Envy-freeness in 3D Hedonic Games

TL;DR

This paper investigates the existence and computational complexity of envy-free partitions in 3-agent coalitions within hedonic games, revealing that weaker solution concepts are more likely to exist under less restrictive preference conditions.

Contribution

It provides a comprehensive complexity classification for envy-free solutions in 3-agent coalition hedonic games, highlighting the impact of preference restrictions.

Findings

Existence of envy-free partitions depends on preference restrictions.

Polynomial-time algorithms exist under certain preference restrictions.

Weaker envy-freeness concepts are more computationally tractable.

Abstract

We study the problem of partitioning a set of agents into coalitions based on the agents' additively separable preferences, which can also be viewed as a hedonic game. We apply three successively weaker solution concepts, namely envy-freeness, weakly justified envy-freeness, and justified envy-freeness. In a model in which coalitions may have any size, trivial solutions exist for these concepts, which provides a strong motivation for placing restrictions on coalition size. In this paper, we require feasible coalitions to have size three. We study the existence of partitions that are envy-free, weakly justified envy-free, and justified envy-free, and the computational complexity of finding such partitions, if they exist. We present a comprehensive complexity classification, in terms of the restrictions placed on the agents' preferences. From this, we identify a general trend that for the three successively weaker solution concepts, existence and polynomial-time solvability hold under successively weaker restrictions.