  Though competitive analysis is often a very good tool for the analysis of
online algorithms, sometimes it does not give any insight and sometimes it
gives counter-intuitive results. Much work has gone into exploring other
performance measures, in particular targeted at what seems to be the core
problem with competitive analysis: the comparison of the performance of an
online algorithm is made to a too powerful adversary. We consider a new
approach to restricting the power of the adversary, by requiring that when
judging a given online algorithm, the optimal offline algorithm must perform as
well as the online algorithm, not just on the entire final request sequence,
but also on any prefix of that sequence. This is limiting the adversary's usual
advantage of being able to exploit that it knows the sequence is continuing
beyond the current request. Through a collection of online problems, including
machine scheduling, bin packing, dual bin packing, and seat reservation, we
investigate the significance of this particular offline advantage.
