The weak aggregating algorithm and weak mixability

This paper resolves the problem of predicting as well as the best expert up to an additive term of the order o(n), where n is the length of a sequence of letters from a finite alphabet. We call the games that permit this weakly mixable and give a geometrical characterisation of the class of weakly mixable games. Weak mixability turns out to be equivalent to convexity of the finite part of the set of superpredictions. For bounded games we introduce the Weak Aggregating Algorithm that allows us to obtain additive terms of the form C root n. (C) 2007 Elsevier Inc. All rights reserved.