  For a federated learning model to perform well, it is crucial to have a
diverse and representative dataset. However, the data contributors may only be
concerned with the performance on a specific subset of the population, which
may not reflect the diversity of the wider population. This creates a tension
between the principal (the FL platform designer) who cares about global
performance and the agents (the data collectors) who care about local
performance. In this work, we formulate this tension as a game between the
principal and multiple agents, and focus on the linear experiment design
problem to formally study their interaction. We show that the statistical
criterion used to quantify the diversity of the data, as well as the choice of
the federated learning algorithm used, has a significant effect on the
resulting equilibrium. We leverage this to design simple optimal federated
learning mechanisms that encourage data collectors to contribute data
representative of the global population, thereby maximizing global performance.
