Stopping Criteria for Value Iteration on Stochastic Games with Quantitative Objectives

A classic solution technique for Markov decision processes (MDP) and stochastic games (SG) is value iteration (VI). Due to its good practical performance, this approximative approach is typically preferred over exact techniques, even though no practical bounds on the imprecision of the result could be given until recently. As a consequence, even the most used model checkers could return arbitrarily wrong results. Over the past decade, different works derived stopping criteria, indicating when the precision reaches the desired level, for various settings, in particular MDP with reachability, total reward, and mean payoff, and SG with reachability. In this paper, we provide the first stopping criteria for VI on SG with total reward and mean payoff, yielding the first anytime algorithms in these settings. To this end, we provide the solution in two flavours: First through a reduction to the MDP case and second directly on SG. The former is simpler and automatically utilizes any advances on MDP. The latter allows for more local computations, heading towards better practical efficiency. Our solution unifies the previously mentioned approaches for MDP and SG and their underlying ideas. To achieve this, we isolate objective-specific subroutines as well as identify objective-independent concepts. These structural concepts, while surprisingly simple, form the very essence of the unified solution.