Percentile Queries in Multi-Dimensional Markov Decision Processes

Markov decision processes (MDPs) with multi-dimensional weights are useful to analyze systems with multiple objectives that may be conflicting and require the analysis of trade-offs. We study the complexity of percentile queries in such MDPs and give algorithms to synthesize strategies that enforce such constraints. Given a multi-dimensional weighted MDP and a quantitative payoff function f, thresholds vi (one per dimension), and probability thresholds αi, we show how to compute a single strategy to enforce that for all dimensions i, the probability of outcomes ρ satisfying fi(ρ) ≥ vi is at least αi. We consider classical quantitative payoffs from the literature (sup, inf, lim sup, lim inf, mean-payoff, truncated sum, discounted sum). Our work extends to the quantitative case the multi-objective model checking problem studied by Etessami et al. in unweighted MDPs.