Envy-freeness in 3D Hedonic Games

TL;DR

The paper investigates partitioning a set of $N$ agents into coalitions of fixed size $3$ under additively separable hedonic preferences, focusing on envy-free, weakly justified envy-free, and justified envy-free notions. It delivers a comprehensive complexity classification across binary/ternary, symmetric/non-symmetric preferences and varying maximum degrees, revealing that tractability improves under weaker restrictions and that hardness persists under many natural generalizations. Key results include an $O(|N|)$-time algorithm for envy-freeness when underlying graphs have max degree $2$, NP-completeness for max degree $3$, a matching $O(|N|)$ algorithm for weakly justified envy-freeness under degree-2, and NP-hardness for higher-arity or non-binary symmetric cases for justified envy-freeness. The work highlights a frontier between polynomial-time solvability and NP-hardness in fixed-size coalitions, providing both constructive algorithms and robust hardness reductions that hinge on gadget-based reductions from XSAT, DTC, and PIT. This advances understanding of how coalition size and preference structure shape the feasibility of fair partitions in hedonic games with applications to team formation and roommate-style allocations.

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.

Figures (7)

• Figure 1: An example ASHG $(N, V)$ containing six agents and a partition into triples $\pi$, marked by the dashed enclosure. The weighted arcs depict the agents' valuations. Unless otherwise specified, $v_{\alpha_i}(\alpha_j) = 0$ for any $\alpha_i$ and $\alpha_j$ in $N$.
• Figure 2: Part of the known hierarchy of solution concepts in hedonic games hgch15BY19Bilo22. In the diagram, an arrow points from one concept to another if any partition that satisfies the former must also satisfy the latter. This figure is adapted from the Handbook of Computational Social Choicehgch15.
• Figure 3: The reduction from $\text{X3SAT}_{+}^{\,=3}$ to the problem of deciding if a given ASHG contains an envy-free partition into triples. A variable gadget $W_i$ and clause gadget $D_r$ are represented as undirected graphs.
• Figure 4: The reduction from $\text{X3SAT}_{+}^{\,=3}$ to the problem of deciding if a given ASHG contains a wj-envy-free partition into triples. A variable gadget $W_i$, clause gadget $D_r$, and garbage collector gadget $G_i$ are represented as undirected graphs.
• Figure 5: The reduction from DTC to the problem of deciding if a given ASHG with ternary preferences contains a j-envy-free partition into triples

Theorems & Definitions (122)

• Lemma 2.1
• proof
• Theorem 2.2
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Lemma 2.7