Against in adaptive adversary, we show that the power of randomization
in on-line algorithms is severely limited! We prove the existence of
an efficient ''simulation'' of randomized on-line algorithms by determ
inistic ones, which is best possible in general. The proof of the uppe
r bound is existential. We deal with the issue of computing the effici
ent deterministic algorithm, and show that this is possible in very ge
neral cases.