The Multiple Dimensions of Mean-Payoff Games

Outline We consider quantitative game models for the design of reactive systems working in resource-constrained environment. The game is played on a finite weighted graph where some resource (e.g., battery) can be consumed or recharged along the edges of the graph. In mean-payoff games, the resource usage is computed as the long-run average resource consumption. In energy games, the resource usage is the initial amount of resource necessary to maintain the resource level always positive. We review fundamental results about mean-payoff games that show the existence of memoryless optimal strategies, and the equivalence of mean-payoff games with finite-duration reachability games, as well as with energy games (which can also be viewed as safety games). These results provide conceptually simple backward-induction algorithms for solving mean-payoff games, and for constructing memoryless optimal strategies. It follows that mean-payoff games can be solved in NP ∩ coNP. Then we consider games with multiple mean-payoff conditions for systems using multiple resources. In multi-dimension mean-payoff games, memory is necessary for optimal strategies, and the previous equivalence results with reachability and energy (safety) games no longer hold. First, infinite memory is necessary in general for optimal strategies. With infinite memory, the limit of the long-run average resource consumption may not exist, and it is necessary to distinguish between the limsup and the liminf of the long-run average resource consumption. Second, the equivalence with a multi-dimensional version of energy games holds only if the players are restricted to use finite-memory strategies, and in that case the limsupand the liminf-value coincide. The complexity of solving multi-dimension mean-payoff games is as follows, depending on which class of strategies is given to the player: NP-complete for memoryless strategies, coNP-complete for finite-memory strategies, NP ∩ coNP for infinite-memory strategies and a conjunction of limsup objectives, and coNPcomplete for infinite-memory strategies and a conjunction of liminf objectives.