Nash Equilibria with Derangement Degree Probabilities
This provides a theoretical construction for games with equilibrium probabilities of arbitrarily high algebraic degree, relevant to game theory and algebraic complexity.
The authors prove the existence of an n-player normal-form game with integer payoffs that has a unique fully mixed Nash equilibrium, where each probability weight is an algebraic number of degree equal to the derangement number !n, with minimal polynomial having Galois group S_{!n} and !n+1 nonzero coefficients.
We prove for every $n\ge4$ the existence of an $n$-player game in normal form with integer payoffs that has a unique Nash equilibrium, which is fully mixed. In the equilibrium, each probability weight is an algebraic number of degree $\mathbin{!n}$ (the derangement number), and its minimal polynomial has Galois group $S_{\mathbin{!n}}$ and $\mathbin{!n}+1$ nonzero coefficients.