I’ve been asked about what happens when there’s a third party. There’s a general multi-category version of the Brier Score that allows for >2 outcomes. There’s a nice section in the first link above. I may put a mafia example in writing in this post in the future, for now here’s a screenshot of it from the article mentioned.
Again in reference to considering >2 possible alignments, there’s also the more straightforward solution of just having your probabilities consider town vs. not town, then you can use the simpler formulation.
A follow up math question that came up when I was discussing Brier Scores with a friend was “Why square? What happens when you use absolute value instead as a way of forcing non-negativity”. There’s a decent answer to this question, but it’s something that could be fun to think about on your own for a bit. Do message me if you have thoughts or just want to know why. It’s fairly interesting.
Edited in a couple of notes on multi-category Brier Scores (for games with more than 2 alignments) and an interesting math question about why Brier Scores are formulated the way it is.
