Two Views on Multiple Mean-Payoff Objectives in Markov Decision Processes

We study Markov decision processes (MDPs) with multiple limit-average (or mean-payoff) functions. We consider two different objectives, namely, expectation and satisfaction objectives. Given an MDP with k reward functions, in the expectation objective the goal is to maximize the expected limit-average value, and in the satisfaction objective the goal is to maximize the probability of runs such that the limit-average value stays above a given vector. We show that under the expectation objective, in contrast to the single-objective case, both randomization and memory are necessary for strategies, and that finite-memory randomized strategies are sufficient. Under the satisfaction objective, in contrast to the single-objective case, infinite memory is necessary for strategies, and that randomized memoryless strategies are sufficient for epsilon-approximation, for all epsilon>0. We further prove that the decision problems for both expectation and satisfaction objectives can be solved in polynomial time and the trade-off curve (Pareto curve) can be epsilon-approximated in time polynomial in the size of the MDP and 1/epsilon, and exponential in the number of reward functions, for all epsilon>0. Our results also reveal flaws in previous work for MDPs with multiple mean-payoff functions under the expectation objective, correct the flaws and obtain improved results.