  We consider approximating the minmax value of a multi-player game in
strategic form. Tightening recent bounds by Borgs et al., we observe that
approximating the value with a precision of epsilon log n digits (for any
constant epsilon>0 is NP-hard, where n is the size of the game. On the other
hand, approximating the value with a precision of c log log n digits (for any
constant c >= 1) can be done in quasi-polynomial time. We consider the
parameterized complexity of the problem, with the parameter being the number of
pure strategies k of the player for which the minmax value is computed. We show
that if there are three players, k=2 and there are only two possible rational
payoffs, the minmax value is a rational number and can be computed exactly in
linear time. In the general case, we show that the value can be approximated
with any polynomial number of digits of accuracy in time n^(O(k)). On the other
hand, we show that minmax value approximation is W[1]-hard and hence not likely
to be fixed parameter tractable. Concretely, we show that if k-CLIQUE requires
time n^(Omega(k)) then so does minmax value computation.
