Aram asked how the random algorithm does for paging: at each time just pick a random page to evict from the cache. Jason sent in an example where the optimum is 1, but the expected cost for RANDOM is k. The sequence is:
The optimal action is to evict when you see the first , with an optimal cost of 1. But the random algorithm, on each cache miss, will pick a random page to evict, until it finally hits on . Each time it has a 1-in-k chance of success, which leads to an expected cost of .
In fact, Raghavan and Snir show that any memoryless randomized algorithm for paging cannot have competitive ratio less than . They have a similar construction, and give sequences where the ratio remains even when the optimal costs go to infinity. (Else we could give a weak-competitiveness bound with an additive factor of .)