On the price of anarchy of two-stage machine scheduling games

  • Deshi YeEmail author
  • Lin Chen
  • Guochuan Zhang


We consider a scheduling game, in which both the machines and the jobs are players. Machines are controlled by different selfish agents and attempt to maximize their workloads by choosing a scheduling policy among the given set of policies, while each job is controlled by a selfish agent that attempts to minimize its completion time by selecting a machine. Namely, this game was done in two-stage. In the first stage, every machine simultaneously chooses a policy from some given set of policies, and in the second stage, every job simultaneously chooses a machine. In this work, we use the price of anarchy to measure the efficiency of such equilibria where each machine is allowed to use one of the at most two policies. We provide nearly tight bounds for every combination of two deterministic scheduling policies with respect to two social objectives: minimizing the maximum job completion, and maximizing the minimum machine completion time.


Price of anarchy Scheduling Coordination mechanisms 



The authors thank anonymous referees for helpful comments and suggestions to improve the presentation of this paper.


  1. Ashlagi I, Tennenholtz M, Zohar A (2010) Competing schedulers. In: Proceedings of the 24th AAAI conference on artificial intelligence (AAAI), pp. 691–696Google Scholar
  2. Azar Y, Jain K, Mirrokni V (2008) (Almost) optimal coordination mechanisms for unrelated machine scheduling. In: Proceedings of the 19th annual ACM-SIAM symposium on Discrete algorithms (SODA), pp 323–332Google Scholar
  3. Caragiannis I (2009) Efficient coordination mechanisms for unrelated machine scheduling. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 815–824Google Scholar
  4. Chen X, Epstein L, Kleiman E, van Stee R (2013) Maximizing the minimum load: the cost of selfishness. Theoret Comput Sci 482:9–19MathSciNetCrossRefGoogle Scholar
  5. Chen X, Hu X, Ma W, Wang C (2012) Efficiency of dual equilibria in selfish task allocation to selfish machines. In: Proceedings of the 6th international conference on combinatorial optimization and applications (COCOA), pp. 312–323Google Scholar
  6. Christodoulou G, Koutsoupias E, Nanavati A (2004) Coordination mechanisms. In: Proceedings of the 31st international colloquium on automata, languages, and programming (ICALP), pp 45–56Google Scholar
  7. Cohen J, Dürr C, Nguyen Kim T (2011) Non-clairvoyant scheduling games. Theory Comput Syst 1–21Google Scholar
  8. Cole R, Correa JR, Gkatzelis V, Mirrokni V, Olver N (2011) Inner product spaces for minsum coordination mechanisms. In: Proceedings of the 43rd annual ACM symposium on theory of computing (STOC), pp 539–548Google Scholar
  9. Deuermeyer BL, Friesen DK, Langston MA (1982) Scheduling to maximize the minimum processor finish time in a multiprocessor system. SIAM J Algebraic Discrete Methods 3(2):190–196MathSciNetCrossRefGoogle Scholar
  10. Dobson G (1984) Scheduling independent tasks on uniform processors. SIAM J Comput 13:705–716MathSciNetCrossRefGoogle Scholar
  11. Dubey P (1986) Inefficiency of Nash equilibria. Math Oper Res 11(1):1–8MathSciNetCrossRefGoogle Scholar
  12. Epstein L, Kleiman E, Stee R (2009) Maximizing the minimum load: the cost of selfishness. In: Proceedings of the 5th international workshop on internet and network economics (WINE), pp 232–243Google Scholar
  13. Epstein L, Kleiman E, Stee R (2014) The cost of selfishness for maximizing the minimum load on uniformly related machines. J Comb Optim 27(4):767–777MathSciNetCrossRefGoogle Scholar
  14. Epstein L, Stee R (2012) The price of anarchy on uniformly related machines revisited. Inf Comput 212:37–54MathSciNetCrossRefGoogle Scholar
  15. Fabrikant A, Luthra A, Maneva E, Papadimitriou C, Shenker S (2003) On a network creation game. In: ACM Symposium on principles of distributed computing (PODC), pp 347–351Google Scholar
  16. Feldman M, Immorlica N, Lucier B, Roughgarden T, Syrgkanis V (2016) The price of anarchy in large games. In: Proceedings of the 48th annual ACM symposium on theory of computing (STOC), pp 963–976Google Scholar
  17. Finn G, Horowitz E (1979) A linear time approximation algorithm for multiprocessor scheduling. BIT Numer Math 19(3):312–320MathSciNetCrossRefGoogle Scholar
  18. Friesen DK (1987) Tighter bounds for LPT scheduling on uniform processors. SIAM J Comput 16:554–560MathSciNetCrossRefGoogle Scholar
  19. Graham RL (1966) Bounds for certain multiprocessing anomalies. Bell System Tech J 45:1563–1581CrossRefGoogle Scholar
  20. Graham RL (1969) Bounds on multiprocessing timing anomalies. SIAM J Appl Math 17:263–269MathSciNetzbMATHGoogle Scholar
  21. Heydenreich B, Müller R, Uetz M (2007) Games and mechanism design in machine scheduling—an introduction. Prod Oper Manag 16(4):437–454CrossRefGoogle Scholar
  22. Hoeksma R, Uetz M (2011) The price of anarchy for minsum related machine scheduling. In: Proceedings of the 9th international conference on approximation and online algorithms (WAOA), pp 261–273Google Scholar
  23. Immorlica N, Li LE, Mirrokni VS, Schulz AS (2009) Coordination mechanisms for selfish scheduling. Theor Comput Sci 410:1589–1598MathSciNetCrossRefGoogle Scholar
  24. Koutsoupias E, Papadimitriou C (1999) Worst-case equilibria. In: Proceedings of the 16th symposium on theoretical aspects of computer science (STACS), pp 404–413Google Scholar
  25. Lin L, Tan Z (2014) Inefficiency of Nash equilibrium for scheduling games with constrained jobs: a parametric analysis. Theor Comput Sci 521:123–134MathSciNetCrossRefGoogle Scholar
  26. Lucier B, Borodin A (2010) Price of anarchy for greedy auctions. In: Proceedings of the 21st annual ACM-SIAM symposium on discrete algorithms (SODA), pp 537–553Google Scholar
  27. Nisan N, Roughgarden T, Tardos E, Vazirani VV (2007) Algorithmic game theory. Cambridge University Press, Cambridge CrossRefGoogle Scholar
  28. Roughgarden T, Tardos E (2004) Bounding the inefficiency of equilibria in nonatomic congestion games. Games Econ Behav 47(2):389–403MathSciNetCrossRefGoogle Scholar
  29. Roughgarden T, Tardos E (2002) How bad is selfish routing? J ACM 49(2):236–259MathSciNetCrossRefGoogle Scholar
  30. Tan Z, Wan L, Zhang Q, Ren W (2012) Inefficiency of equilibria for the machine covering game on uniform machines. Acta Inform 49(6):361–379MathSciNetCrossRefGoogle Scholar
  31. Vetta AR (2002) Nash equilibria in competitive societies with applications to facility location, traffic routing and auctions. In: Symposium on the foundations of computer science (FOCS), pp 416–425Google Scholar

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Authors and Affiliations

  1. 1.College of Computer ScienceZhejiang UniversityHangzhouChina
  2. 2.Department of Computer ScienceTexas Tech UniversityLubbockUSA

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