Skip to main content
SpringerLink
Log in

Competitive Cost Sharing with Economies of Scale

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We consider a general class of non-cooperative buy-at-bulk cost sharing games, in which k players make investments to purchase a set of resources. Each resource has a certain cost and must bought to be available to the players. Each player has a certain constraint on the number and types of resources that she needs to have available, and she can specify payments to make a resource available to her. She strives to fulfill her constraint with the smallest investment possible. Our model includes a natural economy of scale: for a subset of players capacity must be installed at the resources, and the cost increase for a resource r is composed of a fixed price c(r) and a global concave capacity function g. This cost can be shared arbitrarily between players.

We consider the existence and total cost of pure-strategy exact and approximate Nash equilibria. In general, prices of anarchy and stability depend heavily on the economy of scale and are Θ(k/g(k)). For non-linear functions g pure Nash equilibria might not exist, and deciding their existence is \(\textsf{NP}\)-hard. For subclasses of games corresponding to covering problems, primal-dual methods can be applied to derive cheap and stable approximate Nash equilibria in polynomial time. In addition, for singleton games optimal Nash equilibria exist. In this case expensive exact as well as cheap approximate Nash equilibria can be computed in polynomial time. Most of these results can be extended to games based on facility location problems.

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. Albers, S.: On the value of coordination in network design. SIAM J. Comput. 38(6), 2273–2302 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrews, M.: Hardness of buy-at-bulk network design. In: Proc. of the 45th Symp. on Foundations of Computer Science (FOCS), pp. 115–124 (2004)

  3. Anshelevich, E., Caskurlu, B.: Exact and approximate equilibria for optimal groupnetwork formation. In: Proc. of the 17th European Symposium on Algorithms (ESA), pp. 239–250 (2009)

  4. Anshelevich, E., Caskurlu, B.: Price of stability in survivable network design. In: Proc. of the 2nd Intl. Symp. on Algorithmic Game Theory (SAGT), pp. 208–219 (2009)

  5. Anshelevich, E., Dasgupta, A., Kleinberg, J., Roughgarden, T., Tardos, É., Wexler, T.: The price of stability for network design with fair cost allocation. SIAM J. Comput. 38(4), 1602–1623 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Anshelevich, E., Dasgupta, A., Tardos, É., Wexler, T.: Near-optimal network design with selfish agents. Theory Comput. 4, 77–109 (2008)

    Article  MathSciNet  Google Scholar 

  7. Awerbuch, B., Azar, Y.: Buy-at-bulk network design. In: Proc. of the 38th Symp. on Foundations of Computer Science (FOCS), pp. 542–547 (1997)

  8. Biló, V., Fanelli, A., Flammini, M., Moscardelli, L.: When ignorance helps: Graphical multicast cost sharing games. In: Proc. of the 33rd Intl. Symp. on Mathematical Foundations of Computer Science (MFCS), pp. 108–119 (2008)

  9. Cardinal, J., Hoefer, M.: Selfish service installation in networks. In: Proc. of the 2nd Intl. Workshop on Internet & Network Economics (WINE), pp. 174–185 (2006)

  10. Charikar, M., Karloff, H., Mathieu, C., Naor, J., Saks, M.: Online multicast with egalitarian cost sharing. In: Proc. of the 20th Symp. on Parallelism in Algorithms and Architectures (SPAA), pp. 70–76 (2008)

  11. Chekuri, C., Taghi Hajiaghayi, M., Kortarz, G., Salavatipour, M.: Approximation algorithms for non-uniform buy-at-bulk network design. In: Proc. of the 47th Symp. on Foundations of Computer Science (FOCS), pp. 677–686 (2006)

  12. Chekuri, C., Chuzhoy, J., Lewin-Eytan, L., Naor, J., Orda, A.: Non-cooperative multicast and facility location games. IEEE J. Sel. Areas Commun. 25(6), 1193–1206 (2007)

    Article  Google Scholar 

  13. Chekuri, C., Taghi Hajiaghayi, M., Kortsarz, G., Salavatipour, M.: Approximation algorithms for node-weighted buy-at-bulk networks. In: Proc. of the 18th Symp. on Discrete Algorithms (SODA) (2007)

  14. Chen, H.-L., Roughgarden, T.: Network design with weighted players. Theory Comput. Syst. 45(2), 302–324 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen, H.-L., Roughgarden, T., Valiant, G.: Designing networks with good equilibria. In: Proc. of the 19th Symp. on Discrete Algorithms (SODA), pp. 854–863 (2008)

