On-line choice of on-line algorithms

Let {Al,Az,... ,Am} be a set of on-line algorithms for a problem P with input set I. We assume that P can be represented as a metrical task system. Each A; has a competitive ratio si with respect to the optimum offline algorithm, but only for a subset of the possible inputs such that the union of these subsets covers I. Given this setup, we construct a generic deterministic on-line algorithm and a generic randomized on-line algorithm for P that are competitive over all possible inputs. We show that their competitive ratios are optimal up to constant factors. Our analysis proceeds via an amusing card game.