Stopping Criteria for Value Iteration on Concurrent Stochastic Reachability and Safety Games

TL;DR

This paper introduces a bounded value iteration method for concurrent stochastic games that guarantees approximation precision by using over- and under-approximations, improving upon traditional stopping criteria.

Contribution

It proposes a new bounded value iteration approach for CSGs that ensures convergence within a specified error bound, addressing limitations of traditional stopping criteria.

Findings

Bounded VI provides guaranteed approximation bounds.

The method converges reliably within specified error margins.

It outperforms traditional VI stopping criteria in accuracy.

Abstract

We consider two-player zero-sum concurrent stochastic games (CSGs) played on graphs with reachability and safety objectives. These include degenerate classes such as Markov decision processes or turn-based stochastic games, which can be solved by linear or quadratic programming; however, in practice, value iteration (VI) outperforms the other approaches and is the most implemented method. Similarly, for CSGs, this practical performance makes VI an attractive alternative to the standard theoretical solution via the existential theory of reals. VI starts with an under-approximation of the sought values for each state and iteratively updates them, traditionally terminating once two consecutive approximations are $\epsilon$-close. However, this stopping criterion lacks guarantees on the precision of the approximation, which is the goal of this work. We provide bounded (a.k.a. interval) VI for CSGs: it complements standard VI with a converging sequence of over-approximations and terminates once the over- and under-approximations are $\epsilon$-close.