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 NP∩coNP, matching the current known bound for single objectives; and in general the decision problem is PSPACE-hard and can be solved in 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.
Formal Methods in System Design
Tobias Winkler and Joost-Pieter Katoen are supported by the DFG RTG 2236 UnRAVeL and the innovation programme under the Marie Skłodowska-Curie grant agreement No. 101008233 (Mission). Krishnendu Chatterjee is supported by the ERC CoG 863818 (ForM-SMArt) and the Vienna Science and Technology Fund (WWTF) Project ICT15-003. Maximilian Weininger is supported by the DFG projects 383882557 Statistical Unbounded Verification (SUV) and 427755713 Group-By Objectives in Probabilistic Verification (GOPro). Stefanie Mohr is supported by the DFG RTG 2428 CONVEY. Open Access funding enabled and organized by Projekt DEAL.
Chatterjee K, Katoen JP, Mohr S, Weininger M, Winkler T. Stochastic games with lexicographic objectives. Formal Methods in System Design. 2023. doi:10.1007/s10703-023-00411-4
Chatterjee, K., Katoen, J. P., Mohr, S., Weininger, M., & Winkler, T. (2023). Stochastic games with lexicographic objectives. Formal Methods in System Design. Springer Nature. https://doi.org/10.1007/s10703-023-00411-4
Chatterjee, Krishnendu, Joost P Katoen, Stefanie Mohr, Maximilian Weininger, and Tobias Winkler. “Stochastic Games with Lexicographic Objectives.” Formal Methods in System Design. Springer Nature, 2023. https://doi.org/10.1007/s10703-023-00411-4.
K. Chatterjee, J. P. Katoen, S. Mohr, M. Weininger, and T. Winkler, “Stochastic games with lexicographic objectives,” Formal Methods in System Design. Springer Nature, 2023.
Chatterjee K, Katoen JP, Mohr S, Weininger M, Winkler T. 2023. Stochastic games with lexicographic objectives. Formal Methods in System Design.
Chatterjee, Krishnendu, et al. “Stochastic Games with Lexicographic Objectives.” Formal Methods in System Design, Springer Nature, 2023, doi:10.1007/s10703-023-00411-4.
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