We prove an exact, i.e., formulated without Delta-expansions,
Ramsey principle for infinite block sequences in vector spaces over
countable fields, where the two sides of the dichotomic principle are
represented by respectively winning strategies in Gowers' block sequence
game and winning strategies in the infinite asymptotic game. This allows
us to recover Gowers' dichotomy theorem for block sequences in normed
vector spaces by a simple application of the basic determinacy theorem for
infinite asymptotic games.