Online Allocation and Pricing with Economies of Scale

  • Avrim Blum
  • Yishay Mansour
  • Liu Yang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9470)


Allocating multiple goods to customers in a way that maximizes some desired objective is a fundamental part of Algorithmic Mechanism Design. We consider here the problem of offline and online allocation of goods that have economies of scale, or decreasing marginal cost per item for the seller. In particular, we analyze the case where customers have unit-demand and arrive one at a time with valuations on items, sampled iid from some unknown underlying distribution over valuations. Our strategy operates by using an initial sample to learn enough about the distribution to determine how best to allocate to future customers, together with an analysis of structural properties of optimal solutions that allow for uniform convergence analysis. We show, for instance, if customers have \(\{0,1\}\) valuations over items, and the goal of the allocator is to give each customer an item he or she values, we can efficiently produce such an allocation with cost at most a constant factor greater than the minimum over such allocations in hindsight, so long as the marginal costs do not decrease too rapidly. We also give a bicriteria approximation to social welfare for the case of more general valuation functions when the allocator is budget constrained.



Avrim is supported in part by the National Science Foundation under grants CCF-1101215, CCF-1116892, and IIS-1065251. Yishay is supported in part by a grant from the Israel Science Foundation, a grant from the United States-Israel Binational Science Foundation (BSF), a grant by Israel Ministry of Science and Technology and the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). For part of this work, Liu was supported in part by NSF grant IIS-1065251 and a Google Core AI grant.


  1. 1.
    Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: The online set cover problem. SIAM J. Comput. 39(2), 361–370 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blum, A., Gupta, A., Mansour, Y., Sharma, A.: Welfare and profit maximization with production costs. In: FOCS, pp. 77–86 (2011)Google Scholar
  3. 3.
    Devanur, N.R., Hayes, T.P.: The adwords problem: Online keyword matching with budgeted bidders under random permutations. In: Proc. ACM EC, EC 2009, pp. 71–78 (2009)Google Scholar
  4. 4.
    Devanur, N.R., Jain, K.: Online matching with concave returns. In: Proc. STOC, pp. 137–144 (2012)Google Scholar
  5. 5.
    Goel, G., Mehta, A.: Online budgeted matching in random input models with applications to adwords. In: Proc. SODA, pp. 982–991 (2008)Google Scholar
  6. 6.
    Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for on-line bipartite matching. In: Proc. STOC, pp. 352–358 (1990)Google Scholar
  7. 7.
    Khuller, S., Moss, A., Naor, J.: The budgeted maximum coverage problem. Inf. Process. Lett. 70(1), 39–45 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mehta, A., Saberi, A., Vazirani, U., Vazirani, V.: Adwords and generalized online matching. J. ACM 54(5), 22 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V.: Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Open Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA
  2. 2.Computer Science DepartmentTel Aviv UniversityTel AvivIsrael
  3. 3.IBM T. J. Watson Research CenterYorktown HeightsUSA

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