Is the "angle=arccos(spherical)" scoring rule strictly proper?

Question

The Spherical scoring rule is known to be strictly proper.

However, it is not very intuitive.

Its arccos, however, is the angle between the prediction (p,1-p) and actual outcome (a,1-a) vectors (where a=1 or a=0).

Is it strictly proper? (arccos is decreasing, so we need to replace min with max in the definition, which is fine).

Answer 1

No, it is not:

as one can see, 0.2 is the only max for the spherical rule, but it is not the max for the angle.

The intuition is that the angle has the wrong balance between "left" and "right" parts: it grows to fast on the left and drops too slowly on the right.