Profit Sharing with Thresholds and Non-monotone Player Utilities

  • Elliot Anshelevich
  • John Postl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8768)


We study profit sharing games in which players select projects to participate in and share the reward resulting from that project equally. Unlike most existing work, in which it is assumed that the player utility is monotone in the number of participants working on their project, we consider non-monotone player utilities. Such utilities could result, for example, from “threshold” or “phase transition” effects, when the total benefit from a project improves slowly until the number of participants reaches some critical mass, then improves rapidly, and then slows again due to diminishing returns.

Non-monotone player utilities result in a lot of instability: strong Nash equilibrium may no longer exist, and the quality of Nash equilibria may be far away from the centralized optimum. We show, however, that by adding additional requirements such as players needing permission to leave a project from the players currently on this project, or instead players needing permission to a join a project from players on that project, we ensure that strong Nash equilibrium always exists. Moreover, just the addition of permission to leave already guarantees the existence of strong Nash equilibrium within a factor of 2 of the social optimum. In this paper, we provide results on the existence and quality of several different coalitional solution concepts, focusing especially on permission to leave and join projects, and show that such requirements result in the existence of good stable solutions even for the case when player utilities are non-monotone.


Nash Equilibrium Social Optimum Potential Game Contractual Obligation Stability Concept 
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.


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  1. 1.
    Anshelevich, E., Postl, J.: Profit Sharing with Thresholds and Non-monotone Player Utilities,
  2. 2.
    Augustine, J., Chen, N., Elkind, E., Fanelli, A., Gravin, N., Shiryaev, D.: Dynamics of profit-sharing games. In: IJCAI 2011, pp. 37–42 (2011)Google Scholar
  3. 3.
    Awerbuch, B., Azar, Y., Epstein, A., Mirrkoni, V.S., Skopalik, A.: Fast convergence to nearly optimal solutions in potential games. In: EC 2008, pp. 264–273 (2008)Google Scholar
  4. 4.
    Aziz, H.: Stable marriage and roommate problems with individual-based stability. arXiv: 1204.1628Google Scholar
  5. 5.
    Aziz, H., Brandl, F.: Existence of Stability in Hedonic Coalition Formation Games. In: AAMAS 2012, pp. 763–770 (2012)Google Scholar
  6. 6.
    Bloch, F., Diamantoudi, E.: Noncooperative formation of coalitions in hedonic games. International Journal of Game Theory 40(2), 263–280 (2010)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bogomolnaia, A., Jackson, M.O.: The stability of hedonic coalition structures. Games and Economic Behavior 38(2), 201–230 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Chalkiadakis, G., Elkind, E., Markakis, E., Polukarov, M., Jennings, N.R.: Cooperative Games with Overlapping Coalitions. Journal of Artificial Intelligence Research 39(1), 179–216 (2010)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Drèze, J.H., Greenberg, J.: Hedonic Coalitions: Optimality and Stability. Econometrica 48(4), 987–1003 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Feldman, M., Lewin-Eytan, L., Naor, J.S.: Hedonic clustering games. In: SPAA 2012, pp. 267–276 (2012)Google Scholar
  11. 11.
    Goemans, M.X., Li, L., Mirrokni, V.S.: M. Thottan. Market sharing games applied to content distribution in ad-hoc networks. JSAC 2006 24(5), 1020–1033 (2006)Google Scholar
  12. 12.
    Hajduková, J.: Coalition formation games: A survey. International Game Theory Review 8(4), 613–641 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Harks, T., Klimm, M., Möhring, R.H.: Strong Nash Equilibria in Games with the Lexicographical Improvement Property. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 463–470. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Karakaya, M.: Hedonic coalition formation games: A new stability notion. Mathematical Social Sciences 61(3), 157–165 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Kleinberg, J., Oren, S.: Mechanisms for (Mis)allocating Scientific Credit. In: STOC 2011, pp. 529–538 (2011)Google Scholar
  16. 16.
    Kutten, S., Lavi, R., Trehan, A.: Composition Games for Distributed Systems: the EU Grant Games. In: AAAI 2013, pp. 1–16 (2013)Google Scholar
  17. 17.
    Mirrokni, V.S., Vetta, A.: Convergence issues in competitive games. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 183–194. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Tardos, É., Wexler, T.: Network formation games. In: Nisan, N., Tardos, É., Roughgarden, T., Vazirani, V. (eds.) Algorithmic Game Theory, ch. 19, Cambridge University Press (2007)Google Scholar
  19. 19.
    Vetta, A.: Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions. In: FOCS 2002, pp. 416–425 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Elliot Anshelevich
    • 1
  • John Postl
    • 1
  1. 1.Rensselaer Polytechnic InstituteTroyUSA

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