XX On the Complexity of Approximating a Nash Equilibrium

We show that computing a relatively (i.e. multiplicatively as opposed to additively) approximate Nash equilibrium in two-player games is PPAD-complete, even for constant values of the approximation. Our result is the first constant inapproximability result for Nash equilibrium, since the original results on the computational complexity of the problem [DGP06; CD06]. Moreover, it provides an apparent—assuming that PPAD is not contained in TIME(nO(log n))—dichotomy between the complexities of additive and relative approximations, as for constant values of additive approximation a quasi-polynomial-time algorithm is known [LMM03]. Such a dichotomy does not exist for values of the approximation that scale inverse-polynomially with the size of the game, where both relative and additive approximations are PPADcomplete [CDT06]. As a byproduct, our proof shows that (unconditionally) the [LMM03] sparse-support lemma cannot be extended to relative notions of constant approximation.