  We often assume that robots which collaborate with humans should behave in
ways that are transparent (e.g., legible, explainable). These transparent
robots intentionally choose actions that convey their internal state to nearby
humans: for instance, a transparent robot might exaggerate its trajectory to
indicate its goal. But while transparent behavior seems beneficial for
human-robot interaction, is it actually optimal? In this paper we consider
collaborative settings where the human and robot have the same objective, and
the human is uncertain about the robot's type (i.e., the robot's internal
state). We extend a recursive combination of Bayesian Nash equilibrium and the
Bellman equation to solve for optimal robot policies. Interestingly, we
discover that it is not always optimal for collaborative robots to be
transparent; instead, human and robot teams can sometimes achieve higher
rewards when the robot is opaque. In contrast to transparent robots, opaque
robots select actions that withhold information from the human. Our analysis
suggests that opaque behavior becomes optimal when either (a) human-robot
interactions have a short time horizon or (b) users are slow to learn from the
robot's actions. We extend this theoretical analysis to user studies across 43
total participants in both online and in-person settings. We find that --
during short interactions -- users reach higher rewards when working with
opaque partners, and subjectively rate opaque robots as about equal to
transparent robots. See videos of our experiments here:
https://youtu.be/u8q1Z7WHUuI
