Termination Criteria for Solving Concurrent Safety and Reachability Games

TL;DR

This paper introduces a strategy improvement algorithm for concurrent safety games on graphs, enabling the approximation of game values from below and facilitating the computation of bounds from both directions.

Contribution

It presents the first strategy improvement algorithm that approximates the value of concurrent safety games from below, enhancing solution methods for these complex games.

Findings

The algorithm produces a sequence of strategies with monotonically increasing probabilities of winning.

Combining this with existing methods allows for converging bounds on game values.

Previous methods could only approximate from one side without known convergence rates.

Abstract

We consider concurrent games played on graphs. At every round of a game, each player simultaneously and independently selects a move; the moves jointly determine the transition to a successor state. Two basic objectives are the safety objective to stay forever in a given set of states, and its dual, the reachability objective to reach a given set of states. We present in this paper a strategy improvement algorithm for computing the value of a concurrent safety game, that is, the maximal probability with which player~1 can enforce the safety objective. The algorithm yields a sequence of player-1 strategies which ensure probabilities of winning that converge monotonically to the value of the safety game. Our result is significant because the strategy improvement algorithm provides, for the first time, a way to approximate the value of a concurrent safety game from below. Since a value iteration algorithm, or a strategy improvement algorithm for reachability games, can be used to approximate the same value from above, the combination of both algorithms yields a method for computing a converging sequence of upper and lower bounds for the values of concurrent reachability and safety games. Previous methods could approximate the values of these games only from one direction, and as no rates of convergence are known, they did not provide a practical way to solve these games.