  We introduce the concept of strategically reproducible bases in Banach spaces
and show that operators which have large diagonal with respect to strategically
reproducible bases are factors of the identity. We give several examples of
classical Banach spaces in which the Haar system is strategically reproducible:
multi-parameter Lebesgue spaces, mixed-norm Hardy spaces and most significantly
the space $L^1$. Moreover, we show the strategical reproducibility is inherited
by unconditional sums.
