Abstract
We consider the classical machine scheduling, where n jobs need to be scheduled on m machines, and where job j scheduled on machine i contributes \(p_{ij}\in \mathbb {R}\) to the load of machine i, with the goal of minimizing the makespan, i.e., the maximum load of any machine in the schedule. We study the inefficiency of schedules that are obtained when jobs arrive sequentially one by one, and the jobs choose the machine on which they will be scheduled, aiming at being scheduled on a machine with a small load. We measure the inefficiency of a schedule as the ratio of the makespan obtained in the worst-case equilibrium schedule, and of the optimum makespan. This ratio is known as the sequential price of anarchy (SPoA). We also introduce two alternative inefficiency measures, which allow for a favorable choice of the order in which the jobs make their decisions. As our first result, we disprove the conjecture of Hassin and Yovel (Oper Res Lett 43(5):530–533, 2015) claiming that the sequential price of anarchy for \(m=2\) machines is at most 3. We show that the sequential price of anarchy grows at least linearly with the number n of players, assuming arbitrary tie-breaking rules. That is, we show \({\textbf {SPoA}} \in \Omega (n)\). At the end of the paper, we show that if an authority can change the order of the jobs adaptively to the decisions made by the jobs so far (but cannot influence the decisions of the jobs), then there exists an adaptive ordering in which the jobs end up in an optimum schedule.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
When each player chooses deterministically one machine, this definition is obvious. When equilibria are mixed or randomized, each player chooses one machine according to some probability distribution, and the social cost is the expected makespan of the resulting schedule.
References
Abed, F., Correa, J. R., & Huang, C. C. (2014). Optimal coordination mechanisms for multi-job scheduling games. In Proceedings of the 22nd annual european symposium on algorithms (ESA) (pp. 13–24). Springer.
Andelman, N., Feldman, M., & Mansour, Y. (2009). Strong price of anarchy. Games and Economic Behavior, 2(65), 289–317.
Angelucci, A., Bilò, V., Flammini, M., & Moscardelli, L. (2015). On the sequential price of anarchy of isolation games. Journal of Combinatorial Optimization, 29(1), 165–181.
Anshelevich, E., Dasgupta, A., Kleinberg, J. M., Tardos, É., Wexler, T., & Roughgarden, T. (2008). The price of stability for network design with fair cost allocation. SIAM Journal on Computing, 38(4), 1602–1623.
Awerbuch, B., Azar, Y., Richter, Y., & Tsur, D. (2006). Tradeoffs in worst-case equilibria. Theoretical Computer Science, 361(2), 200–209.
Azar, Y., Fleischer, L., Jain, K., Mirrokni, V., & Svitkina, Z. (2015). Optimal coordination mechanisms for unrelated machine scheduling. Operations Research, 63(3), 489–500.
Bilò, V., Flammini, M., Monaco, G., & Moscardelli, L. (2015). Some anomalies of farsighted strategic behavior. Theory of Computing Systems, 56(1), 156–180.
Caragiannis, I. (2013). Efficient coordination mechanisms for unrelated machine scheduling. Algorithmica, 66(3), 512–540.
Caragiannis, I., & Fanelli, A. (2016). An almost ideal coordination mechanism for unrelated machine scheduling. In Proceedings of the 9th international symposium on algorithmic game theory (SAGT) (pp. 315–326). Springer.
Chen, B., Potts, C. N., & Woeginger, G. J. (1999). Handbook of combinatorial optimization, chap. A review of machine scheduling: Complexity, algorithms and approximability (pp. 1493–1641). Berlin: Springer.
Christodoulou, G., Koutsoupias, E., & Nanavati, A. (2009). Coordination mechanisms. Theoretical Computer Science, 410(36), 3327–3336. graphs, Games and Computation: Dedicated to Professor Burkhard Monien on the Occasion of his 65th Birthday
Cole, R., Correa, J. R., Gkatzelis, V., Mirrokni, V., & Olver, N. (2015). Decentralized utilitarian mechanisms for scheduling games. Games and Economic Behavior, 92, 306–326.
Correa, J. R., de Jong, J., de Keijzer, B., & Uetz, M. (2019). The inefficiency of nash and subgame perfect equilibria for network routing. Mathematics of Operations Research, 44(4), 1286–1303.
