Non-clairvoyant Scheduling Games
In a scheduling game, each player owns a job and chooses a machine to execute it. While the social cost is the maximal load over all machines (makespan), the cost (disutility) of each player is the completion time of its own job. In the game, players may follow selfish strategies to optimize their cost and therefore their behaviors do not necessarily lead the game to an equilibrium. Even in the case there is an equilibrium, its makespan might be much larger than the social optimum, and this inefficiency is measured by the price of anarchy – the worst ratio between the makespan of an equilibrium and the optimum. Coordination mechanisms aim to reduce the price of anarchy by designing scheduling policies that specify how jobs assigned to a same machine are to be scheduled. Typically these policies define the schedule according to the processing times as announced by the jobs. One could wonder if there are policies that do not require this knowledge, and still provide a good price of anarchy. This would make the processing times be private information and avoid the problem of truthfulness. In this paper we study these so-called non-clairvoyant policies. In particular, we study the RANDOM policy that schedules the jobs in a random order without preemption, and the EQUI policy that schedules the jobs in parallel using time-multiplexing, assigning each job an equal fraction of CPU time.
For these models we study two important questions, the existence of Nash equilibria and the price of anarchy. We show under some restrictions that the game under RANDOM policy is a potential game for two unrelated machines but it is not for three or more; for uniform machines, we prove that the game under this policy always possesses a Nash equilibrium by using a novel potential function with respect to a refinement of best-response dynamic. Moreover, we show that the game under the EQUI policy is a potential game.
Next, we analyze the inefficiency of EQUI policy. Interestingly, the (strong) price of anarchy of EQUI, a non-clairvoyant policy, is asymptotically the same as that of the best strongly local policy – policies in which a machine may look at the processing time of jobs assigned to it. The result also indicates that knowledge of jobs’ characteristics is not necessarily needed.
KeywordsNash Equilibrium Completion Time Coordination Mechanism Potential Game Pure Nash Equilibrium
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- 5.Azar, Y., Jain, K., Mirrokni, V.S.: (Almost) Optimal Coordination Mechanisms for Unrelated Machine Scheduling. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 323–332 (2008)Google Scholar
- 7.Caragiannis, I.: Efficient coordination mechanisms for unrelated machine scheduling. In: SODA, pp. 815–824 (2009)Google Scholar
- 9.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)Google Scholar
- 10.Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. In: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 413–420 (2002)Google Scholar
- 12.Durr, C., Kim, T.N.: Non-clairvoyant scheduling games, http://www.lix.polytechnique.fr/~thang/Papers/EQUI.pdf
- 13.Edmonds, J.: Scheduling in the dark. In: Proceedings of the 31st ACM Symposium on Theory of Computing, STOC (1999)Google Scholar
- 14.Fiat, A., Kaplan, H., Levy, M., Olonetsky, S.: Strong Price of Anarchy for Machine Load Balancing. In: Proceedings of the 34th International Colloquium on Automata, Languages and Programming, pp. 583–594 (2007)Google Scholar
- 17.Gairing, M., Lücking, T., Mavronicolas, M., Monien, B.: Computing Nash equilibria for scheduling on restricted parallel links. In: 36th ACM Symposium on Theory of Computing, pp. 613–622 (2004)Google Scholar
- 18.Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell System Technical Journal 45, 1563–1581 (1966)Google Scholar
- 21.Immorlica, N., Li, L., Mirrokni, V.S., Schulz, A.: Coordination Mechanisms for Selfish Scheduling. In: Proceedings of the 1st International Workshop on Internet and Network Economics, pp. 55–69 (2005)Google Scholar
- 24.Vredeveld, T.: Combinatorial Approximation Algorithms: Guaranteed Versus Experimental Performance. PhD thesis, Technische Universiteit Eindhoven, The Netherlands (2002)Google Scholar