Stability in Online Coalition Formation

TL;DR

The paper investigates stability in online coalition formation within additively separable hedonic games, where agents arrive sequentially and must be assigned to coalitions immediately. It analyzes a range of stability notions (NS, IS, CNS, CIS, CR, SCR) plus Pareto optimality, showing a dichotomy: deterministic online algorithms can achieve CNS in symmetric games with certain utility restrictions and PO in strict ASHGs, while any randomized online algorithm generally cannot guarantee a positive probability of stability for several notions. The results highlight fundamental limitations of online stability compared to offline settings, using Yao's principle and carefully constructed adversarial instances. These findings inform both theoretical understanding and practical design of online multi-agent systems, emphasizing the challenge of achieving robust stability under online arrivals without welfare optimization.

Abstract

Coalition formation is concerned with the question of how to partition a set of agents into disjoint coalitions according to their preferences. Deviating from most of the previous work, we consider an online variant of the problem, where agents arrive in sequence. Whenever an agent arrives, they must be assigned to a coalition immediately and irrevocably. The scarce existing literature on online coalition formation has focused on maximizing social welfare, a demanding requirement, even in the offline setting. Instead, we seek to achieve \emph{stable} coalition structures online and treat the most common stability concepts based on deviations by single agents and groups of agents. We present a comprehensive picture in additively separable hedonic games, leading to dichotomies, where positive results are obtained by deterministic algorithms and negative results even hold for randomized algorithms.

Figures (6)

• Figure 1: Logical relationships between our solution concepts AzSa15a. An arrow from concept $\alpha$ to concept $\beta$ indicates that if a partition satisfies $\alpha$, then it also satisfies $\beta$. For reference, we also depict welfare optimality.
• Figure 2: Adversarial AFGs for computing CNS partitions. Every deterministic algorithm fails for one of the two possible instances.
• Figure 3: Adversarial FENGs for computing CIS partitions. Every deterministic algorithm fails for one of the two possible instances.
• Figure 4: Adversarial instance for achieving individual stability in $\{-y,x\}$-ASHGs for $x,y>0$. We only depict the positive utilities of $x$. All remaining utilities are $-y$.
• Figure 5: Adversarial instance for achieving partitions in the (strict) core in $\{-y,x\}$-ASHGs for $x,y>0$. All such instances have a nonempty strict core. We only depict the positive utilities of $x$. All remaining utilities are $-y$.

Theorems & Definitions (29)

• Proposition 4.1
• proof
• Theorem 4.2
• proof
• Claim 1
• proof
• Theorem 4.3
• proof
• Theorem 4.4
• proof