Advertisement

Theory of Computing Systems

, Volume 59, Issue 3, pp 440–475 | Cite as

Assignment Games with Conflicts: Robust Price of Anarchy and Convergence Results via Semi-Smoothness

  • Elliot Anshelevich
  • John Postl
  • Tom Wexler
Article

Abstract

We study assignment games in which jobs select machines, and in which certain pairs of jobs may conflict, which is to say they may incur an additional cost when they are both assigned to the same machine, beyond that associated with the increase in load. Questions regarding such interactions apply beyond allocating jobs to machines: when people in a social network choose to align themselves with a group or party, they typically do so based upon not only the inherent quality of that group, but also who amongst their friends (or enemies) chooses that group as well. We show how semi-smoothness, a recently introduced generalization of smoothness, is necessary to find tight bounds on the robust price of anarchy, and thus on the quality of correlated and Nash equilibria, for several natural job-assignment games with interacting jobs. For most cases, our bounds on the robust price of anarchy are either exactly 2 or approach 2. We also prove new convergence results implied by semi-smoothness for our games. Finally we consider coalitional deviations, and prove results about the existence and quality of strong equilibrium.

Keywords

Price of anarchy Group formation Smooth games Congestion games Cut games Games in networks 

Notes

Acknowledgments

This work was supported in part by NSF grants CCF-0914782, CCF-1101495, and CNS-1017932. We thank Carlos Varela for illuminating discussions on the subject of job assignment.

References

  1. 1.
    Augustine, J., Chen, N., Elkind, E., Fanelli, A., Gravin, N., Shiryaev, D.: Dynamics of profit-sharing games. In: IJCAI, pp. 37–42 (2011)Google Scholar
  2. 2.
    Awerbuch, B., Azar, Y., Epstein, A., Mirrkoni, V.S., Skopalik, A.: Fast convergence to nearly optimal solutions in potential games. In: Proceedings of EC, pp. 264–273 (2008)Google Scholar
  3. 3.
    Bachrach, Y., Syrgkanis, V., Tardos, É., Vojnović, M.: Strong Price of Anarchy, Utility Games, and Coalitional Dynamics. In: Proceedings of SAGT (2014)Google Scholar
  4. 4.
    Bhalgat, A., Chakraborty, T., Khanna, S.: Approximating pure nash equilibrium in cut, party affiliation, and satisfiability games. In: Proceedings of EC, pp. 73–82 (2010)Google Scholar
  5. 5.
    Blum, A., Hajiaghayi, M., Ligett, K., Roth, A.: Regret minimization and the price of total anarchy. In: Proceedings of STOC, pp. 373–382 (2008)Google Scholar
  6. 6.
    Caragiannis, I., Flammini, M., Kaklamansis, C., Kanellopoulos, P., Moscardelli, L.: Tight bounds for selfish and greedy load balancing. In: Proceedings of ICALP, pp. 311–322 (2006)Google Scholar
  7. 7.
    Caragiannis, I., Kaklamansis, C., Kanellopoulos, P., Kyropoulou, M., Lucier, B., Leme, R.P., Tardos, É.: Bounding the inefficiency of outcomes in generalized second price auctions. J. Econ. Theory 156, 343–388 (2015)Google Scholar
  8. 8.
    Christodoulou, G., Koutsoupias, E.: The price of anarchy of finite congestion games. In: Proceedings of STOC, pp. 67–73 (2005)Google Scholar
  9. 9.
    Christodoulou, G., Mirrokni, V.S., Sidiropoulos, A.: Convergence and approximation in potential games. In: Proceedings of STACS, pp. 349–360 (2006)Google Scholar
  10. 10.
    Fanelli, A., Flammini, M., Moscardelli, L.: The speed of convergence in congestion games under best-response dynamics. In: Proceedings of ICALP, pp. 796–807 (2008)Google Scholar
  11. 11.
    Fanelli, A., Moscardelli, L.: On best response dynamics in weighted congestion games with polynomial delays. In: Proceedings of WINE, pp. 55–66 (2009)Google Scholar
  12. 12.
    Feldman, M., Lewin-Eytan, L., Naor, J.S.: Hedonic clustering games. In: Proceedings of SPAA, pp. 267–276 (2012)Google Scholar
  13. 13.
    Goemans, M.X., Li, L., Mirrokni, V.S., Thottan, M.: Market sharing games applied to content distribution in ad-hoc networks. In: Proceedings of JSAC, vol. 24, no. 5, pp. 1020–1033 (2006)Google Scholar
  14. 14.
    Gourvès, L., Monnot, J.: On strong equilibria in the max cut game. In: Proceedings of WINE, pp. 608–615 (2009)Google Scholar
  15. 15.
    Gourvès, L., Monnot, J.: The max k-cut game and its strong equilibria. In: Proceedings of TAMC, pp. 234–246 (2010)Google Scholar
  16. 16.
    Hoefer, M.: Cost Sharing and Clustering under Distributed Competition. PhD Thesis, Universität Konstanz (2007)Google Scholar
  17. 17.
    Jackson, M.O., Yariv, L.: Diffusion of behavior and equilibrium properties in network games. Am. Econ. Rev. 97(2), 92–98 (2007)CrossRefGoogle Scholar
  18. 18.
    Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: KDD ’03, pp 137–146Google Scholar
  19. 19.
    Kleinberg, J.: Cascading behavior in networks: algorithmic and economic issues. In: Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V. (eds.) Algorithmic Game Theory, pp. 613–632 (2007)Google Scholar
  20. 20.
    Kothari, A., Suri, S., Tóth, C.D., Zhou, Y.: Congestion games, load balancing, and the price of anarchy. In: Proceedings of CAAN, pp. 13–27 (2004)Google Scholar
  21. 21.
    Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: Proceedings of STACS, pp. 404–413 (1999)Google Scholar
  22. 22.
    Lucier, B., Leme, R.P.: GSP auctions with correlated types. In: Proceedings of EC, pp. 71–80 (2011)Google Scholar
  23. 23.
    Mirrokni, V.S., Vetta, A.: Convergence issues in competitive games. In: Proceedings of RANDOM-APPROX, pp. 183–194 (2014)Google Scholar
  24. 24.
    Monderer, D., Shapley, L.S.: Potential games. Games Econ. Behav. 14(1), 124–143 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Morris, S.: Contagion. Rev. Econ. Stud. 67(1), 57–78 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V. (eds.): Algorithmic Game Theory. Cambridge University Press (2007)Google Scholar
  27. 27.
    Roughgarden, T.: Intrinsic robustness of the price of anarchy. In: Proceedings of STOC, pp. 513–522 (2009)Google Scholar
  28. 28.
    Roughgarden, T., Schoppmann, F.: Local smoothness and the price of anarchy in atomic splittable congestion games. In: Proceedings of SODA, pp. 255–267 (2011)Google Scholar
  29. 29.
    Suri, S., Tóth, C.D., Zhou, Y.: Selfish load balancing and atomic congestion games. Algorithmica 47(1), 79–96 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Vöcking, B.: Selfish load balancing. In: Nisan, N., Roughgarden, T., Tardos, É., Vazirani, V. (eds.) Algorithmic Game Theory, pp. 517–542 (2007)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Rensselaer Polytechnic InstituteTroyUSA
  2. 2.Oberlin CollegeOberlinUSA

Personalised recommendations