## LESSWRONGLW

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In a true/false question that is true with probability p, if you assign probability q, your probability of losing is p(1−q)^2+(1−p)q^2. (The probabily the answer is true and you spin false twice plus the probability the answer is false and you spin true twice.)

This probability is minimized when its derivative with respect to q is 0, or at the boundary. This derivative is −2p(1−q)+2(1−p)q, whis is 0 when q=p. We now know the minimum is achieved when q is 0, 1, or p. The probability of losing when q=0 is p. The probability of losing when q=1 is 1−p. The probability of losing when q=p is p(1−p), which is the lowest of the three options.

# 108

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.