Quasi-Proportional Mechanisms: Prior-Free Revenue Maximization

  • Vahab Mirrokni
  • S. Muthukrishnan
  • Uri Nadav
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6034)


Inspired by Internet ad auction applications, we study the problem of allocating a single item via an auction when bidders place very different values on the item. We formulate this as the problem of prior-free auction and focus on designing a simple mechanism that always allocates the item. Rather than designing sophisticated pricing methods like prior literature, we design better allocation methods. In particular, we propose quasi-proportional allocation methods in which the probability that an item is allocated to a bidder depends (quasi-proportionally) on the bids.

We prove that corresponding games for both all-pay and winners-pay quasi-proportional mechanisms admit pure Nash equilibria and this equilibrium is unique. We also give an algorithm to compute this equilibrium in polynomial time. Further, we show that the revenue of the auctioneer is promisingly high compared to the ultimate, i.e., the highest value of any of the bidders, and show bounds on the revenue of equilibria both analytically, as well as using experiments for specific quasi-proportional functions. This is the first known revenue analysis for these natural mechanisms (including the special case of proportional mechanism which is common in network resource allocation problems).


Nash Equilibrium Reserve Price Price Auction Pure Nash Equilibrium Reserve Prex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abernethy, J., Hazan, E., Rakhlin, A.: Competing in the Dark: An Efficient Algorithm for Bandit Linear Optimization. In: COLT 2008 (2008)Google Scholar
  2. 2.
    Baliga, S., Vohra, R.: Market Research and Market Design (2003)Google Scholar
  3. 3.
    Baye, M., Kovenock, D., de Vried, C.: The all-pay auction with Complete Information. Economic Theory 8, 291–305Google Scholar
  4. 4.
    Che, Y., Gale, I.: Expected revenue of all-pay auctions and first-price sealed-bid auctions with budget constraints. Economic Letters, 373–379 (1996)Google Scholar
  5. 5.
    Clarke, E.: Multipart pricing of public goods. Public Choice 11, 17–33 (1971)CrossRefGoogle Scholar
  6. 6.
    Even Dar, E., Mansour, Y., Nadav, U.: On the convergence of regret minimization dynamics in concave games. In: STOC 2009 (2009)Google Scholar
  7. 7.
    Fiat, A., Goldberg, A.V., Hartline, J.D., Karlin, A.R.: Competitive generalized auctions. In: STOC 2002, pp. 72–81 (2002)Google Scholar
  8. 8.
    Groves, T.: Incentives in teams. Econometrica 41(4), 617–631 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hajek, B., Gopalakrishnan, G.: Do greedy autonomous systems make for a sensible internet? Presented at the Conference on Stochastic Networks, Stanford University (2002)Google Scholar
  10. 10.
    Hartline, J., Karline, A.: Profit Maximization in Mechanism Design. In: Algorithmic Game Theory (October 2007)Google Scholar
  11. 11.
    Johari, R., Tsitsiklis, J.N.: Efficiency loss in a network resource allocation game. Mathematics of Operations Research 29(3), 407–435 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kalai, A., Vempala, S.: Efficient algorithms for online decision problems. J. Comput. Syst. Sci. 71(3), 291–307 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kelly, F.: Charging and rate control for elastic traffic. European Transactions on Telecommunications 8, 33–37 (1997)CrossRefGoogle Scholar
  14. 14.
    Liu, D., Chen, J.: Designing online auctions with past performance information. Decision Support Systems 42, 1307–1320 (2006)CrossRefGoogle Scholar
  15. 15.
    Lu, P., Teng, S.-H., Yu, C.: Truthful Auctions with Optimal Profit. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 27–36. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Myerson, R.: Optimal auction design. Mathematics of Operations Research 6, 58–73 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Rosen, J.: Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 520–534 (1965)Google Scholar
  18. 18.
    Segal, I.: Optimal Pricing Mechanisms with Unknown Demand. American Economic Review 93(3), 509–529 (2003)CrossRefGoogle Scholar
  19. 19.
    Vickrey, W.: Counterspeculation, auctions and competitive-sealed tenders. Finance 16(1), 8–37 (1961)CrossRefGoogle Scholar
  20. 20.
    Zinkevich, M.: Online convex programming and generalized infinitesimal gradient ascent. In: Twentieth International Conference on Machine Learning (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Vahab Mirrokni
    • 1
  • S. Muthukrishnan
    • 1
  • Uri Nadav
    • 2
  1. 1.Google ResearchNew York
  2. 2.Tel-aviv University 

Personalised recommendations