Solve the online unweighted 0-1 knapsack problem efficiently
Well as you already noticed, or at least didn't mention, it is easy to see that there is no deterministic competitive Algorithm for your problem. (Counterexamples only need two items, and you can use the fact that the algorithm is deterministic.)
Your approach taking randomization actually isn't enough as some difficulties cannot be overcome by randomization. But there are a few relaxation models of your problem.
Some of those, and a few details to my answer can be found in the paper by Susanne Albers, Arindam Khan and Leon Ladewig named &quot;Improved Online Algorithms for Knapsack and GAP in the Random Order Mode&quot; (see section 1.1).