Beyond Winning Strategies: Admissible and Admissible Winning Strategies for Quantitative Reachability Games

TL;DR

This paper explores admissible and admissible winning strategies in quantitative reachability games, addressing synthesis challenges, existence conditions, and proposing algorithms with practical examples.

Contribution

It introduces the concept of admissible strategies for quantitative games, proves their existence, and develops synthesis algorithms for practical applications.

Findings

Admissible strategies always exist in quantitative reachability games.

Both admissible and admissible winning strategies are history-dependent and non-memoryless.

The paper provides necessary and sufficient conditions for strategy existence.

Abstract

Classical reactive synthesis approaches aim to synthesize a reactive system that always satisfies a given specifications. These approaches often reduce to playing a two-player zero-sum game where the goal is to synthesize a winning strategy. However, in many pragmatic domains, such as robotics, a winning strategy does not always exist, yet it is desirable for the system to make an effort to satisfy its requirements instead of "giving up". To this end, this paper investigates the notion of admissible strategies, which formalize "doing-your-best", in quantitative reachability games. We show that, unlike the qualitative case, quantitative admissible strategies are history-dependent even for finite payoff functions, making synthesis a challenging task. In addition, we prove that admissible strategies always exist but may produce undesirable optimistic behaviors. To mitigate this, we propose admissible winning strategies, which enforce the best possible outcome while being admissible. We show that both strategies always exist but are not memoryless. We provide necessary and sufficient conditions for the existence of both strategies and propose synthesis algorithms. Finally, we illustrate the strategies on gridworld and robot manipulator domains.