*In which we show how to use expert advice, and introduce the powerful “multiplicative weight” algorithm.*

We study the following online problem. We have “experts” that, at each time step , suggest a strategy about what to do at that time (for example, they might be advising on what technology to use, on what investments to make, they might make predictions on whether something is going to happen, thus requiring certain actions, and so on). Based on the quality of the advice that the experts offered in the past, we decide which advice to follow, or with what fraction of our investment to follow which strategy. Subsequently, we find out which loss or gain was associated to each strategy, and, in particular, what loss or gain we personally incurred with the strategy or mix of strategies that we picked, and we move to step .

We want to come up with an algorithm to use the expert advice such that, at the end, that is, at time , we are about as well off as if we had known in advance which expert was the one that gave the best advice, and we had always followed the strategy suggested by that expert at each step. Note that we make no probabilistic assumption, and our analysis will be a worst-case analysis over all possible sequences of events.

The “multiplicative update” algorithm provides a very good solution to this problem, and the analysis of this algorithm is a model for the several other applications of this algorithm, in rather different contexts.