  Proper scoring rules incentivize experts to accurately report beliefs,
assuming predictions cannot influence outcomes. We relax this assumption and
investigate incentives when predictions are performative, i.e., when they can
influence the outcome of the prediction, such as when making public predictions
about the stock market. We say a prediction is a fixed point if it accurately
reflects the expert's beliefs after that prediction has been made. We show that
in this setting, reports maximizing expected score generally do not reflect an
expert's beliefs, and we give bounds on the inaccuracy of such reports. We show
that, for binary predictions, if the influence of the expert's prediction on
outcomes is bounded, it is possible to define scoring rules under which optimal
reports are arbitrarily close to fixed points. However, this is impossible for
predictions over more than two outcomes. We also perform numerical simulations
in a toy setting, showing that our bounds are tight in some situations and that
prediction error is often substantial (greater than 5-10%). Lastly, we discuss
alternative notions of optimality, including performative stability, and show
that they incentivize reporting fixed points.
