Non-Obvious Manipulability in Additively Separable and Fractional Hedonic Games
Diodato Ferraioli, Giovanna Varricchio
TL;DR
This work addresses the design of Non-Obvious Manipulable (NOM) mechanisms for Additively Separable and Fractional Hedonic Games, where agents’ utilities are derived from pairwise scores and social welfare is the sum of individual utilities. It introduces scale-independence (SI) as a sufficient condition for NOM and shows how a Repr-based transformation can convert any rho-approximation mechanism into an SI (and hence NOM) mechanism without losing the approximation ratio. A matching-based NOM mechanism is proposed, yielding an $n$-approximation for ASHGs and a $2$-approximation for FHGs, and its NOM property follows from SI. The paper also provides a detailed discrete-settings analysis with duplex valuations, identifying thresholds on $x$ (the magnitude of negative edges) where NOM is compatible with optimality, revealing a nuanced landscape that depends on the discrete value set. Overall, the results illuminate how NOM can achieve near-optimal outcomes in hedonic games under bounded rationality, offering practical mechanism designs and guiding future extensions to broader game classes and stronger notions of rationality.
Abstract
In this work, we consider the design of Non-Obviously Manipulable (NOM) mechanisms, mechanisms that bounded rational agents may fail to recognize as manipulable, for two relevant classes of succinctly representable Hedonic Games: Additively Separable and Fractional Hedonic Games. In these classes, agents have cardinal scores towards other agents, and their preferences over coalitions are determined by aggregating such scores. This aggregation results in a utility function for each agent, which enables the evaluation of outcomes via the utilitarian social welfare. We first prove that, when scores can be arbitrary, every optimal mechanism is NOM; moreover, when scores are limited in a continuous interval, there exists an optimal mechanism that is NOM. Given the hardness of computing optimal outcomes in these settings, we turn our attention to efficient and NOM mechanisms. To this aim, we first prove a characterization of NOM mechanisms that simplifies the class of mechanisms of interest. Then, we design a NOM mechanism returning approximations that asymptotically match the best-known approximation achievable in polynomial time. Finally, we focus on discrete scores, where the compatibility of NOM with optimality depends on the specific values. Therefore, we initiate a systematic analysis to identify which discrete values support this compatibility and which do not.