  Much of machine learning research focuses on predictive accuracy: given a
task, create a machine learning model (or algorithm) that maximizes accuracy.
In many settings, however, the final prediction or decision of a system is
under the control of a human, who uses an algorithm's output along with their
own personal expertise in order to produce a combined prediction. One ultimate
goal of such collaborative systems is "complementarity": that is, to produce
lower loss (equivalently, greater payoff or utility) than either the human or
algorithm alone. However, experimental results have shown that even in
carefully-designed systems, complementary performance can be elusive. Our work
provides three key contributions. First, we provide a theoretical framework for
modeling simple human-algorithm systems and demonstrate that multiple prior
analyses can be expressed within it. Next, we use this model to prove
conditions where complementarity is impossible, and give constructive examples
of where complementarity is achievable. Finally, we discuss the implications of
our findings, especially with respect to the fairness of a classifier. In sum,
these results deepen our understanding of key factors influencing the combined
performance of human-algorithm systems, giving insight into how algorithmic
tools can best be designed for collaborative environments.
