One rule with this proper property is to pay a dollar minus the squared error of the bet, rather than the bet itself—if you bet 30 cents on the winning light, your error would be 70 cents, your squared error would be 49 cents ((0.7)^2 = 0.49), and a dollar minus your squared error would be 51 cents.[3] (Presumably your play money is denominated in the square root of cents, so that the squared error is a monetary sum.)
Isn’t the squared-error rule only proper for N=2? For example, the frequencies f1=0.5,f2=f3=0.25 give p1=0.6,p2=p3=0.2 as an optimal bet when minimizing ∑ifi(1−pi)2.
Isn’t the squared-error rule only proper for N=2? For example, the frequencies f1=0.5,f2=f3=0.25 give p1=0.6,p2=p3=0.2 as an optimal bet when minimizing ∑ifi(1−pi)2.