Let x be your estimation of propability of some event.
Let’s define penalty t=(1−x)2 if event happened, t=x2 otherwise.
If p is propability of event, expected penalty equals:
p(1−x)2+(1−p)x2=p−2px+px2+x2−px2=p−2px+x2
And expected penalty derivative equals −2p+2x
So, if you try to minimize average penalty, you are motivated to give your best possible estimate. It’s possible that you can use it to grade your own predictions, I don’t know for sure.
Let x be your estimation of propability of some event.
Let’s define penalty t=(1−x)2 if event happened, t=x2 otherwise.
If p is propability of event, expected penalty equals:
p(1−x)2+(1−p)x2=p−2px+px2+x2−px2=p−2px+x2
And expected penalty derivative equals −2p+2x
So, if you try to minimize average penalty, you are motivated to give your best possible estimate. It’s possible that you can use it to grade your own predictions, I don’t know for sure.