Egalitarianism in Online Coalition Formation

We investigate the online coalition formation problem, where agents arrive one by one and must be assigned to coalitions, with their utilities for others revealed upon arrival. Our focus lies on additively separable hedonic games, where agents assign cardinal utilities to others, assumed to be controlled by an adversary in our online context. This paper introduces the evaluation of partitions based on their egalitarian social welfare, with the goal of maximizing the minimum utility of any agent. This objective strikes balance between fairness and efficiency by prioritizing the satisfaction of the least well-off agents. For various real-life scenarios, we establish tight or nearly tight upper bounds on the competitive ratio and complement these findings with optimal or near-optimal algorithms. However, we also demonstrate that in some cases, no competitive algorithm is feasible. In particular, under the classic worst-case adversarial model, where agents arrive in an arbitrary order, we show that no algorithm has a non-trivial competitive ratio, if at all.