Skip to main content
SpringerLink
Log in

Cost Sharing Mechanisms for Fair Pricing of Resource Usage

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We propose a simple and intuitive cost mechanism which assigns costs for the competitive usage of m resources by n selfish agents. Each agent has an individual demand; demands are drawn according to some probability distribution. The cost paid by an agent for a resource it chooses is the total demand put on the resource divided by the number of agents who chose that same resource. So, resources charge costs in an equitable, fair way, while each resource makes no profit out of the agents.

We call our model the Fair Pricing model. Its fair cost mechanism induces a non-cooperative game among the agents. To evaluate the Nash equilibria of this game, we introduce the Diffuse Price of Anarchy, as an extension of the Price of Anarchy that takes into account the probability distribution on the demands. We prove:

  1. •

    Pure Nash equilibria may not exist, unless all chosen demands are identical.

  2. •

    A fully mixed Nash equilibrium exists for all possible choices of the demands. Further on, the fully mixed Nash equilibrium is the unique Nash equilibrium in case there are only two agents.

  3. •

    In the worst-case choice of demands, the Price of Anarchy is Θ(n); for the special case of two agents, the Price of Anarchy is less than \(2-\frac{1}{m}\) .

  4. •

    Assume now that demands are drawn from a bounded, independent probability distribution, where all demands are identically distributed, and each demand may not exceed some (universal for the class) constant times its expectation. It happens that the constant is just 2 when each demand is distributed symmetrically around its expectation.

    We prove that, for asymptotically large games where the number of agents tends to infinity, the Diffuse Price of Anarchy is at most that universal constant. This implies the first separation between Price of Anarchy and Diffuse Price of Anarchy.

Towards the end, we consider two closely related cost sharing models, namely the Average Cost Pricing and the Serial Cost Sharing models, inspired by Economic Theory. In contrast to the Fair Pricing model, we prove that pure Nash equilibria do exist for both these models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, É., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 295–304 (October 2004)

  2. Billera, L., Heath, D.: Allocation of shared costs: a set of axioms yielding a unique procedure. Math. Oper. Res. 7, 32–39 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  3. Christodoulou, G., Koutsoupias, E., Nanavati, A.: Coordination mechanisms. In: Diaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) Proceedings of the 31st International Colloquium on Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 3142, pp. 345–357. Springer, New York (2004)

    Google Scholar 

  4. Fotakis, D., Kontogiannis, S., Koutsoupias, E., Mavronicolas, M., Spirakis, P.: The structure and complexity of Nash equilibria for a selfish routing game. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) Proceedings of the 29th International Colloquium on Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 2380, pp. 123–134. Springer, New York (2002)

    Chapter  Google Scholar 

  5. Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Rode, M.: Nash equilibria in discrete routing games with convex latency functions. In: Diaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) Proceedings of the 31st International Colloquium on Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 3142, pp. 645–657. Springer, New York (2004)

    Google Scholar 

  6. Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Spirakis, P.: Extreme Nash equilibria. In: Blundo, C., Laneve, C. (eds.) Proceedings of the 8th Italian Conference on Theoretical Computer Science. Lecture Notes in Computer Science, vol. 2841, pp. 1–20. Springer, New York (2003)

    Google Scholar 

  7. Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Spirakis, P.: Structure and complexity of extreme Nash equilibria. Theor. Comput. Sci. 343(1–2), 133–157 (2005). Special Issue titled Game Theory Meets Theoretical Computer Science, M. Mavronicolas and S. Abramsky, Guest eds.

    Article  MATH  Google Scholar 

  8. Hayrapetyan, A., Tardos, É., Wexler, T.: A network pricing game for selfish traffic. In: Proceedings of the 24th Annual ACM Symposium on Principles of Distributed Computing, pp. 284–291 (July 2005)

  9. Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58, 13–30 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  10. Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. In: Meinel, G., Tison, S. (eds.) Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 1563, pp. 404–413. Springer, New York (1999)

    Google Scholar 

  11. Koutsoupias, E., Papadimitriou, C.H.: Beyond competitive analysis. SIAM J. Comput. 30(1), 300–317 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Koutsoupias, E., Mavronicolas, M., Spirakis, P.: Approximate equilibria and ball fusion. Theory Comput. Syst. 36(6), 683–693 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lücking, T., Mavronicolas, M., Monien, B., Rode, M.: A new model for selfish routing. In: Diekert, V., Habib, M. (eds.) Proceedings of the 21st International Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 2996, pp. 547–558. Springer, New York (2004). Also, accepted to Theor. Comput. Sci.

    Google Scholar 

  14. Lücking, T., Mavronicolas, M., Monien, B., Rode, M., Spirakis, P., Vrto, I.: Which is the worst-case Nash equilibrium? In: Rovan, B., Vojtás, P. (eds.) Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 2747, pp. 551–561. Springer, New York (2003)

    Google Scholar 

  15. Mavronicolas, M., Spirakis, P.: The price of selfish routing. Algorithmica 48(1), 91–126 (June 2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Moulin, H., Shenker, S.: Serial cost sharing. Econometrica 60(5), 1009–1037 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  17. Moulin, H., Shenker, S.: Average cost pricing versus serial cost sharing: an axiomatic comparison. J. Econ. Theory 64(1), 178–201 (1994)

    Article  MATH  Google Scholar 

  18. Nash, J.F.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)

    Article  MathSciNet  Google Scholar 

  19. Papadimitriou, C.H.: Algorithms, games and the Internet. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 749–753 (July 2001)

  20. Shapiro, C., Varian, H.R.: Information Rules—A Strategic Guide to the Network Economy. Harvard Business School Press (1998)

  21. Spence, M.: Nonlinear prices and welfare. J. Mark. Res. 8(1), 1–18 (1977)

    Google Scholar 

  22. Varian, H.R.: Price discrimination and social welfare. Am. Econ. Rev. 75(4), 870–875 (1985)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Cyprus, Nicosia, 1678, Cyprus

    Marios Mavronicolas

  2. Department of Computer Engineering and Informatics, Patras University, Rion, Patras, 26500, Greece

    Panagiota N. Panagopoulou & Paul G. Spirakis

  3. Research Academic Computer Technology Institute, Patras University, Rion, Patras, 26500, Greece

    Panagiota N. Panagopoulou & Paul G. Spirakis

Authors
  1. Marios Mavronicolas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Panagiota N. Panagopoulou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paul G. Spirakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Panagiota N. Panagopoulou.

Additional information

A preliminary version of this work appeared in the Proceedings of the 1st International Workshop on Internet and Network Economics, X. Deng and Y. Ye, eds., Lecture Notes in Computer Science, vol. 3828, pp. 210–224, Springer, December 2005. This work has been partially supported by the EU within the 6th Framework Programme under contract 001907 “Dynamically Evolving, Large Scale Information Systems” ( \(\mathsf{DELIS}\) ), by the General Secretariat for Research and Technology of the Greek Ministry of Development within the programme \(\mathsf{PENED 2003}\) , and by research funds at University of Cyprus.

M. Mavronicolas is currently visiting Faculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, 33102 Paderborn, Germany.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mavronicolas, M., Panagopoulou, P.N. & Spirakis, P.G. Cost Sharing Mechanisms for Fair Pricing of Resource Usage. Algorithmica 52, 19–43 (2008). https://doi.org/10.1007/s00453-007-9108-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-007-9108-4

Keywords

Search

Navigation