Nash Equilibria with Irradical Probabilities

TL;DR

This paper constructs specific multi-player games demonstrating Nash equilibria with algebraic probabilities that cannot be expressed in closed form, highlighting complex solution structures in game theory.

Contribution

It introduces explicit examples of games with unique, fully mixed Nash equilibria having irradical probabilities, expanding understanding of equilibrium algebraic complexity.

Findings

Existence of games with irradical equilibrium probabilities for all n≥4

Unique fully mixed Nash equilibria with algebraic but non-closed form probabilities

Demonstrates complexity of equilibrium solutions in multi-player games

Abstract

We present for every $n\ge4$ an $n$-player game in normal form with payoffs in $\{0,1,2\}$ that has a unique, fully mixed, Nash equilibrium in which all the probability weights are irradical (i.e., algebraic but not closed form expressible even with $m$-th roots for any integer $m$).