English: The dashed black line represents the forecaster's true belief, the x-axis is their actual prediction, and the y-axis the expected score given their true belief.

Note also: • The red line is the linear scoring rule, which just gives you the probability you assigned to the right outcome as score.

We see that it is not "proper" -- that is, the highest value is usually given by predicting close to 0 or 1, as opposed to your true belief.

For the other three functions, note that they all have their maxima at the true belief -- your best strategy is to be truthful. They are: • Orange line: spherical scoring rule • Purple line: Brier score • Green line: log score (you may recognize the shape of this curve. Because the expected log score is in fact the Shannon entropy of the underlying variable)

Generated using GIFsmos. http://www.gifsmos.com/