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A Regret Analysis of Bilateral Trade

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Listed:
  • Nicol`o Cesa-Bianchi

    (TSE)

  • Tommaso Cesari

    (TSE)

  • Roberto Colomboni

    (IIT)

  • Federico Fusco
  • Stefano Leonardi

Abstract

Bilateral trade, a fundamental topic in economics, models the problem of intermediating between two strategic agents, a seller and a buyer, willing to trade a good for which they hold private valuations. Despite the simplicity of this problem, a classical result by Myerson and Satterthwaite (1983) affirms the impossibility of designing a mechanism which is simultaneously efficient, incentive compatible, individually rational, and budget balanced. This impossibility result fostered an intense investigation of meaningful trade-offs between these desired properties. Much work has focused on approximately efficient fixed-price mechanisms, i.e., Blumrosen and Dobzinski (2014; 2016), Colini-Baldeschi et al. (2016), which have been shown to fully characterize strong budget balanced and ex-post individually rational direct revelation mechanisms. All these results, however, either assume some knowledge on the priors of the seller/buyer valuations, or a black box access to some samples of the distributions, as in D{\"u}tting et al. (2021). In this paper, we cast for the first time the bilateral trade problem in a regret minimization framework over rounds of seller/buyer interactions, with no prior knowledge on the private seller/buyer valuations. Our main contribution is a complete characterization of the regret regimes for fixed-price mechanisms with different models of feedback and private valuations, using as benchmark the best fixed price in hindsight. More precisely, we prove the following bounds on the regret: $\bullet$ $\widetilde{\Theta}(\sqrt{T})$ for full-feedback (i.e., direct revelation mechanisms); $\bullet$ $\widetilde{\Theta}(T^{2/3})$ for realistic feedback (i.e., posted-price mechanisms) and independent seller/buyer valuations with bounded densities; $\bullet$ $\Theta(T)$ for realistic feedback and seller/buyer valuations with bounded densities; $\bullet$ $\Theta(T)$ for realistic feedback and independent seller/buyer valuations; $\bullet$ $\Theta(T)$ for the adversarial setting.

Suggested Citation

  • Nicol`o Cesa-Bianchi & Tommaso Cesari & Roberto Colomboni & Federico Fusco & Stefano Leonardi, 2021. "A Regret Analysis of Bilateral Trade," Papers 2102.08754, arXiv.org.
  • Handle: RePEc:arx:papers:2102.08754
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    References listed on IDEAS

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    1. Nicolò Cesa-Bianchi & Gábor Lugosi & Gilles Stoltz, 2006. "Regret Minimization Under Partial Monitoring," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 562-580, August.
    2. Maxime C. Cohen & Ilan Lobel & Renato Paes Leme, 2020. "Feature-Based Dynamic Pricing," Management Science, INFORMS, vol. 66(11), pages 4921-4943, November.
    3. William Vickrey, 1961. "Counterspeculation, Auctions, And Competitive Sealed Tenders," Journal of Finance, American Finance Association, vol. 16(1), pages 8-37, March.
    4. Myerson, Roger B. & Satterthwaite, Mark A., 1983. "Efficient mechanisms for bilateral trading," Journal of Economic Theory, Elsevier, vol. 29(2), pages 265-281, April.
    5. Josef Broder & Paat Rusmevichientong, 2012. "Dynamic Pricing Under a General Parametric Choice Model," Operations Research, INFORMS, vol. 60(4), pages 965-980, August.
    6. Devanur, Nikhil R. & Peres, Yuval & Sivan, Balasubramanian, 2019. "Perfect Bayesian Equilibria in repeated sales," Games and Economic Behavior, Elsevier, vol. 118(C), pages 570-588.
    7. Gábor Bartók & Dean P. Foster & Dávid Pál & Alexander Rakhlin & Csaba Szepesvári, 2014. "Partial Monitoring---Classification, Regret Bounds, and Algorithms," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 967-997, November.
    8. Arnoud V. den Boer & N. Bora Keskin, 2020. "Discontinuous Demand Functions: Estimation and Pricing," Management Science, INFORMS, vol. 66(10), pages 4516-4534, October.
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    1. Nicol`o Cesa-Bianchi & Tommaso Cesari & Roberto Colomboni & Federico Fusco & Stefano Leonardi, 2021. "Bilateral Trade: A Regret Minimization Perspective," Papers 2109.12974, arXiv.org.

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