Using the Nucleolus for Incentive Allocation in Vertical Federated Learning

Vertical federated learning (VFL) is a promising approach for collaboratively training machine learning models using private data partitioned vertically across different parties. Ideally in a VFL setting, the active party (party possessing features of samples with labels) benefits by improving its machine learning model through collaboration with some passive parties (parties possessing additional features of the same samples without labels) in a privacy-preserving manner. However, motivating passive parties to participate in VFL can be challenging. In this paper, we focus on the problem of allocating incentives to the passive parties by the active party based on their contributions to the VFL process. We address this by formulating the incentive allocation problem as a bankruptcy game, a concept from cooperative game theory. Using the Talmudic division rule, which leads to the Nucleolus as its solution, we ensure a fair distribution of incentives. We evaluate our proposed method on synthetic and real-world datasets and show that it ensures fairness and stability in incentive allocation among passive parties who contribute their data to the federated model. Additionally, we compare our method to the existing solution of calculating Shapley values and show that our approach provides a more efficient solution with fewer computations.