Structure-Preserving Physics-Informed Neural Networks with Energy or Lyapunov Structure

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, an optimal mechanism that is NOM exists. 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.