fundamental economic scenario comprising a strategically acting buyer and seller (holding an item), each holding valuations for the item, drawn from publicly known distributions. It was recently shown that the only mechanisms that are simultaneously dominant strategy incentive compatible, strongly budget balanced, and ex-post individually rational, are fixed price mechanisms, i.e., mechanisms that are parametrised by a price p, and trade occurs if and only if the valuation of the buyer is at least p and the valuation of the seller is at most p. The gain from trade (GFT) is the increase in welfare that results from applying a mechanism. We study the GFT achievable by fixed price mechanisms. We explore this question for both the bilateral trade setting and a double auction setting where there are multiple i.i.d. unit demand buyers and sellers. We first identify a fixed price mechanism that achieves a GFT of at least 2 / r times the optimum, where r is the probability that the seller’s valuation does not exceed that of the buyer’s valuation. This extends a previous result by McAfee. Subsequently, we improve this approximation factor in an asymptotic sense, by showing that a more sophisticated rule for setting the fixed price results in a GFT within a factor \(O(\log (1/r))\) of the optimum. This is asymptotically the best approximation factor possible. For the double auction setting, we present a fixed price mechanism that achieves for all \(\epsilon > 0\) a gain from trade of at least \((1-\epsilon )\) times the optimum with probability \(1 - 2/e^{\#T \epsilon ^2 /2}\), where \(\#T\) is the expected number of trades of the mechanism. This can be interpreted as a “large market” result: Full efficiency is achieved in the limit, as the market gets thicker.

Notes

Acknowledgements

We thank Tim Roughgarden for helpful discussions at the early stages of this work.

References

  1. 1.
    Blumrosen, L., Dobzinski, S.: Reallocation mechanisms. In: Proceedings of the 15th ACM Conference on Economics and Computation (EC), p. 617. ACM (2014)Google Scholar
  2. 2.
    Blumrosen, L., Dobzinski, S.: (Almost) Efficient Mechanisms for Bilateral Trading. ArXiv/CoRR, abs/1604.04876 (2016)Google Scholar
  3. 3.
    Blumrosen, L., Mizrahi, Y.: Approximating gains-from-trade in bilateral trading. In: Cai, Y., Vetta, A. (eds.) WINE 2016. LNCS, vol. 10123, pp. 400–413. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-54110-4_28 CrossRefGoogle Scholar
  4. 4.
    Brustle, J., Cai, Y., Wu, F., Zhao, M.: Approximating gains from trade in two-sided markets via simple mechanisms. In: Proceedings of the 2017 ACM Conference on Economics and Computation (EC), pp. 589–590 (2017)Google Scholar
  5. 5.
    Colini-Baldeschi, R., de Keijzer, B., Leonardi, S., Turchetta, S.: Approximately efficient double auctions with strong budget balance. In: Proceedings of the 27th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1424–1443. SIAM (2016)Google Scholar
  6. 6.
    Colini-Baldeschi, R., Goldberg, P.W., de Keijzer, B., Leonardi, S., Turchetta, S., Roughgarden, T.: Approximately efficient two-sided combinatorial auctions. In: Proceedings of the 18th ACM Conference on Economics and Computation (EC), pp. 591–608. ACM (2017)Google Scholar
  7. 7.
    Deng, X., Goldberg, P.W., Tang, B., Zhang, J.: Revenue maximization in a Bayesian double auction market. Theor. Comput. Sci. 539, 1–12 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Deshmukh, K., Goldberg, A.V., Hartline, J.D., Karlin, A.R.: Truthful and competitive double auctions. In: Möhring, R., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 361–373. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-45749-6_34 CrossRefGoogle Scholar
  9. 9.
    Feldman, M., Gonen, R.: Markets with strategic multi-minded mediators. ArXiv/ CoRR, abs/1603.08717 (2016)Google Scholar
  10. 10.
    Feldman, M., Gonen, R.: Online truthful mechanisms for multi-sided markets. CoRR, abs/1604.04859 (2016)Google Scholar
  11. 11.
    Gerstgrasser, M., Goldberg, P.W., Koutsoupias, E.: Revenue maximization for market intermediation with correlated priors. In: Gairing, M., Savani, R. (eds.) SAGT 2016. LNCS, vol. 9928, pp. 273–285. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53354-3_22 CrossRefGoogle Scholar
  12. 12.
    Giannakopoulos, Y., Koutsoupias, E., Lazos, P.: Online Market Intermediation. ArXiv/CoRR, abs/1703.09279 (2017)Google Scholar
  13. 13.
    Gresik, T.A., Satterthwaite, M.A.: The rate at which a simple market converges to efficiency as the number of traders increases: an asymptotic result for optimal trading mechanisms. J. Econ. Theory 48(1), 304–332 (1989)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    McAfee, P.R.: A dominant strategy double auction. J. Econ. Theory 56(2), 434–450 (1992)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    McAfee, R.P.: The gains from trade under fixed price mechanisms. Appl. Econ. Res. Bull. 1, 1–10 (2008)Google Scholar
  16. 16.
    Myerson, R.B., Satterthwaite, M.A.: Efficient mechanisms for bilateral trading. J. Econ. Theory 29(2), 265–281 (1983)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Rustichini, A., Satterthwaite, M.A., Williams, S.R.: Convergence to efficiency in a simple market with incomplete information. Econometrica 62(5), 1041–1063 (1994)CrossRefMATHGoogle Scholar
  18. 18.
    Satterthwaite, M.A., Williams, S.R.: The optimality of a simple market mechanism. Econometrica 70(5), 1841–1863 (2002)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Segal-Halevi, E., Hassidim, A., Aumann, Y.: A random-sampling double-auction mechanism. ArXiv/CoRR, abs/1604.06210 (2016)Google Scholar
  20. 20.
    Segal-Halevi, E., Hassidim, A., Aumann, Y.: SBBA: a strongly-budget-balanced double-auction mechanism. In: Gairing, M., Savani, R. (eds.) SAGT 2016. LNCS, vol. 9928, pp. 260–272. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53354-3_21 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Riccardo Colini-Baldeschi
    • 1
  • Paul Goldberg
    • 2
  • Bart de Keijzer
    • 3
  • Stefano Leonardi
    • 4
  • Stefano Turchetta
    • 5
  1. 1.LUISS RomeRomeItaly
  2. 2.University of OxfordOxfordEngland
  3. 3.Centrum Wiskunde & Informatica (CWI)AmsterdamNetherlands
  4. 4.Sapienza University of RomeRomeItaly
  5. 5.KPMG ItalyRomeItaly

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