More on Brier Scores: A Brief on Brier Scores  University of Virginia Library Research Data Services + Sciences, Brier score  Wikipedia, among others.
Additional Notes:
 I’ve been asked about what happens when there’s a third party. There’s a general multicategory 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 nonnegativity”. 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.