  We study methods for improving fairness to subgroups in settings with
overlapping populations and sequential predictions. Classical notions of
fairness focus on the balance of some property across different populations.
However, in many applications the goal of the different groups is not to be
predicted equally but rather to be predicted well. We demonstrate that the task
of satisfying this guarantee for multiple overlapping groups is not
straightforward and show that for the simple objective of unweighted average of
false negative and false positive rate, satisfying this for overlapping
populations can be statistically impossible even when we are provided
predictors that perform well separately on each subgroup. On the positive side,
we show that when individuals are equally important to the different groups
they belong to, this goal is achievable; to do so, we draw a connection to the
sleeping experts literature in online learning. Motivated by the one-sided
feedback in natural settings of interest, we extend our results to such a
feedback model. We also provide a game-theoretic interpretation of our results,
examining the incentives of participants to join the system and to provide the
system full information about predictors they may possess. We end with several
interesting open problems concerning the strength of guarantees that can be
achieved in a computationally efficient manner.
