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Concurrent Stochastic Games with Stateful-Discounted and Parity Objectives: Complexity and Algorithms

Authors Ali Asadi , Krishnendu Chatterjee , Raimundo Saona , Jakub Svoboda

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Author Details

Ali Asadi
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Krishnendu Chatterjee
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Raimundo Saona
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Jakub Svoboda
  • Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria

Funding

This research was partially supported by ERC CoG 863818 (ForM-SMArt), Austrian Science Fund (FWF) 10.55776/COE12, and French Agence Nationale de la Recherche (ANR) ANR-21-CE40-0020 (CONVERGENCE project).

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Ali Asadi, Krishnendu Chatterjee, Raimundo Saona, and Jakub Svoboda. Concurrent Stochastic Games with Stateful-Discounted and Parity Objectives: Complexity and Algorithms. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FSTTCS.2024.5

Abstract

We study two-player zero-sum concurrent stochastic games with finite state and action space played for an infinite number of steps. In every step, the two players simultaneously and independently choose an action. Given the current state and the chosen actions, the next state is obtained according to a stochastic transition function. An objective is a measurable function on plays (or infinite trajectories) of the game, and the value for an objective is the maximal expectation that the player can guarantee against the adversarial player. We consider: (a) stateful-discounted objectives, which are similar to the classic discounted-sum objectives, but states are associated with different discount factors rather than a single discount factor; and (b) parity objectives, which are a canonical representation for ω-regular objectives. For stateful-discounted objectives, given an ordering of the discount factors, the limit value is the limit of the value of the stateful-discounted objectives, as the discount factors approach zero according to the given order.
The computational problem we consider is the approximation of the value within an arbitrary additive error. The above problem is known to be in EXPSPACE for the limit value of stateful-discounted objectives and in PSPACE for parity objectives. The best-known algorithms for both the above problems are at least exponential time, with an exponential dependence on the number of states and actions. Our main results for the value approximation problem for the limit value of stateful-discounted objectives and parity objectives are as follows: (a) we establish TFNP[NP] complexity; and (b) we present algorithms that improve the dependency on the number of actions in the exponent from linear to logarithmic. In particular, if the number of states is constant, our algorithms run in polynomial time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
Keywords
  • Concurrent Stochastic Games
  • Parity Objectives
  • Discounted-sum Objectives

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