You probably already know that you can incentivise honest reporting of probabilities using a proper scoring rule like log score, but did you know that you can also incentivize honest reporting of confidence intervals?

To incentize reporting of a $90\%$ confidence interval, take the score $-S-20\cdot D$, where $S$ is the size of your confidence interval, and $D$ is the distance between the true value and the interval. $D$ is $0$ whenever the true value is in the interval.

This incentivizes not only giving an interval that has the true value