The complexity of multi-mean-payoff and multi-energy games

Yaron Velner, Krishnendu Chatterjee*, Laurent Doyen, Thomas A. Henzinger, Alexander Rabinovich, Jean François Raskin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

64 Scopus citations


In mean-payoff games, the objective of the protagonist is to ensure that the limit average of an infinite sequence of numeric weights is nonnegative. In energy games, the objective is to ensure that the running sum of weights is always nonnegative. Multi-mean-payoff and multi-energy games replace individual weights by tuples, and the limit average (resp., running sum) of each coordinate must be (resp., remain) nonnegative. We prove finite-memory determinacy of multi-energy games and show inter-reducibility of multi-mean-payoff and multi-energy games for finite-memory strategies. We improve the computational complexity for solving both classes with finite-memory strategies: we prove coNP-completeness improving the previous known EXPSPACE bound. For memoryless strategies, we show that deciding the existence of a winning strategy for the protagonist is NP-complete. We present the first solution of multi-mean-payoff games with infinite-memory strategies: we show that mean-payoff-sup objectives can be decided in NP∩coNP, whereas mean-payoff-inf objectives are coNP-complete.

Original languageEnglish
Pages (from-to)177-196
Number of pages20
JournalInformation and Computation
StatePublished - 1 Apr 2015


FundersFunder number
Microsoft267989, 279499, FP7-601148
European Research Council279307
Austrian Science FundS11402-N23, S11407-N23, P23499-N23


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