we were scored according to the log of our probability density on the true answer, except for the red-card drawer, who got the negative of this number.

This part still needs to be improved by someone. Log probability densities are only defined up to an additive constant log of a scaling factor. A player could get a high score by drawing the red card for a question with an answer in small units.

To normalize the scores, you could subtract the average of the log probability densities across groups.