Single-Deviation Stability in Additively Separable Hedonic Games with Constrained Coalition Sizes

TL;DR

This paper investigates stability in additively separable hedonic games with constrained coalition sizes, focusing on four single-agent deviation notions ($NS$, $IS$, $CNS$, $CIS$) and their feasible variants. It provides a complete existence/classification picture for fixed bounds and delivers a detailed complexity landscape: polynomial-time algorithms for CIS and CNS when the upper bound is $2$, polynomial-time CIS$^*$ constructions under nonzero or nonnegative valuations, and NP-hardness for most other combinations in the $(\lambda,\mu)$-bounded regime. The authors also establish foundational results on the existence of feasible partitions, and show that when only upper bounds are present, feasibility and stability interact in nuanced ways, especially for lower bounds $\lambda\ge 2$. The work advances understanding of how size constraints affect stability and computability in hedonic games, with implications for practical group formation problems where minimum/maximum group sizes must be respected.

Abstract

We study stability in additively separable hedonic games when coalition sizes have to respect fixed size bounds. We consider four classic notions of stability based on single-agent deviations, namely, Nash stability, individual stability, contractual Nash stability, and contractual individual stability. For each stability notion, we consider two variants: in one, the coalition left behind by a deviator must still be of a valid size, and in the other there is no such constraint. We provide a full picture of the existence of stable outcomes with respect to given size parameters. Additionally, when there are only upper bounds, we fully characterize the computational complexity of the associated existence problem. In particular, we obtain polynomial-time algorithms for contractual individual stability and contractual Nash stability, where the latter requires an upper bound of 2. We obtain further results for Nash stability and contractual individual stability, when the lower bound is at least 2.

Figures (7)

• Figure 1: Overview and logical implications of stability concepts. Each standard stability concept is associated with a feasible stability concept indicated by a star.
• Figure 2: Illustration of the game $(N,v)$ in \ref{['ex:intro']} for the case $k = 3$. Black and red edges indicate a symmetric valuation of $1$ and $-1$, respectively. The partition $\mathfrak C$ is indicated in blue.
• Figure 3: ASHG constructed in the proof of \ref{['prop:nonexistCNSasy']}. Red edges indicate valuations of $-1$, while missing edges indicate a valuation of $0$.
• Figure 4: Schematic of the reduction from the proof of \ref{['thm:CNS-Uge3']}. We depict the reduced instance for the instance $\langle R,\mathfrak S{}\rangle$ of X3C where $R = \{1,2,3,4,5,6\}$, and $\mathfrak S{} = \{S,T,U\}$, with $S = \{1,2,3\}$, $T = \{2,3,4\}$, and $U = \{4,5,6\}$. Exact valuations are defined in the appendix. Thin edges represent a valuation of $1$, thick edges represent a valuation of $2$, and red edges represent a valuation of $-1$. Dashed edges represent a valuation of $0$. If the edge is undirected, the same valuation applies in both directions. Omitted edges represent a negative utility of $-3$. In blue, we highlight the nonsingleton coalitions of the CNS partition corresponding to the cover $\{S,U\}$.
• Figure 5: Illustration of the ASHG of \ref{['ex:CISflawed']}. We only depict the nonzero valuations.

Theorems & Definitions (47)

• Proposition 4.0
• proof
• Proposition 4.0
• proof
• Proposition 4.0
• proof
• Example 5.1
• Proposition 5.1
• proof
• Proposition 5.1