Malicious experts versus the multiplicative weights algorithm in online prediction

We consider a prediction problem with two experts and a forecaster. We assume that one of the experts is honest and makes correct prediction with probability $\mu $ at each round. The other one is malicious, who knows true outcomes at each round and makes predictions in order to maximize the loss of the forecaster. Assuming the forecaster adopts the classical multiplicative weights algorithm, we find an upper bound (5) for the value function of the malicious expert, and also a lower bound (19) . Our results imply that the multiplicative weights algorithm cannot resist the corruption of malicious experts. We also show that an adaptive multiplicative weights algorithm is asymptotically optimal for the forecaster, and hence more resistant to the corruption of malicious experts.