A *strictly proper scoring rule* is one in which a forecaster
maximizes (or minimizes, depending on the nature of the score) by
forecasting exactly his or her true beliefs about the situation.

A *proper scoring rule* is one in which the forecaster cannot
gets the best score by forecasting his or her true beliefs, although
it may be possible to get the same score by forecasting something
else.

To test if a score is strictly proper, we calculate the expected score a forecaster will receive for a forecast and find what forecast, f, yields the best score as a function of his or her true belief, p. If the score is strictly proper, then the score will be maximized (minimized) iff p = f.

Let's test the family of scores that look like the half Brier score, expect for the exponent, defined by

B_{n}=(|f -x|)^{n},

where f is the forecast probability and x=0 or 1, depending on whether the event does not or does occur, respectively. The expected value is given by

E[B_{n}] = p(1-f)^{n} + (1-p)f ^{n}

where p is the forecaster's true belief. To find the value of f
that minimizes E[B_{n}], we take the partial derivative with
respect to f and set it equal to 0. That is,

dE/df = -np(1-f)^{n-1} + n(1-p)f ^{n-1} = 0.

or

p(1-f)^{n-1} = (1-p)f ^{n-1}

Thus, we want

p{(1-f)^{n-1} + f ^{n-1 }} = f ^{n-1}

or

p = f ^{n-1 }/{(1-f)^{n-1} + f ^{n-1 }}

**For n=1** (a linear scoring rule), this gives p=1/2, so that
the maximized by forecasting the extreme values of f, 0 or 1, if
p<0.5 or p>0.5, respectively.

**For n=2** (quadratic scoring rule--the Brier score), this
gives p=f, which is strictly proper.

**For n>2**, this yields a complicated function that, in
effect, says that a forecaster should forecast values closer to 0.5
than his or her true belief, unless p is very close to 0 or 1.