Stability in Random Hedonic Games

TL;DR

This work studies random additively separable hedonic games (ASHGs) with i.i.d. utilities to understand stability beyond worst-case guarantees. It introduces a three-stage clustering algorithm that, with high probability, outputs a partition that is individually rational and entry-/exit-denying, with coalitions of size $\Theta(\log n)$ and count $\Theta(n/\log n)$. The results show that while independent single-agent deviations are responsible for much instability, coordinated stability (IS/CNS) can be achieved in random models, and Nash stability tends to vanish when utilities follow $U(-1,1)$. The findings illuminate the practical feasibility of stable coalition structures in large, randomly-structured groups and clarify the roles of different stability notions. The analysis leverages probabilistic combinatorics and random-graph techniques to bridge game theory and algorithmic stability in a mean-zero, adversarially unbiased setting.

Abstract

Partitioning a large group of employees into teams can prove difficult because unsatisfied employees may want to transfer to other teams. In this case, the team (coalition) formation is unstable and incentivizes deviation from the proposed structure. Such a coalition formation scenario can be modeled in the framework of hedonic games and a significant amount of research has been devoted to the study of stability in such games. Unfortunately, stable coalition structures are not guaranteed to exist in general and their practicality is further hindered by computational hardness barriers. We offer a new perspective on this matter by studying a random model of hedonic games. For three prominent stability concepts based on single-agent deviations, we provide a high probability analysis of stability in the large agent limit. Our first main result is an efficient algorithm that outputs an individually and contractually Nash-stable partition with high probability. Our second main result is that the probability that a random game admits a Nash-stable partition tends to zero. Our approach resolves the two major downsides associated with individual stability and contractual Nash stability and reveals agents acting single-handedly are usually to blame for instabilities.

Figures (3)

• Figure 1: Stability concepts for hedonic games, where arrows represent logical relationships; see \ref{['sec:stab']} for formal definitions. Contractual individual stability and individual rationality can always be satisfied while there exist No-instances for all other solution concepts.
• Figure 2: Visualization of our coalition formation algorithm. In the first stage, illustrated by the vertical sets, we consider $G^i$ for $i\in [20]$ and form clique coalitions $C^i_1,\dots,C^1_t$ by \ref{['alg:greedycol']}. In each of the $G^i$, we have a set $R^i_1$ of remaining vertices. In the second stage, illustrated by horizontal coalitions, we merge the cliques obtained in the first stage by \ref{['alg:greedy']} form the merged coalitions $C_1,\dots, C_{t'}$. We are left with the remainder set $R$ of agents that we add to the obtained coalitions in the final stage.
• Figure 3: Visualization of utility revelation in Stage 1. Assume that $C$ is the tentative coalition. When checking whether $a$ can join $C$, we reveal a utility smaller than $\frac{1}{2}$ and exit the if-condition. For $b$, all utilities are large enough and therefore $b$ is added to $C$.

Theorems & Definitions (53)

• Proposition 4.0
• proof
• Lemma 4.1: Hoef63a
• Proposition 4.1
• proof
• Theorem 5.1
• proof
• Theorem 5.2
• proof
• Definition 5.3