Fair incentive allocation in vertical federated learning using nucleolus

Federated learning (FL) enables collaborative model training on decentralized data, overcoming privacy barriers that prevent direct sharing. Vertical federated learning (VFL) is an FL setting where different parties hold distinct features of the same users and jointly train machine learning models without sharing raw data. In this setting, with one label-holding active party and multiple passive parties, an important challenge is how the active party can fairly reward the passive parties for their contributions. We address this problem by proposing NIAM (Nucleolus-based Incentive Allocation Mechanism), which frames incentive allocation as a bankruptcy game and computes the nucleolus efficiently through the Talmudic division rule, to distribute incentives. In NIAM, the total performance gain is treated as an “estate” and the marginal contribution of each passive party as an individual claim; the computed nucleolus is a unique payoff that balances these claims, ensuring fair and stable rewards. We evaluate NIAM on public benchmark datasets covering both regression and classification tasks and conclude that it produces allocations that satisfy fairness axioms like efficiency, symmetry, and null player. Additionally, we compare NIAM with existing VFL incentive mechanisms, which shows that it avoids excluding contributors by ensuring all positively contributing parties receive rewards, treats equally contributing parties the same, and provides a more equitable basis for long-term collaboration.