Online algorithms and changing past decisions We know that for some online problems, algorithms can decrease their competitive ratio greatly if they are allowed to change some of their past decisions (see http://epubs.siam.org/doi/pdf/10.1137/1.9781611973402.35).

I wonder if there are instances of online problems such that the competitive ratio does not decrease even if the algorithm is allowed to change its decisions ?
Every problem where the online algorithm is optimal, i.e. has competitive ratio 1, is a (trivial) example. Such algorithms exist, for example for some scheduling problems.