Correa, J. R., & Queyranne, M. (2012). Efficiency of equilibria in restricted uniform machine scheduling with total weighted completion time as social cost. Naval Research Logistics (NRL), 59(5), 384–395.
Czumaj, A., & Vöcking, B. (2007). Tight bounds for worst-case equilibria. ACM Transactions Algorithms, 3(1), 4:1-4:17.
de Jong, J., & Uetz, M.(2014) The sequential price of anarchy for atomic congestion games. In Proceedings of the 10th international conference on web and internet economics (WINE). LNCS (Vol. 8877, pp. 429–434).
Epstein, L. (2010). Equilibria for two parallel links: The strong price of anarchy versus the price of anarchy. Acta Informatica, 47(7), 375–389.
Even-Dar, E., Kesselman, A., & Mansour, Y. (2003) Convergence time to Nash equilibria. In Proceedings of the 30th international colloquium on automata, languages, and programming (ICALP) (pp. 502–513). Springer.
Feldmann, R., Gairing, M., Lücking, T., Monien, B., &Rode, M. (2003) Nashification and the coordination ratio for a selfish routing game. In ICALP 2003 (Vol. 30, pp. 514–526). Springer.
Fiat, A., Kaplan, H., Levy, M., & Olonetsky, S. (2007). Strong price of anarchy for machine load balancing. In ICALP 2007 (Vol. 4596, pp. 583–594). Springer.
Finn, G., & Horowitz, E. (1979). A linear time approximation algorithm for multiprocessor scheduling. BIT Numerical Mathematics, 19(3), 312–320.
Fotakis, D., Kontogiannis, S., Koutsoupias, E., Mavronicolas, M., & Spirakis, P.(2002). The structure and complexity of Nash equilibria for a selfish routing game. In Proceedings of the 29th international conference on automata, languages and programming (ICALP) (pp. 123–134). Springer.
Hassin, R., & Yovel, U. (2015). Sequential scheduling on identical machines. Operations Research Letters, 43(5), 530–533.
Hoeksma, R., & Uetz, M.(2012). The price of anarchy for minsum related machine scheduling. In Proceedings of the 9th international workshop on approximation and online algorithms (WAOA) (pp. 261–273). Springer.
Immorlica, N., Li, L. E., Mirrokni, V. S., & Schulz, A. S. (2009). Coordination mechanisms for selfish scheduling. Theoretical Computer Science, 410(17), 1589–1598.
Koutsoupias, E., Mavronicolas, M., & Spirakis, P. (2003). Approximate equilibria and ball fusion. Theory of Computing Systems, 36(6), 683–693.
Koutsoupias, E., & Papadimitriou, C. H. (1999). Worst-case equilibria. In Proceedings of the 16th annual symposium on theoretical aspects of computer science (STACS). LNCS (Vol. 1563, pp. 404–413).
Leme, R. P., Syrgkanis, V., & Tardos, É. (2012). The curse of simultaneity. In Proceedings of innovations in theoretical computer science (ITCS) (pp. 60–67).
Pinedo, M. L. (2012). Scheduling: Theory, algorithms, and systems. Springer.
Tranæs, T. (1998). Tie-breaking in games of perfect information. Games and Economic Behavior, 22(1), 148–161.
Vöcking, B. (2007). Selfish load balanching. In Algorithmic game theory.
Zhang, L., Zhang, Y., Du, D., & Bai, Q. (2018). Improved price of anarchy for machine scheduling games with coordination mechanisms. Optimization Letters.
Acknowledgements
We are grateful to Thomas Erlebach for spotting a mistake in an earlier proof of Theorem 5 and for suggesting a fix of the proof. We thank Paul Dütting for valuable discussions. We also thank anonymous reviewers and seminar participants of OR 2016 for suggestions that improved the paper. An earlier version of the paper appeared in the proceedings of the 16th Web and Internet Economics (WINE 2020).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Chen, C., Giessler, P., Mamageishvili, A. et al. Sequential solutions in machine scheduling games. J Sched (2024). https://doi.org/10.1007/s10951-024-00810-3
Accepted:
Published:
DOI: https://doi.org/10.1007/s10951-024-00810-3