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Average-energy games

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Abstract

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.

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Notes

  1. We slightly abuse the notation as we see cycles as sequences of edges. The concatenation of cycles \(\mathcal {C}_{a} = s\,s'\ldots {} s\) and \(\mathcal {C}_{b} = s\,s'' \ldots {} s\) is to be understood as its natural interpretation \(\mathcal {C}_{a} \cdot \mathcal {C}_{b} = s\,s'\ldots {} s\,s'' \ldots {} s\): the origin state s only appears once in the middle and not twice as it would with \(\mathcal {C}_{a}\) and \(\mathcal {C}_{b}\) seen as true sequences of states.

  2. Observe that given a state in \(G''\), it is indeed possible to build any neighboring state using only E and w from the original game: one can effectively build the graph \(G''\) on-the-fly.

  3. In \({ EG}_{{ LU}}\) games with only \(\mathcal {P}_{2} \) (i.e., \(S_{1} = \emptyset \)), \(\mathcal {P}_{2} \) does not need memory to play as he can pick beforehand which of the energy bounds (lower or upper) he will transgress, and then do so with a memoryless strategy.

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Author information

Authors and Affiliations

  1. LSV, CNRS and ENS Cachan, Avenue du Président Wilson, 61, 94230, Cachan, France

    Patricia Bouyer & Nicolas Markey

  2. Computer Science Department, Université libre de Bruxelles (ULB), Avenue Franklin Roosevelt, 50, 1050, Brussels, Belgium

    Mickael Randour

  3. Aalborg University, Fredrik Bajers Vej 5, 9100, Aalborg, Denmark

    Kim G. Larsen & Simon Laursen

Authors
  1. Patricia Bouyer
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  2. Nicolas Markey
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  3. Mickael Randour
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  4. Kim G. Larsen
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  5. Simon Laursen
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Correspondence to Mickael Randour.

Additional information

Work partially supported by European project CASSTING (FP7-ICT-601148) and ERC project EQualIS (StG-308087). Mickael Randour is an F.R.S.-FNRS Postdoctoral Researcher.

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Bouyer, P., Markey, N., Randour, M. et al. Average-energy games. Acta Informatica 55, 91–127 (2018). https://doi.org/10.1007/s00236-016-0274-1

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