  We study the structural complexity of bimatrix games, formalized via rank,
from an empirical perspective. We consider a setting where we have data on
player behavior in diverse strategic situations, but where we do not observe
the relevant payoff functions. We prove that high complexity (high rank) has
empirical consequences when arbitrary data is considered. Additionally, we
prove that, in more restrictive classes of data (termed laminar), any
observation is rationalizable using a low-rank game: specifically a zero-sum
game. Hence complexity as a structural property of a game is not always
testable. Finally, we prove a general result connecting the structure of the
feasible data sets with the highest rank that may be needed to rationalize a
set of observations.
