  Humans frequently make decisions with the aid of artificially intelligent
(AI) systems. A common pattern is for the AI to recommend an action to the
human who retains control over the final decision. Researchers have identified
ensuring that a human has appropriate reliance on an AI as a critical component
of achieving complementary performance. We argue that the current definition of
appropriate reliance used in such research lacks formal statistical grounding
and can lead to contradictions. We propose a formal definition of reliance,
based on statistical decision theory, which separates the concepts of reliance
as the probability the decision-maker follows the AI's recommendation from
challenges a human may face in differentiating the signals and forming accurate
beliefs about the situation. Our definition gives rise to a framework that can
be used to guide the design and interpretation of studies on human-AI
complementarity and reliance. Using recent AI-advised decision making studies
from literature, we demonstrate how our framework can be used to separate the
loss due to mis-reliance from the loss due to not accurately differentiating
the signals. We evaluate these losses by comparing to a baseline and a
benchmark for complementary performance defined by the expected payoff achieved
by a rational decision-maker facing the same decision task as the behavioral
decision-makers.
