Deviation Dynamics in Cardinal Hedonic Games

TL;DR

This paper develops meta-theorems that connect the existence of No-instances in cardinal hedonic games to hardness results for deciding possible and necessary convergence of deviation dynamics across ASHGs, FHGs, and MFHGs. It unifies a broad family of stability notions, including voting-based and anonymous, monotone conditions, and provides RX3C-based reductions to show hardness, while also offering a constructive CIS dynamics approach that can efficiently yield individually rational CIS partitions from the singleton start. The results demonstrate a dichotomy: CIS dynamics guarantee convergence, while many other stability notions lead to hardness unless the dynamics follow well-structured paths, with linear convergence guaranteed from singleton starts but potential exponential behavior in worst cases. These insights advance understanding of decentralized coalition formation by delivering a general hardness framework and practical dynamics for contracting into stable partitions in dynamic hedonic games.

Abstract

Computing stable partitions in hedonic games is a challenging task because there exist games in which stable outcomes do not exist. Even more, these No-instances can often be leveraged to prove computational hardness results. We make this impression rigorous in a dynamic model of cardinal hedonic games by providing meta theorems. These imply hardness of deciding about the possible or necessary convergence of deviation dynamics based on the mere existence of No-instances. Our results hold for additively separable, fractional, and modified fractional hedonic games (ASHGs, FHGs, and MFHGs). Moreover, they encompass essentially all reasonable stability notions based on single-agent deviations. In addition, we propose dynamics as a method to find individually rational and contractually individual stable (CIS) partitions in ASHGs. In particular, we find that CIS dynamics from the singleton partition possibly converge after a linear number of deviations but may require an exponential number of deviations in the worst case.

Figures (9)

• Figure 1: Illustration of an $\text{ASHG}$. Blue boxes indicate the initial partition. A straight arrow from an agent $x$ to an agent $y$ indicates $v\xspace_x(y) = 1$ while a dashed arrow indicates $v\xspace_x(y) = -1$. Missing arrows indicate a valuation of $0$.
• Figure 2: Illustration of the reduction. A covering instance $(\mathcal{U}, \mathcal{M})$ is represented by agents $N_{\mathcal{U}}$ and $N_{\mathcal{M}}$. Here, we have $\mathcal{U} = \{a, \ldots, f\}$ and $\mathcal{M} = \{X, Y, Z\}$ with $X = \{a, b, c\}$, $Y = \{b, c, d\}$, and $Z = \{d, e, f\}$. Black and red arrows indicate potential utility increases and decreases, respectively. Important coalitions of the starting partition are indicated in blue. In Yes-instances, dynamics can lead to agent $\gamma$ ending up in a singleton coalition.
• Figure 3: Illustration of the reduction. For the sake of simplicity, the depicted reduction is from X3C instead of RX3C. However, the schematic is analogous apart from a small change to the valuation $v\xspace_\beta(\alpha)$. The reduced instance for the source instance $(\mathcal{U}, \mathcal{M})$ is displayed, where $\mathcal{U} = \{a, \ldots, f\}$, and $\mathcal{M} = \{X, Y, Z\}$ with $X = \{a, b, c\}$, $Y = \{b, c, d\}$, and $Z = \{d, e, f\}$. A directed edge from agent $p$ to agent $d$ represents the valuation $v\xspace_p(d)$. Whenever two or more displayed agents belong to the same coalition in the starting partition $\pi_0$, we indicate this by blue boxes.
• Figure 4: Illustration of the reduction. For the sake of simplicity, the depicted reduction is from X3C instead of RX3C. However, the schematic is analogous apart from a small change to the valuation $v\xspace_\beta(\alpha)$. The reduced instance for the source instance $(\mathcal{U}, \mathcal{M})$ is displayed, where $\mathcal{U} = \{a, \ldots, f\}$, and $\mathcal{M} = \{X, Y, Z\}$ with $X = \{a, b, c\}$, $Y = \{b, c, d\}$, and $Z = \{d, e, f\}$. A directed edge from agent $p$ to agent $d$ represents the valuation $v\xspace_p(d)$. Whenever two or more displayed agents belong to the same coalition in the starting partition $\pi_0$, we indicate this by blue boxes.
• Figure 5: Illustration of the reduction for the proof of \ref{['thm:unified_PCD']}.

Theorems & Definitions (53)

• Example 1
• Proposition 1
• Theorem 3
• Theorem 4
• Corollary 5
• Theorem 6
• Theorem 8
• Theorem 9
• proof
• Claim 9