Acta Informatica

, Volume 55, Issue 2, pp 91–127 | Cite as

Average-energy games

  • Patricia Bouyer
  • Nicolas Markey
  • Mickael Randour
  • Kim G. Larsen
  • Simon Laursen
Original Article


Two-player quantitative zero-sum games provide a natural framework to synthesize controllers with performance guarantees for reactive systems within an uncontrollable environment. Classical settings include mean-payoff games, where the objective is to optimize the long-run average gain per action, and energy games, where the system has to avoid running out of energy. We study average-energy games, where the goal is to optimize the long-run average of the accumulated energy. We show that this objective arises naturally in several applications, and that it yields interesting connections with previous concepts in the literature. We prove that deciding the winner in such games is in \(\mathsf{NP}\cap \mathsf{coNP}\) and at least as hard as solving mean-payoff games, and we establish that memoryless strategies suffice to win. We also consider the case where the system has to minimize the average-energy while maintaining the accumulated energy within predefined bounds at all times: this corresponds to operating with a finite-capacity storage for energy. We give results for one-player and two-player games, and establish complexity bounds and memory requirements.


  1. 1.
    Aminof, B., Rubin, S.: First cycle games. In: Proceedings of SR, EPTCS 146, pp. 83–90 (2014)Google Scholar
  2. 2.
    Björklund, H., Sandberg, S., Vorobyov, S.: Memoryless determinacy of parity and mean payoff games: a simple proof. Theor. Comput. Sci. 310(1–3), 365–378 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bloem, R., Chatterjee, K., Henzinger, T.A., Jobstmann, B.: Better quality in synthesis through quantitative objectives. In: Proceedings of CAV, LNCS 5643, pp 140–156. Springer, Berlin (2009)Google Scholar
  4. 4.
    Bohy, A., Bruyère, V., Filiot, E., Raskin, J.-F.: Synthesis from LTL specifications with mean-payoff objectives. In: Proceedings of of TACAS, LNCS 7795, pp. 169–184. Springer, Berlin (2013)Google Scholar
  5. 5.
    Boros, E., Elbassioni, K., Gurvich, V., Makino, K.: Markov decision processes and stochastic games with total effective payoff. In: Proceedings of STACS, LIPIcs 30, pp. 103–115. Schloss Dagstuhl—LZI (2015)Google Scholar
  6. 6.
    Bouyer, P., Fahrenberg, U., Larsen, K.G., Markey, N., Srba, J.: Infinite runs in weighted timed automata with energy constraints. In: Proceedings of FORMATS, LNCS 5215, pp. 33–47. Springer, Berlin (2008)Google Scholar
  7. 7.
    Bouyer, P., Markey, N., Randour, M., Larsen, K.G., Laursen, S.: Average-energy games. In: Proceedings of GandALF, EPTCS 193, pp. 1–15 (2015)Google Scholar
  8. 8.
    Brázdil, T., Klaška, D., Kučera, A., Novotný, P.: Minimizing running costs in consumption systems. In: Proceedings of CAV, LNCS 8559, pp. 457–472. Springer, Berlin (2014)Google Scholar
  9. 9.
    Brim, L., Chaloupka, J., Doyen, L., Gentilini, R., Raskin, J.-F.: Faster algorithms for mean-payoff games. Formal Methods Syst. Des. 38(2), 97–118 (2011)CrossRefMATHGoogle Scholar
  10. 10.
    Cassez, F., Jensen, J.J., Larsen, K.G., Raskin, J.-F., Reynier, P.-A.: Automatic synthesis of robust and optimal controllers—an industrial case study. In: Proceedings of HSCC, LNCS 5469, pp. 90–104. Springer, Berlin (2009)Google Scholar
  11. 11.
    Chakrabarti, A., de Alfaro, L., Henzinger, T.A., Stoelinga, M.: Resource interfaces. In: Proceedings of EMSOFT, LNCS 2855, pp. 117–133. Springer, Berlin (2003)Google Scholar
  12. 12.
    Chatterjee, K., Doyen, L.: Energy parity games. In: Proceedings of ICALP, LNCS 6199, pp. 599–610. Springer, Berlin (2010)Google Scholar
  13. 13.
    Chatterjee, K., Velner, Y.: Mean-payoff pushdown games. In: Proceedings of LICS, pp. 195–204. IEEE (2012)Google Scholar
  14. 14.
    Chatterjee, K., Prabhu, V.S.: Quantitative timed simulation functions and refinement metrics for real-time systems. In: Proceedings of HSCC, pp. 273–282. ACM (2013)Google Scholar
  15. 15.
