Fairness in model-sharing games

In many real-world situations, data is distributed across multiple self-interested agents. These agents can collaborate to build a machine learning model based on data from multiple agents, potentially reducing the error each experiences. However, sharing models in this way raises questions of fairness: to what extent can the error experienced by one agent be significantly lower than the error experienced by another agent in the same coalition? In this work, we consider two notions of fairness that each may be appropriate in different circumstances: egalitarian fairness (which aims to bound how dissimilar error rates can be) and proportional fairness (which aims to reward players for contributing more data). We similarly consider two common methods of model aggregation, one where a single model is created for all agents (uniform), and one where an individualized model is created for each agent. For egalitarian fairness, we obtain a tight multiplicative bound on how widely error rates can diverge between agents collaborating (which holds for both aggregation methods). For proportional fairness, we show that the individualized aggregation method always gives a small player error that is upper bounded by proportionality. For uniform aggregation, we show that this upper bound is guaranteed for any individually rational coalition (where no player wishes to leave to do local learning).