Question: Ade assigns scores to meteorologists (because he has nothing better to do) in the following manner: if a meteorologist says there is a probability p of rain tomorrow, then they get the score 1 − (1 − p)^2 , if it rains tomorrow and 1 − p^2 , if it doesn’t rain tomorrow.
This way, the score for a wrong prediction is very low if the prediction was confident (p close to 0 or 1), but not too punishing if the prediction was uncertain (p close to 1/2). Suppose I am a meteorologist who somehow always knows the correct probability of rain (I'm a rainmaker). I know there is a chance of 3/10 of rain tomorrow. What should I tell Ade the probability is, in order to maximize my expected score from him?
This way, the score for a wrong prediction is very low if the prediction was confident (p close to 0 or 1), but not too punishing if the prediction was uncertain (p close to 1/2). Suppose I am a meteorologist who somehow always knows the correct probability of rain (I'm a rainmaker). I know there is a chance of 3/10 of rain tomorrow. What should I tell Ade the probability is, in order to maximize my expected score from him?
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