Online Coalition Formation under Random Arrival or Coalition Dissolution

TL;DR

This paper studies online coalition formation with agents arriving sequentially, analyzing algorithms' competitive ratios under random arrival and coalition dissolution, improving upon basic models by achieving near-optimal performance.

Contribution

It introduces new algorithms and bounds for online coalition formation, including models with random arrival and coalition dissolution, achieving near-optimal competitive ratios.

Findings

Greedy algorithm achieves optimal ratio in basic model.

Random arrival model reduces ratio to Θ(1/n^2).

Waiting-greedy algorithm improves ratio to Θ(1/n).

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 $\Theta\left(\frac{1}{n^2}\right)$. In contrast, an alternative algorithm, which is based on alternating between waiting and greedy phases, can achieve a competitive ratio of $\Theta\left(\frac{1}{n}\right)$. Second, we relax the irrevocability of decisions by allowing the dissolution of coalitions into singleton coalitions. We achieve an asymptotically optimal competitive ratio of $\Theta\left(\frac 1n\right)$ 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.