Stochastic games with lexicographic objectives

Formal Methods in System Design, vol. 63, pp. 40–80

Abstract

Abstract We study turn-based stochastic zero-sum games with lexicographic preferences over objectives. Stochastic games are standard models in control, verification, and synthesis of stochastic reactive systems that exhibit both randomness as well as controllable and adversarial non-determinism. Lexicographic order allows one to consider multiple objectives with a strict preference order. To the best of our knowledge, stochastic games with lexicographic objectives have not been studied before. For a mixture of reachability and safety objectives, we show that deterministic lexicographically optimal strategies exist and memory is only required to remember the already satisfied and violated objectives. For a constant number of objectives, we show that the relevant decision problem is in $$\textsf{NP}\cap \textsf{coNP}$$ NP ∩ coNP , matching the current known bound for single objectives; and in general the decision problem is $$\textsf{PSPACE}$$ PSPACE -hard and can be solved in $$\textsf{NEXPTIME}\cap \textsf{coNEXPTIME}$$ NEXPTIME ∩ coNEXPTIME . We present an algorithm that computes the lexicographically optimal strategies via a reduction to the computation of optimal strategies in a sequence of single-objectives games. For omega-regular objectives, we restrict our analysis to one-player games, also known as Markov decision processes. We show that lexicographically optimal strategies exist and need either randomization or finite memory. We present an algorithm that solves the relevant decision problem in polynomial time. We have implemented our algorithms and report experimental results on various case studies.

Authors 5

1. Krishnendu Chatterjee

   Institute of Science and Technology Austria

   Affiliation as printed

   IST Austria, Klosterneuburg, Austria

2. Joost-Pieter Katoen Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, Aachen, Germany

3. Stefanie Mohr

   Technical University of Munich

   Affiliation as printed

   Technical University of Munich, Munich, Germany

4. Maximilian Weininger

   Technical University of Munich

   Affiliation as printed

   Technical University of Munich, Munich, Germany

5. Tobias Winkler corresponding Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, Aachen, Germany