    Chatterjee, K., Randour, M., Raskin, J.-F.: Strategy synthesis for multi-dimensional quantitative objectives. Acta Inf. 51(3–4), 129–163 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Chatterjee, K., Doyen, L., Randour, M., Raskin, J.-F.: Looking at mean-payoff and total-payoff through windows. Inf. Comput. 242, 25–52 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ehrenfeucht, A., Mycielski, J.: Positional strategies for mean payoff games. Int. J. Game Theory 8(2), 109–113 (1979)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fearnley, J., Jurdziński, M.: Reachability in two-clock timed automata is PSPACE-complete. In: Proceedings of ICALP, LNCS 7966, pp. 212–223. Springer, Berlin (2013)Google Scholar
  19. 19.
    Filar, J., Vrieze, K.: Competitive Markov Decision Processes. Springer, Berlin (1997)MATHGoogle Scholar
  20. 20.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)MATHGoogle Scholar
  21. 21.
    Gawlitza, T., Seidl, H.: Games through nested fixpoints. In: Proceedings of CAV, LNCS 5643, pp. 291–305. Springer, Berlin (2009)Google Scholar
  22. 22.
    Gimbert, H., Zielonka, W.: When can you play positionnaly? In: Proceedings of MFCS, LNCS 3153, pp. 686–697. Springer, Berlin (2004)Google Scholar
  23. 23.
    Gimbert, H., Zielonka, W.: Games where you can play optimally without any memory. In: Proceedings of CONCUR, LNCS 3653, pp. 428–442. Springer, Berlin (2005)Google Scholar
  24. 24.
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games: A Guide to Current Research, LNCS 2500. Springer, Berlin (2002)Google Scholar
  25. 25.
    Juhl, L., Larsen, K.G., Raskin, J.-F.: Optimal bounds for multiweighted and parametrised energy games. In: Theories of Programming and Formal Methods, LNCS 8051, pp. 244–255. Springer, Berlin (2013)Google Scholar
  26. 26.
    Jurdziński, M.: Deciding the winner in parity games is in UP\(\cap \)co-UP. Inf. Process. Lett. 68(3), 119–124 (1998)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Jurdziński, M., Sproston, J., Laroussinie, F.: Model checking probabilistic timed automata with one or two clocks. Logical Methods Comput. Sci. 4(3), 1–28 (2008)MathSciNetMATHGoogle Scholar
  28. 28.
    Karp, R.M.: A characterization of the minimum cycle mean in a digraph. Discrete Math. 23(3), 309–311 (1978)Google Scholar
  29. 29.
    Kopczynski, E.: Half-positional determinacy of infinite games. In: Proceedings of ICALP, LNCS 4052, pp. 336–347. Springer, Berlin (2006)Google Scholar
  30. 30.
    Lafourcade, P., Lugiez, D., Treinen, R.: Intruder deduction for AC-like equational theories with homomorphisms. Research Report LSV-04-16, Laboratoire Spécification et Vérification, ENS Cachan, France (2004)Google Scholar
  31. 31.
    Lafourcade, P., Lugiez, D., Treinen, R.: Intruder deduction for AC-like equational theories with homomorphisms. In: Proceedings of RTA, LNCS 3467, pp. 308–322. Springer, Berlin (2005)Google Scholar
  32. 32.
    Larsen, K.G., Laursen, S., Zimmermann, M.: Limit your consumption! Finding bounds in average-energy games. In: Proceedings of QAPL, EPTCS (2016)Google Scholar
  33. 33.
    Randour, M.: Automated synthesis of reliable and efficient systems through game theory: a case study. In: Proceedings of the European Conference on Complex Systems 2012, Springer Proceedings in Complexity XVII, pp. 731–738. Springer, Berlin (2013)Google Scholar
  34. 34.
    Randour, M.: Synthesis in Multi-Criteria Quantitative Games. Ph.D. Thesis, Université de Mons, Belgium (2014)Google Scholar
  35. 35.
    Sipser, M.: Introduction to the Theory of Computation. PWS Publishing Company, Boston (1997)MATHGoogle Scholar
  36. 36.
    Thuijsman, F., Vrieze, O.J.: The bad match; a total reward stochastic game. OR Spektrum 9(2), 93–99 (1987)Google Scholar
  37. 37.
    Velner, Y., Chatterjee, K., Doyen, L., Henzinger, T.A., Rabinovich, A.M., Raskin, J.-F.: The complexity of multi-mean-payoff and multi-energy games. Inf. Comput. 241, 177–196 (2015)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Zwick, U., Paterson, M.: The complexity of mean payoff games on graphs. Theor. Comput. Sci. 158(1–2), 343–359 (1996)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Patricia Bouyer
    • 1
  • Nicolas Markey
    • 1
  • Mickael Randour
    • 2
  • Kim G. Larsen
    • 3
  • Simon Laursen
    • 3
  1. 1.LSV, CNRS and ENS CachanCachanFrance
  2. 2.Computer Science DepartmentUniversité libre de Bruxelles (ULB)BrusselsBelgium
  3. 3.Aalborg UniversityAalborgDenmark

Personalised recommendations