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 confidence interval, take the score , where is the size of your confidence interval, and is the distance between the true value and the interval. is whenever the true value is in the interval.

This incentivizes not only giving an interval that has the true value 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 and in log space. So if the true value is and the interval is , then is and is for underestimates and for overestimates. Of course, you need questions with positive answers to do this.

To do a confidence interval, take the score .

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.

This is an underappreciated fact! I like how simple the rule is when framed in terms of size and distance.

You mention both the linear and log rules. The log rule has the benefit of being scale-invariant, so your score isn't affect by the units the answer is measured in, but it can't deal with negatives and gets overly sensitive around zero. The linear rule doesn't blow up around zero, is shift-invariant, and can handle negative values fine. The best generic scoring rule would have all these properties.

Turns out (based on Lambert and Shoham, ... (read more),,,,,,,,,,,,,,,,,,,,