Quantitative Fairness Games

TL;DR

This paper studies two-player games on finite colored graphs with goals related to the frequency of colors in infinite paths, providing complexity results for the existence of winning strategies.

Contribution

It introduces quantitative fairness properties as refinements of classical fairness and establishes the CoNP-completeness of the strategy existence problem.

Findings

Strategies for fairness properties are CoNP-complete to verify.

Quantitative fairness properties generalize classical fairness notions.

Results apply to systems requiring balanced or prioritized treatment.

Abstract

We consider two-player games played on finite colored graphs where the goal is the construction of an infinite path with one of the following frequency-related properties: (i) all colors occur with the same asymptotic frequency, (ii) there is a constant that bounds the difference between the occurrences of any two colors for all prefixes of the path, or (iii) all colors occur with a fixed asymptotic frequency. These properties can be viewed as quantitative refinements of the classical notion of fair path in a concurrent system, whose simplest form checks whether all colors occur infinitely often. In particular, the first two properties enforce equal treatment of all the jobs involved in the system, while the third one represents a way to assign a given priority to each job. For all the above goals, we show that the problem of checking whether there exists a winning strategy is CoNP-complete.