Price of anarchy and price of stability in multi-agent project scheduling

  • Alessandro AgnetisEmail author
  • Cyril Briand
  • Sandra Ulrich Ngueveu
  • Přemysl Šůcha
S.I.: Project Management and Scheduling 2018


We consider a project scheduling environment in which the activities are partitioned among a set of agents. The owner of each activity can decide its length, which is linearly related to its cost within a minimum (crash) and a maximum (normal) length. For each day the project makespan is reduced with respect to its normal value, a reward is offered to the agents, and each agent receives a given ratio of the reward. As in classical game theory, we assume that the agents’ parameters are common knowledge. We study the Nash equilibria of the corresponding non-cooperative game as a desired state where no agent is motivated to change his/her decision. Regarding project makespan as an overall measure of efficiency, here we consider the worst and the best Nash equilibria (i.e., for which makespan is maximum and, respectively, minimum among Nash equilibria). We show that the problem of finding the worst Nash equilibrium is NP-hard (finding the best Nash equilibrium is already known to be strongly NP-hard), and propose an ILP formulation for its computation. We then investigate the values of the price of anarchy and the price of stability in a large sample of realistic size problems and get useful insights for the project owner.


Multi-agent project scheduling Nash equilibria Flow networks Price of anarchy Price of stability 



This work was supported by the European Regional Development Fund under the project AI&Reasoning (Reg. No. CZ.02.1.01/0.0/0.0/15_003/0000466).

Supplementary material


  1. Agnetis, A., Briand, C., Billaut, J.-C., & Šucha, P. (2015). Nash equilibria for the multi-agent project scheduling problem with controllable processing times. Journal of Scheduling, 18(1), 15–27.CrossRefGoogle Scholar
  2. Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network flows. Upper Saddle River, NJ: Prentice-Hall.Google Scholar
  3. Anshelevich, E., Dasgupta, A., Kleinberg, J., 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.CrossRefGoogle Scholar
  4. Averbakh, I. (2010). Nash equilibria in competitive project scheduling. European Journal of Operational Research, 205(3), 552–556.CrossRefGoogle Scholar
  5. Briand, C., Ngueveu, S. U., & Šucha, P. (2017). Finding an optimal Nash equilibrium to the multi-agent project scheduling problem. Journal of Scheduling, 20, 475–491.CrossRefGoogle Scholar
  6. Christodoulou, G., & Koutsoupias, E. (2005). On the price of anarchy and stability of correlated equilibria of linear congestion games. In G. S. Brodal, & S. Leonardi (Eds.), Algorithms—ESA 2005. ESA 2005. Lecture Notes in Computer Science (Vol. 3669, pp. 59–70). Berlin: Springer.Google Scholar
  7. Ciurea, E., & Ciupalâ, L. (2004). Sequential and parallel algorithms for minimum flows. Journal of Applied Mathematics and Computing, 15, 53–75.CrossRefGoogle Scholar
  8. Confessore, G., Giordani, S., & Rismondo, S. (2007). A market-based multi-agent system model for decentralized multi-project scheduling. Annals of Operations Research, 150, 115–135.CrossRefGoogle Scholar
  9. De Ita Luna, G., Zacarias-Flores, F., & Altamirano-Robles, L. C. (2015). Finding pure Nash equilibrium for the resource-constrained project scheduling problem. Computación y Sistemas, 19(1), 17–27.CrossRefGoogle Scholar
  10. De Reyck, B., & Herroelen, W. (1999). The multi-mode resource-constrained project scheduling problem with generalized precedence relations. European Journal of Operational Research, 119, 538–556.CrossRefGoogle Scholar
  11. Demeulemeester, E. L., & Herroelen, W. S. (2002). Project scheduling—A research handbook. Dordrecht: Kluwer.Google Scholar
  12. Demeulemeester, E., Vanhoucke, M., & Herroelen, W. (2003). Rangen: A random network generator for activity-on-the-node networks. Journal of Scheduling, 6, 17–38.CrossRefGoogle Scholar
  13. Estévez-Fernández, A. (2012). A game theoretical approach to sharing penalties and rewards in projects. European Journal of Operational Research, 216(3), 647–657.CrossRefGoogle Scholar
  14. Herroelen, W. S., & De Reyck, B. (1999). Phase transitions in project scheduling. Journal of the Operational Research Society, 50(2), 148–156.CrossRefGoogle Scholar
  15. Kolisch, R., Sprecher, A., & Drexl, A. (1992). Characterization and generation of a general class of resource-constrained project scheduling problems. Institut für Betriebswirtschaftslehre Christian-Albrechts-Universität zu Kiel, working paper no. 301.Google Scholar
  16. Phillips, S., & Dessouky, M. I. (1977). Solving the project time/cost tradeoff problem using the minimal cut concept. Management Science, 24(4), 393–400.CrossRefGoogle Scholar
  17. Van Eynde, R. (2017). Multi-project scheduling—The application of a decoupled schedule generation scheme and a game mechanic, Master’s Dissertation in Business Engineering, Universiteit Gent.Google Scholar
  18. Varakantham, P., & Fu, N. (2017). Mechanism design for strategic project scheduling, research collection school of information systems. Singapore Management University, 8-2017.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria dell’Informazione e Scienze MatematicheUniversità degli Studi di SienaSienaItaly
  2. 2.LAAS-CNRS, Université de Toulouse, CNRS, UPSToulouseFrance
  3. 3.LAAS-CNRS, Université de Toulouse, CNRS, INPToulouseFrance
  4. 4.Czech Institute of Informatics, Robotics, and CyberneticsCzech Technical University in PraguePragueCzech Republic

Personalised recommendations