Abstract
A Nash Equilibriun (NE) is a strategy profile that is resilient to unilateral deviations, and is predominantly used in analysis of competitive games. A downside of NE is that it is not necessarily stable against deviations by coalitions. Yet, as we show in this paper, in some cases, NE does exhibit stability against coalitional deviations, in that the benefits from a joint deviation are bounded. In this sense, NE approximates strong equilibrium (SE) [6].
We provide a framework for quantifying the stability and the performance of various assignment policies and solution concept in the face of coalitional deviations. Within this framework we evaluate a given configuration according to three measurements: (i) IR min: the maximal number α, such that there exists a coalition in which the minimum improvement ratio among the coalition members is α (ii) IR max: the maximum improvement ratio among the coalition’s members. (iii) DRmax: the maximum possible damage ratio of an agent outside the coalition.
This framework can be used to study the proximity between different solution concepts, as well as to study the existence of approximate SE in settings that do not possess any such equilibrium. We analyze these measurements in job scheduling games on identical machines. In particular, we provide upper and lower bounds for the above three measurements for both NE and the well-known assignment rule Longest Processing Time (LPT) (which is known to yield a NE). Most of our bounds are tight for any number of machines, while some are tight only for three machines. We show that both NE and LPT configurations yield small constant bounds for IRmin and DRmax. As for IRmax, it can be arbitrarily large for NE configurations, while a small bound is guaranteed for LPT configurations. For all three measurements, LPT performs strictly better than NE.
With respect to computational complexity aspects, we show that given a NE on m ≥ 3 identical machines and a coalition, it is NP-hard to determine whether the coalition can deviate such that every member decreases its cost. For the unrelated machines settings, the above hardness result holds already for m ≥ 2 machines.
Keywords
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Albers, S.: On the value of coordination in network design. In: SODA (2008)
Albers, S., Elits, S., Even-Dar, E., Mansour, Y., Roditty, L.: On Nash Equilibria for a Network Creation Game. In: SODA (2006)
Andelman, N., Feldman, M., Mansour, Y.: Strong Price of Anarchy. In: SODA (2007)
Anshelevich, E., Dasgupta, A., Kleinberg, J.M., Tardos, É., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. In: FOCS, pp. 295–304 (2004)
Anshelevich, E., Dasgupta, A., Tardos, E., Wexler, T.: Near-Optimal Network Design with Selfish Agents. In: STOC (2003)
Aumann, R.: Acceptable Points in General Cooperative n-Person Games. In: Conti, R., Ruberti, A. (eds.) Optimization Techniques 1973. LNCS, vol. 4, p. 1959. Springer, Heidelberg (1973)
Azar, Y., Tsur, D., Richter, Y., Awerbuch, B.: Tradeoffs in Worst-Case Equilibria. In: Solis-Oba, R., Jansen, K. (eds.) WAOA 2003. LNCS, vol. 2909, pp. 41–52. Springer, Heidelberg (2004)
Bernheim, D.B., Peleg, B., Whinston, M.D.: Coalition-proof nash equilibria: I concepts. Journal of Economic Theory 42, 1–12 (1987)
Christodoulou, G., Koutsoupias, E.: On the Price of Anarchy and Stability of Correlated Equilibria of Linear Congestion Games. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 59–70. Springer, Heidelberg (2005)
Christodoulou, G., Koutsoupias, E., Nanavati, A.: Coordination Mechanisms. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 345–357. Springer, Heidelberg (2004)
Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. In: SODA, pp. 413–420 (2002)
Epstein, A., Feldman, M., Mansour, Y.: Strong Equilibrium in Cost Sharing Connection Games. In: ACMEC (2007)
Fabrikant, A., Luthra, A., Maneva, E., Papadimitriou, C., Shenker, S.: On a network creation game. In: PODC (2003)
Feldman, M., Tamir, T.: Approximate Strong Equilibrium in Job Scheduling Games. http://www.faculty.idc.ac.il/tami/Papers/approxSE.pdf
Fiat, A., Kaplan, H., Levi, M., Olonetsky, S.: Strong Price of Anarchy for Machine Load Balancing. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 583–594. Springer, Heidelberg (2007)
Finn, G., Horowitz, E.: A linear time approximation algorithm for multiprocessor scheduling. BIT Numerical Mathematics 19(3), 312–320 (1979)
Fotakis, D., Kontogiannis, S., Mavronicolas, M., Spiraklis, 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.) ICALP 2002. LNCS, vol. 2380, pp. 510–519. Springer, Heidelberg (2002)
Graham, R.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math. 17, 263–269 (1969)
Holzman, R., Law-Yone, N.: Strong equilibrium in congestion games. Games and Economic Behavior 21, 85–101 (1997)
Holzman, R., Law-Yone, N.: Network structure and strong equilibrium in route selection games. Mathematical Social Sciences 46, 193–205 (2003)
Koutsoupias, E., Papadimitriou, C.H.: Worst-Case Equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999)
Leonardi, S., Sankowski, P.: Network Formation Games with Local Coalitions. In: PODC (2007)
Milchtaich, I.: Crowding games are sequentially solvable. International Journal of Game Theory 27, 501–509 (1998)
Papadimitriou, C.H.: Algorithms, games, and the internet. In: proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 749–753 (2001)
Roughgarden, T., Tardos, E.: How bad is selfish routing? Journal of the ACM 49(2), 236–259 (2002)
Rozenfeld, O., Tennenholtz, M.: Strong and correlated strong equilibria in monotone congestion games. In: working paper, Technion, Israel (2006)
Schuurman, P., Vredeveld, T.: Performance guarantees of local search for multiprocessor scheduling. INFORMS Journal on Computing (to appear)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Feldman, M., Tamir, T. (2008). Approximate Strong Equilibrium in Job Scheduling Games. In: Monien, B., Schroeder, UP. (eds) Algorithmic Game Theory. SAGT 2008. Lecture Notes in Computer Science, vol 4997. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79309-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-540-79309-0_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79308-3
Online ISBN: 978-3-540-79309-0
eBook Packages: Computer ScienceComputer Science (R0)