The Power of Matching for Online Fractional Hedonic Games

TL;DR

This work studies online partitioning into coalitions in fractional hedonic games (FHGs) where agents arrive sequentially and must be assigned irrevocably. It leverages online matching, particularly maximum weight and fractional matching techniques, to obtain constant-competitive welfare guarantees that do not depend on valuation ranges. In two models—random arrival and coalition dissolution with irrevocable decisions—the authors present a $\frac{1}{3}-\frac{1}{n}$-competitive algorithm and a $\frac{1}{6+4\sqrt{2}}$-competitive algorithm, respectively, and prove asymptotic tightness on the tree domain. The results demonstrate that simple, matching-based strategies are asymptotically optimal for online FHGs in these settings, with significant implications for real-time coalition formation problems across economics and AI.

Abstract

We study coalition formation in the framework of fractional hedonic games (FHGs). The objective is to maximize social welfare in an online model where agents arrive one by one and must be assigned to coalitions immediately and irrevocably. A recurrent theme in online coalition formation is that online matching algorithms, where coalitions are restricted to size at most $2$, yield good competitive ratios. For example, computing maximal matchings achieves the optimal competitive ratio for general online FHGs. However, this ratio is bounded only if agents' valuations are themselves bounded. We identify optimal algorithms with constant competitive ratios in two related settings, independent of the range of agent valuations. First, under random agent arrival, we present an asymptotically optimal $(\frac{1}{3}-\frac 1n)$-competitive algorithm, where $n$ is the number of agents. This result builds on our identification of an optimal matching algorithm in a general model of online matching with edge weights and an unknown number of agents. In this setting, we also achieve an asymptotically optimal competitive ratio of $\frac{1}{3}-\frac 1n$. Second, when agents arrive in an arbitrary order but algorithms are allowed to irrevocably and entirely dissolve coalitions, we show that another matching-based algorithm achieves an optimal competitive ratio of $\frac{1}{6 + 4\sqrt{2}}$.

Figures (2)

• Figure 1: Illustration of Phase $i$ in the construction of the adversarial instance in the proof of \ref{['thm:dissolutionbound']}. Each star attached to $a_i$ and $b_i$ contains $\ell_i$ leaves.
• Figure 2: Illustration of the construction in the proof of \ref{['thm:dissolutionbound']} for an exemplary algorithm $\mathit{ALG}$. We display all positive valuations. The remaining valuations within the leaf sets $L_1^a$, $L_1^b$, $L_2^a$, and $L_2^b$ are zero, and all other valuations are large negative numbers. We start with two agents, $a_1$ and $b_1$. We first attempt to dispatch a set $L_1^b$ of leaves towards $b_1$. However, our algorithm might immediately decide to dissolve $\{a_1,b_1\}$ and create a new coalition $\{a'_1,b'_1\}$. We then might be able to have all the leaf agents in $L_1^a$ and $L_1^b$ arrive. This completes the first part of Phase $1$. Now, we start the second part, in which we subsequently increment the valuations. $\mathit{ALG}$ might decide to immediately dissolve $\{a'_1,b'_1\}$ when the next agent arrives. This defines agents $a_2$, $b_2$, and coalition $C_1$. We start with Phase $2$. In the first part, the leaf agents $L_2^a$ and $L_2^b$ might arrive without further interruption. Now assume that $\mathit{ALG}$ would dissolve $\{a_2,b_2\}$ when the next agent arrives (their edge is indicated in bold). This would give rise to the definition of $C_2$ and $D_2$, and we would obtain an inequality for $y_2$ by comparing with the guarantee for the coalition structure containing the nonempty coalitions $C_1$, $C_2$, and $D_2$.

Theorems & Definitions (56)

• Theorem 4.1
• Corollary 4.2
• Proposition 4.2
• Theorem 5.1
• Theorem 5.2
• proof
• Lemma 5.2
• Lemma 5.2
• Lemma 5.2
• Theorem 5.3