Dynamic Pricing in Competitive Markets

  • Paresh Nakhe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10660)


Dynamic pricing of goods in a competitive environment to maximize revenue is a natural objective and has been a subject of research over the years. In this paper, we focus on a class of markets exhibiting the substitutes property with sellers having divisible and replenishable goods. Depending on the prices chosen, each seller observes a certain demand which is satisfied subject to the supply constraint. The goal of the seller is to price her good dynamically so as to maximize her revenue. For the static market case, when the consumer utility satisfies the gross substitutes CES property, we give a \(O(\sqrt{T})\) regret bound on the maximum loss in revenue of a seller using a modified version of the celebrated Online Gradient Descent algorithm by Zinkevich [17]. For a more specialized set of consumer utilities satisfying the iso-elasticity condition, we show that when each seller uses a regret-minimizing algorithm satisfying a certain technical property, the regret with respect to \((1-\alpha )\) times optimal revenue is bounded as \(O(T^{1/4} / \sqrt{\alpha })\). We extend this result to markets with dynamic supplies and prove a corresponding dynamic regret bound, whose guarantee deteriorates smoothly with the inherent instability of the market. As a side-result, we also extend the previously known convergence results of these algorithms in a general game to the dynamic setting.


Dynamic pricing Online convex optimization 



I would like to thank Martin Hoefer and Yun Kuen Cheung for the helpful discussions that helped shape this paper. I would also like to thank the anonymous reviewers for their helpful comments.


  1. 1.
    Besbes, O., Zeevi, A.: Dynamic pricing without knowing the demand function: risk bounds and near-optimal algorithms. Oper. Res. 57(6), 1407?1420 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Besbes, O., Zeevi, A.: On the minimax complexity of pricing in a changing environment. Oper. Res. 59(1), 66?79 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Broder, J., Rusmevichientong, P.: Dynamic pricing under a general parametric choice model. Oper. Res. 60(4), 965?980 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Carvalho, A.X., Puterman, M.L.: Learning and pricing in an internet environment with binomial demands. J. Revenue Pricing Manage. 3(4), 320?336 (2005)CrossRefGoogle Scholar
  5. 5.
    Chen, M., Chen, Z.-L.: Recent developments in dynamic pricing research: multiple products, competition, and limited demand information. Prod. Oper. Manage. 24(5), 704?731 (2015)CrossRefGoogle Scholar
  6. 6.
    den Boer, A.V., Zwart, B.: Simultaneously learning and optimizing using controlled variance pricing. Manage. Sci. 60(3), 770?783 (2013)CrossRefGoogle Scholar
  7. 7.
    Gallego, G., Ming, H.: Dynamic pricing of perishable assets under competition. Manage. Sci. 60(5), 1241?1259 (2014)CrossRefGoogle Scholar
  8. 8.
    Gallego, G., Wang, R.: Multiproduct price optimization and competition under the nested logit model with product-differentiated price sensitivities. Oper. Res. 62(2), 450?461 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Keskin, B., Zeevi, A.: Dynamic pricing with an unknown demand model: asymptotically optimal semi-myopic policies. Oper. Res. 62(5), 1142?1167 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Keskin, N.B., Zeevi, A.: Chasing demand: learning and earning in a changing environment. Math. Oper. Res. 42(2), 277?307 (2017)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kleinberg, R., Leighton, T.: The value of knowing a demand curve: bounds on regret for online posted-price auctions. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003, pp. 594?605. IEEE Computer Society (2003)Google Scholar
  12. 12.
    Mertikopoulos, P.: Learning in concave games with imperfect information (2016)Google Scholar
  13. 13.
    Nakhe, P.: Dynamic pricing in competitive markets. arXiv preprint arXiv:1709.04960 (2017)
  14. 14.
    Rakhlin, A., Sridharan, K.: Online learning with predictable sequences. In: Shai, S.-S., Steinwart, I. (eds.), Proceedings of the 26th Annual Conference on Learning Theory. Proceedings of Machine Learning Research, PMLR, Princeton, NJ, USA, vol. 30, pp. 993?1019, 12?14 June 2013Google Scholar
  15. 15.
    Ramskov, J., Munksgaard, J.: Elasticities-a theoretical introduction. Balmorel Project (2001)Google Scholar
  16. 16.
    Syrgkanis, V., Agarwal, A., Luo, H., Schapire, R.E.: Fast convergence of regularized learning in games. In: Proceedings of the 28th International Conference on Neural Information Processing Systems, NIPS 2015, pp. 2989?2997, Cambridge, MIT Press (2015)Google Scholar
  17. 17.
    Zinkevich, M.: Online convex programming and generalized infinitesimal gradient ascent. In: Proceedings of the Twentieth International Conference on Machine Learning, ICML 2003, pp. 928?935. AAAI Press (2003)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Computer ScienceGoethe UniversityFrankfurt (Main)Germany

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