Quantitative Reductions and Vertex-Ranked Infinite Games

Abstract

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 their qualitative counterparts and additionally retain the optimality of solutions. Moreover, we introduce vertex-ranked games as a general-purpose target 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 ranking of the vertices.

We provide quantitative reductions of quantitative request-response games to vertex-ranked games, thus showing ExpTime-completeness of solving the former games. Moreover, we reduce quantitative Muller games to vertex-ranked games, yielding a new proof of ExpTime-membership of the problem of solving the former games. Furthermore, we exhibit the usefulness and flexibility of vertex-ranked games by showing how to use such games to compute fault-resilient strategies for safety specifications. This work lays the foundation for a general study of fault-resilient strategies for more complex winning conditions.