  This paper studies the design of optimal proper scoring rules when the
principal has partial knowledge of an agent's signal distribution. Recent work
characterizes the proper scoring rules that maximize the increase of an agent's
payoff when the agent chooses to access a costly signal to refine a posterior
belief from her prior prediction, under the assumption that the agent's signal
distribution is fully known to the principal. In our setting, the principal
only knows about a set of distributions where the agent's signal distribution
belongs. We formulate the scoring rule design problem as a max-min optimization
that maximizes the worst-case increase in payoff across the set of
distributions.
  We propose an efficient algorithm to compute an optimal scoring rule when the
set of distributions is finite, and devise a fully polynomial-time
approximation scheme that accommodates various infinite sets of distributions.
We further remark that widely used scoring rules, such as the quadratic and log
rules, as well as previously identified optimal scoring rules under full
knowledge, can be far from optimal in our partial knowledge settings.
