Decidability of rational Nash equilibrium existence

Determine whether the problem of deciding the existence of a rational Nash equilibrium—i.e., a Nash equilibrium whose mixed strategy probabilities are rational numbers—is decidable.

Background

ER-hardness is known for deciding the existence of rational Nash equilibria in 3-player games. However, no algorithmic decidability guarantee has been established.

This question probes the limits of algebraic and logical complexity in equilibrium computation.

References

The problem of deciding whether there exists a rational Nash equilibrium, in which $\sigma_i(s)\in$ for all $i\in{1,2,3}$ and all $s\in\Sigma_i$ is shown to be -hard. It is open whether the problem is even decidable.

— The Existential Theory of the Reals as a Complexity Class: A Compendium  (2407.18006 - Schaefer et al., 2024) in Compendium — Problem 'Irrational Nash Equilibrium'

We leave open whether there is a rational uniform-tie-breaking Blotto game with no rational equilibrium; such examples are known for general multiplayer normal-form games.

— The Complexity of Multiplayer Colonel Blotto Games with Player-Specific Values  (2609.30019 - Bichler et al., 24 Sep 2026) in Section 6, paragraph “Irrational equilibria”