We introduce quantitative reductions, a novel technique for structuring the space of quantitative games and solving them that does not rely on a reduction to qualitative games. We show that such reductions exhibit the same desirable properties as qualitative ones and that they additionally retain the optimality of solutions. Moreover, we introduce vertex-ranked games as general-purpose targets for quantitative reductions and show how to solve them. In such games, the value of a play is determined only by a qualitative winning condition and a vertex-ranking.

We provide quantitative reductions of quantitative request-response games and of quantitative Muller games to vertex-ranked games, thus showing ExpTime-completeness of solving the former two kinds of games. In addition, we exhibit the usefulness and flexibility of vertex-ranked games by using them to compute fault-resilient strategies for safety specifications. This lays the foundation for a general study of fault-resilient strategies for more complex winning conditions.

我们介绍了量化归约法，这是一种用于构造量化博弈空间并解决不依赖于定性博弈归约法的新颖技术。我们表明，这种减少表现出与定性相同的理想特性，并且它们还保留了解决方案的最优性。此外，我们将顶点排序的游戏作为量化减少的通用目标，并展示了解决方法。在这样的游戏中，比赛的价值仅由定性的获胜条件和顶点排名决定。

我们将量化的请求响应游戏和量化的Muller游戏减少到顶点排名的游戏，从而显示ExpTime-解决前两种游戏的完备性。此外，我们通过使用顶点排序游戏计算安全规范的容错策略来展示其有用性和灵活性。这为针对更复杂的获胜条件的故障恢复策略的一般研究奠定了基础。