Linear probability score ($S(p,x)=xp+(1-x)(1-p)$)

The linear probability score pays the reported probability of whichever binary outcome occurs. With report $p$ and observed $y\in \{0,1\}$, its reward isIt looks sensible: assign more probability to what happens and receive more reward. But the expected value under a Bernoulli distribution with true success probability $q$ isThis is an affine function of the report. When $q>1/2$, the maximum over $[0,1]$ is at $p=1$; when $q<1/2$, it is at $p=0$. At $q=1/2$, every report ties. Thus the linear probability score encourages reporting certainty about the more likely outcome and is not a proper scoring rule.

For the stipulated $q=0.7$, the expected value of the reward is $0.3+0.4p$. Honest reporting $p=0.7$ earns $0.58$ in expectation, whereas $p=1$ earns $0.7$. The forecaster improves expected reward by replacing a 70% forecast with a 100% forecast. The Brier score and logarithmic scoring rule avoid this incentive, as the expected-score diagram under proper scoring rule shows.