# Fractional Hedonic Games

## TL;DR

This paper introduces fractional hedonic games, a coalition formation model where players value coalitions based on the average of their members, exploring core stability conditions and computational complexity.

## Contribution

It formally studies fractional hedonic games, providing conditions for core non-emptiness, algorithms for stable outcomes, and complexity results.

## Key findings

- Core is non-empty under certain conditions
- Algorithms can compute core stable outcomes
- Deciding core non-emptiness is computationally hard

## Abstract

The work we present in this paper initiated the formal study of fractional hedonic games, coalition formation games in which the utility of a player is the average value he ascribes to the members of his coalition. Among other settings, this covers situations in which players only distinguish between friends and non-friends and desire to be in a coalition in which the fraction of friends is maximal. Fractional hedonic games thus not only constitute a natural class of succinctly representable coalition formation games, but also provide an interesting framework for network clustering. We propose a number of conditions under which the core of fractional hedonic games is non-empty and provide algorithms for computing a core stable outcome. By contrast, we show that the core may be empty in other cases, and that it is computationally hard in general to decide non-emptiness of the core.