[scottshambaugh]

I had the computer do it for me :)

Really though, the way to do this is to represent each game as a payout matrix A, which for this category of game will be skew-symmetric with -1/0/+1 entries. Then since this is a symmetric zero-sum game, you find the null space of that matrix, impose the constraint that probabilities must sum to 1 and individually be >= 0, and that gives you the endpoints of the Nash Equilibria. The optimal strategies (could be one unique one for odd n!) lie on the convex hull of those points.

[ian_j_butler]

Thanks for writing this. I'm going back to basics on a bunch of this stuff myself as the connections in the AI and ML literature keep heating up. Some stuff from my reading list lately that you might like: https://arxiv.org/abs/2604.27167 https://arxiv.org/pdf/2601.15047 https://arxiv.org/abs/2402.01704

[scottshambaugh]

Thanks for sharing! My background is in control theory, and I think the links between it and game theory and dynamical systems in general are super fruitful but woefully underexplored. Writing this was a good excuse to brush up on my linear algebra basics again.