  16. Deng, X., Ibaraki, T., Nagamochi, H., Zhang, W.: Totally balanced combinatorial optimization games. Math. Program. 87(3), 441–452 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Devanur, N., Mihail, M., Vazirani, V.: Strategyproof cost-sharing mechanisms for set cover and facility location problems. Decis. Support Syst. 39(1), 11–22 (2005)

    Article  Google Scholar 

  18. Eiselt, H., Laporte, G., Thisse, J.-F.: Competitive location models: a framework and bibliography. Transp. Sci. 27, 44–54 (1993)

    Article  MATH  Google Scholar 

  19. Epstein, A., Feldman, M., Mansour, Y.: Strong equilibrium in cost sharing connection games. Games Econ. Behav. 67(1), 51–68 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fanelli, A., Flammini, M., Melideo, G., Moscardelli, L.: Multicast transmissions in non-cooperative networks with a limited number of selfish moves. In: Proc. of the 31st Intl. Symp. on Mathematical Foundations of Computer Science (MFCS), pp. 363–374 (2006)

  21. Goemans, M., Skutella, M.: Cooperative facility location games. J. Algorithms 50(2), 194–214 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gupta, A., Kumar, A., Pál, M., Roughgarden, T.: Approximation via cost sharing: Simpler and better approximation algorithms for network design. J. ACM 54(3), 11 (2007)

    MathSciNet  Google Scholar 

  23. Taghi Hajiaghayi, M., Mahdian, M., Mirrokni, V.: The facility location problem with general cost functions. Networks 42(1), 42–47 (2003)

    Article  MathSciNet  Google Scholar 

  24. Hoefer, M.: Non-cooperative facility location and covering games. In: Proc. of the 17th Intl. Symp. on Algorithms and Computation (ISAAC), pp. 369–378 (2006)

  25. Hoefer, M.: Cost Sharing and Clustering under Distributed Competition. Ph.D. thesis, Lehrstuhl Algorithmik, Universität Konstanz (2007)

  26. Hoefer, M.: Competitive cost sharing with economies of scale. In: Proc. of the 8th Latin American Theoretical Informatics Conference (LATIN), pp. 339–349 (2008)

  27. Hoefer, M.: Non-cooperative tree creation. Algorithmica 53(1), 104–131 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Immorlica, N., Mahdian, M., Mirrokni, V.: Limitations of cross-monotonic cost sharing schemes. ACM Trans. Algorithms 4(2) (2008). Special Issue SODA 2005

  29. Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J. ACM 50(6), 795–824 (2003)

    MathSciNet  Google Scholar 

  30. Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: Proc. of the 16th Symp. on Theoretical Aspects of Computer Science (STACS), pp. 404–413 (1999)

  31. Leonardi, S., Sankowsi, P.: Network formation games with local coalitions. In: Proc. of the 26th Symp. on Principles of Distributed Computing (PODC), pp. 299–305 (2007)

  32. Li, X.-Y., Sun, Z., Wang, W.: Cost sharing and strategyproof mechanisms for set cover games. In: Proc. of the 22nd Symp. on Theoretical Aspects of Computer Science (STACS), pp. 218–230 (2005)

  33. Miller, T., Friesz, T., Tobin, R.: Equilibrium Facility Location in Networks. Springer, Berlin (1996)

    Google Scholar 

  34. Pál, M., Tardos, É.: Group strategyproof mechanisms via primal-dual algorithms. In: Proc. of the 44th Symp. on Foundations of Computer Science (FOCS), pp. 584–593 (2003)

  35. Sun, Z., Li, X.-Y., Wang, W., Chu, X.: Mechanism design for set cover games when elements are agents. In: Proc. of the 1st Intl. Conf. on Algorithmic Applications in Management (AAIM), pp. 360–369 (2005)

  36. Vazirani, V.: Approximation Algorithms. Springer, Berlin (2000)

    Google Scholar 

  37. Vetta, A.: Nash equilibria in competitive societies with application to facility location, traffic routing and auctions. In: Proc. of the 43th Symp. on Foundations of Computer Science (FOCS), pp. 416 (2002)

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, RWTH Aachen, Aachen, Germany

    Martin Hoefer

Authors
  1. Martin Hoefer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Hoefer.

Additional information

Supported by DFG through UMIC Research Centre at RWTH Aachen University.

An extended abstract of this work has appeared in LATIN 2008 [26].

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hoefer, M. Competitive Cost Sharing with Economies of Scale. Algorithmica 60, 743–765 (2011). https://doi.org/10.1007/s00453-009-9367-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-009-9367-3

Keywords

Search

Navigation