  All proper scoring rules incentivize an expert to predict \emph{accurately}
(report their true estimate), but not all proper scoring rules equally
incentivize \emph{precision}. Rather than treating the expert's belief as
exogenously given, we consider a model where a rational expert can endogenously
refine their belief by repeatedly paying a fixed cost, and is incentivized to
do so by a proper scoring rule.
  Specifically, our expert aims to predict the probability that a biased coin
flipped tomorrow will land heads, and can flip the coin any number of times
today at a cost of $c$ per flip. Our first main result defines an
\emph{incentivization index} for proper scoring rules, and proves that this
index measures the expected error of the expert's estimate (where the number of
flips today is chosen adaptively to maximize the predictor's expected payoff).
Our second main result finds the unique scoring rule which optimizes the
incentivization index over all proper scoring rules.
  We also consider extensions to minimizing the $\ell^{th}$ moment of error,
and again provide an incentivization index and optimal proper scoring rule. In
some cases, the resulting scoring rule is differentiable, but not infinitely
differentiable. In these cases, we further prove that the optimum can be
uniformly approximated by polynomial scoring rules.
  Finally, we compare common scoring rules via our measure, and include
simulations confirming the relevance of our measure even in domains outside
where it provably applies.
