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 $90\%$ of the time, but also distributes the remaining 10% equally between overestimates and underestimates.

To keep the lower bound of the interval important, I recommend measuring $S$ and $D$ in log space. So if the true value is $T$ and the interval is $(L,U)$, then $S$ is $\log \left(\frac{U}{L}\right)$ and $D$ is $\log \left(\frac{T}{U}\right)$ for underestimates and $\log \left(\frac{L}{T}\right)$ for overestimates. Of course, you need questions with positive answers to do this.

To do a $P\%$ confidence interval, take the score $-S-\frac{200}{100-P}\cdot D$.

This can be used to make training calibration, using something like Wits and Wagers cards more fun. I also think it could be turned into app, if one could get a large list of questions with numerical values.