  We study the computational complexity of decision problems about Nash
equilibria in $m$-player games. Several such problems have recently been shown
to be computationally equivalent to the decision problem for the existential
theory of the reals, or stated in terms of complexity classes,
$\exists\mathbb{R}$-complete, when $m\geq 3$. We show that, unless they turn
into trivial problems, they are $\exists\mathbb{R}$-hard even for 3-player
zero-sum games.
  We also obtain new results about several other decision problems. We show
that when $m\geq 3$ the problems of deciding if a game has a Pareto optimal
Nash equilibrium or deciding if a game has a strong Nash equilibrium are
$\exists\mathbb{R}$-complete. The latter result rectifies a previous claim of
NP-completeness in the literature. We show that deciding if a game has an
irrational valued Nash equilibrium is $\exists\mathbb{R}$-hard, answering a
question of Bil\`o and Mavronicolas, and address also the computational
complexity of deciding if a game has a rational valued Nash equilibrium. These
results also hold for 3-player zero-sum games.
  Our proof methodology applies to corresponding decision problems about
symmetric Nash equilibria in symmetric games as well, and in particular our new
results carry over to the symmetric setting. Finally we show that deciding
whether a symmetric $m$-player games has a non-symmetric Nash equilibrium is
$\exists\mathbb{R}$-complete when $m\geq 3$, answering a question of Garg,
Mehta, Vazirani, and Yazdanbod.
