Quantifying Core Stability Relaxations in Hedonic Games

TL;DR

The paper introduces a unified α-hedonic game framework to relate two core-stability relaxations—q-size core stability and k-improvement core stability—through a general bound on blocking improvements. The main theorem shows that any $k$-improvement core stable structure for coalitions of sizes up to $q$ is $(m, f(q,m))$-core stable for all $m\ge q+1$, with $f(q,m)$ capturing the interaction between coalition size, the floor and modulo structure, and the α-weighting factors. This result specializes to well-studied classes (fractional, modified fractional, and additively separable hedonic games) and confirms two open conjectures for fractional hedonic games, while also enabling a polynomial-time ILP approach to explore tightness and extreme blocking coalitions. Moreover, the work extends to novel hedonic game models, providing core-existence guarantees and assessing the price of anarchy under these relaxations. The findings unify disparate strands of hedonic game theory, offer practical bounds for approximation quality, and highlight avenues for future research on tightness and dynamic convergence in broader α-hedonic settings.

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 $α$-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.

Figures (8)

• Figure 1: Example of a symmetric hedonic game on the left and of a blocking coalition for an asymmetric game on the left, for which the improvement ratio is unbounded.
• Figure 2: Plotted values of $f(q,m)$ for FHGs (left) and ASHGs (right) such that every $q$-size core stable coalition structure is $(m,f(q,m))$-core stable.
• Figure 3: Examples of maximum-size triangle-free graphs with $4$ to $7$ vertices.
• Figure 4: Plotted values of $f(q,m)$ for FHGs such that every $q$-size core stable coalition structure is $(m,f(q,m))$-core stable, together with the obtained lower bounds (LB) in \ref{['sec:lowerBounds']}.
• Figure 5: Tight example for FHGs for $q = 5$ and $m = 7$. Solid edges indicate a weight of $2$ while dashed edges indicate a weight of $1$. Each agent has an initial utility of $1$.

Theorems & Definitions (37)

• Example 1
• Proposition 1
• proof
• Theorem 1
• proof
• Corollary 1
• Corollary 2
• proof
• Corollary 3
• proof