Online Coalition Formation under Random Arrival or Coalition Dissolution

TL;DR

This work analyzes online coalition formation in additively separable hedonic games under two online variants: random arrival and free dissolution. It shows that the classical greedy approach loses utility-dependent worst-case guarantees under random arrivals, achieving $\Theta\left(\frac{1}{n^{2}}\right)$, while awaiting strategies recover $\Theta\left(\frac{1}{n}\right)$. In the free-dissolution model, dissolving coalitions enables strong connections to online matching, yielding an asymptotically optimal $\Theta\left(\frac{1}{n}\right)$ competitive ratio, and in the deterministic arrival setting the upper bound tightens to $\Theta\left(\frac{1}{n}\right)$ (with a universal $12/n$ cap for any algorithm). Across both models, the results remove the dependence on the utility range seen in the basic model and align online performance with the best known offline approximations, all achieved by deterministic, polynomial-time procedures. The work highlights the central role of online matching techniques in improving online coalition formation and outlines directions for tighter constants and potential randomized improvements.

Abstract

Coalition formation explores how to partition a set of $n$ agents into disjoint coalitions according to their preferences. We consider a cardinal utility model with an additively separable aggregation of preferences and study the online variant of coalition formation, where the agents arrive in sequence. The goal is to achieve competitive social welfare. In the basic model, agents arrive in an arbitrary order and have to be assigned to coalitions immediately and irrevocably. There, the natural greedy algorithm is known to achieve an optimal competitive ratio, which heavily relies on the range of utilities. We complement this result by considering two related models. First, we study a model where agents arrive in a random order. We find that the competitive ratio of the greedy algorithm is $Θ\left(\frac{1}{n^2}\right)$. In contrast, an alternative algorithm, which is based on alternating between waiting and greedy phases, can achieve a competitive ratio of $Θ\left(\frac{1}{n}\right)$. Second, we relax the irrevocability of decisions by allowing the dissolution of coalitions into singleton coalitions. We achieve an asymptotically optimal competitive ratio of $Θ\left(\frac 1n\right)$ by drawing a close connection to a general model of online matching. Hence, in both models, we obtain a competitive ratio that removes the unavoidable utility dependencies in the basic model and essentially matches the best possible approximation ratio by polynomial-time algorithms.

Figures (6)

• Figure 1: Hard instance for online coalition formation algorithms under random arrival. The example contains $n = 2k + 2$ agents. There are $k$ agents in each of the sets $X$ and $Y$. The utility between $a$ and any agent in $X$ and $b$ and any agent in $Y$ is $\epsilon$. All omitted edges represent weights of $-1$.
• Figure 2: Family of instances for the upper bound of the competitive ratio of $\mathit{GDY}$ for matching instances under free dissolution.
• Figure 3: Worst-case performance of $\mathit{GDY}$ with free dissolution in the coalition formation domain. Missing edges represent a utility of $0$. Unlabelled edges represent a utility of $1$.
• Figure 4: Illustration of the adversarial instances in the proof of \ref{['thm:DissolutionStrongUB']}.
• Figure 5: Family of instances for tightness of competitive ratio in \ref{['thm:ggdy']}.

Theorems & Definitions (56)

• Definition 1: Greedy algorithm
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Definition 2: Greedy matching algorithm
• Proposition 1
• proof