# Quantitative Reductions and Vertex-Ranked Infinite Games (Full Version)

## TL;DR

This paper introduces quantitative reductions and vertex-ranked games, providing a new framework for solving quantitative games without reducing to qualitative ones, and demonstrates their effectiveness and computational complexity.

## Contribution

It presents a novel technique for structuring and solving quantitative games directly, introduces vertex-ranked games as a versatile target, and applies these concepts to request-response and Muller games.

## Key findings

- Quantitative reductions retain optimality of solutions.
- Vertex-ranked games can be solved efficiently and are useful for fault-resilient strategies.
- Solving request-response and Muller games is EXPTIME-complete.

## 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 that they 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 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 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.