Quantifying Core Stability Relaxations in Hedonic Games

TL;DR

This paper investigates the stability of coalition formations in hedonic games, providing bounds on coalition improvements, analyzing the effects of coalition size, and proving conjectures for fractional hedonic games.

Contribution

It introduces a unified approach to analyze core stability relaxations in $oldsymbol{ extit{ extalpha}}$-hedonic games, resolving open conjectures for fractional hedonic games.

Findings

Derived an upper bound on coalition improvement factors.

Showed larger coalitions can have lower improvement factors.

Proved two open conjectures for fractional hedonic games.

Abstract

We study relationships between different relaxed notions of core stability in hedonic games, which are a class of coalition formation games. Our unified approach applies to a newly introduced family of hedonic games, called $\alpha$-hedonic games, which contains previously studied variants such as fractional and additively separable hedonic games. In particular, we derive an upper bound on the maximum factor with which a blocking coalition of a certain size can improve upon an outcome in which no deviating coalition of size at most $q$ exists. Counterintuitively, we show that larger blocking coalitions might sometimes have lower improvement factors. We discuss the tightness conditions of our bound, as well as its implications on the price of anarchy of core relaxations. Our general result has direct implications for several well-studied classes of hedonic games, allowing us to prove two open conjectures by Fanelli et al. (2021) for fractional hedonic games.