  We study infinite asymptotic games in Banach spaces with an F.D.D. and prove
that analytic games are determined by characterising precisely the conditions
for the players to have winning strategies. These results are applied to
characterise spaces embeddable into $\ell_p$ sums of finite dimensional spaces,
extending results of Odell and Schlumprecht, and to study various notions of
homogeneity of bases and Banach spaces. These results are related to questions
of rapidity of subsequence extraction from normalised weakly null sequences.
