We introduce for a countable ordinal $\alpha$ the the asymptotic structure $\{X\}_\alpha$, similar as it was done in the finite case by Maurey, Milmann, and Tomczak-Jaegermann. We are doing this using finite games of complexity $\alpha$. With these tools we deduce certain quantitative versions of Gowers' dichotomy result. We also give a different proof of the combinatorical theorem underlying this dichotomy result.