J.D.

[....is at the expense of rewarding knowledge of the correct answer.]

Hmm... I'm not sure that Alice has really more knowledge than Bob in your example.

[EDIT : In fact, in your example, for the quadratic scoring rule, the score of 50:50:$\epsilon:\epsilon$ is better than the score of 40:20:20:20 since $12/25 < 1/2 + 2\epsilon^2$, so that we can indeed say that Alice has more knowledge than Bob after this rule. The following example is, IMHO, more interesting. /EDIT].

Let me propose an other perspective with the following two answers for propositions A:B:C:D :

1) 50:50:0:0

2) 50:25:25:0,

where the correct answer is 100:0:0:0.

In this case, 2) has a better score than 1).

What does 1) know ?... (read more)

[... why do they score more?]

I'm not sure if these are good reasons, but it seems to me that

1) The expected answer to the quiz does not just consist in identifying A as a correct answer but also in identifying the others as incorrect answers. I mean that the expected right answer is 100:0:0:0 (and not, for example, 100:50:0:0 or whatever else).

2) Giving 25:25 for B:C is better than giving 50:0 even if answer C is 0 since 25:25 is closer to 0:0 than 50:0 (for the usual Euclidean distance). In this perspective, a better answer for the 50:50:0:0's guy would have been 50:25:0:0, which is better than 50:25:25:0.

Indeed, 1... (read more)