Quantitative Reductions and Vertex-Ranked Infinite Games

TL;DR

This paper introduces quantitative reductions and vertex-ranked games as new tools for analyzing and solving quantitative games, preserving optimality and enabling fault-resilient strategies, with applications to request-response games.

Contribution

It presents the novel concepts of quantitative reductions and vertex-ranked games, providing methods to solve complex quantitative games without qualitative reductions and demonstrating their effectiveness.

Findings

Quantitative reductions retain optimality and desirable properties.

Vertex-ranked games can be solved efficiently and serve as a versatile target.

Application to fault-resilient strategies for safety specifications.

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. 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