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 $$v_i$$vi (one per dimension), and probability thresholds $$\alpha _i$$αi, we show how to compute a single strategy to enforce that for all dimensions i, the probability of outcomes $$\rho $$ρ satisfying $$f_i(\rho ) \ge v_i$$fi(ρ)≥vi is at least $$\alpha _i$$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. (Log Methods Comput Sci 4(4), 2008) in unweighted MDPs.