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

TL;DR

This paper unifies two perspectives on optimizing multiple mean-payoff objectives in Markov decision processes, providing polynomial algorithms and strategy complexity characterizations for combined expectation and satisfaction goals.

Contribution

It introduces a unified framework for optimizing expectation while satisfying probabilistic guarantees in MDPs, with polynomial algorithms and strategy complexity analysis.

Findings

Polynomial algorithms for decision problems in the unified framework.

Polynomial-time approximation of the Pareto-curve with exponential dependence on dimensions.

Complete characterization of strategy complexity in terms of memory and randomization.

Abstract

We consider Markov decision processes (MDPs) with multiple limit-average (or mean-payoff) objectives. There exist two different views: (i) the expectation semantics, where the goal is to optimize the expected mean-payoff objective, and (ii) the satisfaction semantics, where the goal is to maximize the probability of runs such that the mean-payoff value stays above a given vector. We consider optimization with respect to both objectives at once, thus unifying the existing semantics. Precisely, the goal is to optimize the expectation while ensuring the satisfaction constraint. Our problem captures the notion of optimization with respect to strategies that are risk-averse (i.e., ensure certain probabilistic guarantee). Our main results are as follows: First, we present algorithms for the decision problems which are always polynomial in the size of the MDP. We also show that an approximation of the Pareto-curve can be computed in time polynomial in the size of the MDP, and the approximation factor, but exponential in the number of dimensions. Second, we present a complete characterization of the strategy complexity (in terms of memory bounds and randomization) required to solve our problem.