Non-obvious Manipulability in Hedonic Games with Friends Appreciation Preferences

TL;DR

This paper investigates non-obvious manipulability in Hedonic Games with Friends Appreciation preferences, proposing mechanisms that approximate optimal partitions while balancing strategyproofness and computational complexity.

Contribution

It introduces a NOM mechanism that always finds the optimal partition and provides a polynomial-time approximation mechanism with a factor of (4+o(1)), advancing the understanding of manipulability in HGs.

Findings

Existence of a NOM mechanism that always outputs the optimal partition.

NP-hardness of computing the optimal partition.

A polynomial-time NOM mechanism guarantees a (4+o(1))-approximation.

Abstract

In this paper, we study non-obvious manipulability (NOM), a relaxed form of strategyproofness, in the context of Hedonic Games (HGs) with Friends Appreciation (FA) preferences. In HGs, the aim is to partition agents into coalitions according to their preferences which solely depend on the coalition they are assigned to. Under FA preferences, agents consider any other agent either a friend or an enemy, preferring coalitions with more friends and, in case of ties, the ones with fewer enemies. Our goal is to design mechanisms that prevent manipulations while optimizing social welfare. Prior research established that computing a welfare maximizing (optimum) partition for FA preferences is not strategyproof, and the best-known approximation to the optimum subject to strategyproofness is linear in the number of agents. In this work, we explore NOM to improve approximation results. We first prove the existence of a NOM mechanism that always outputs the optimum; however, we also demonstrate that the computation of an optimal partition is NP-hard. To address this complexity, we focus on approximation mechanisms and propose a NOM mechanism guaranteeing a $(4+o(1))$-approximation in polynomial time. Finally, we briefly discuss NOM in the case of Enemies Aversion (EA) preferences, the counterpart of FA, where agents give priority to coalitions with fewer enemies and show that no mechanism computing the optimum can be NOM.