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

2024-08-23

arXiv.org

AI Summary

In quantitative reachability games where no winning strategy exists, systems must still “do their best” to satisfy specifications. Method: This paper introduces *admissible strategies*—maximizing specification satisfaction—and *admissibly winning strategies*, which jointly optimize admissibility and optimal payoff. We prove that both exist under finite payoffs but are necessarily history-dependent and cannot be realized by memoryless strategies; we further derive necessary and sufficient conditions for their existence and provide a synthesis algorithm. Contribution/Results: Our work is the first to formally establish the intrinsic history-dependence of admissible strategies in quantitative settings and to propose a novel strategy paradigm balancing robustness (via admissibility) and optimality (via payoff maximization). Empirical evaluation on gridworld and robotic arm benchmarks demonstrates the method’s effectiveness and practical applicability.

Technology Category

• Search and Optimization: Adversarial Search
• Multiagent Systems: Adversarial Agents
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Investigates admissible strategies in quantitative reachability games
• Addresses synthesis challenges of history-dependent admissible strategies
• Proposes admissible winning strategies for optimal outcomes

Innovation

• Admissible strategies formalize doing-your-best approach
• Admissible winning strategies enforce best possible outcomes
• Synthesis algorithms for history-dependent strategies provided