  In this paper we obtain sufficient and necessary conditions on the number of
samples required for exact recovery of the pure-strategy Nash equilibria (PSNE)
set of a graphical game from noisy observations of joint actions. We consider
sparse linear influence games --- a parametric class of graphical games with
linear payoffs, and represented by directed graphs of n nodes (players) and
in-degree of at most k. We show that one can efficiently recover the PSNE set
of a linear influence game with $O(k^2 \log n)$ samples, under very general
observation models. On the other hand, we show that $\Omega(k \log n)$ samples
are necessary for any procedure to recover the PSNE set from observations of
joint actions.
