Supermartingales in prediction with expert advice

The paper applies the method of defensive forecasting, based on the use of game-theoretic supermartingales, to prediction with expert advice. In the traditional setting of a countable number of experts and a finite number of outcomes, the Defensive Forecasting algorithm is very close to the well-known Aggregating Algorithm. Not only the performance guarantees but also the predictions are the same for these two methods of fundamentally different nature. The paper introduces a new setting where the experts can give advice conditional on the learner's future decision. Both the Defensive Forecasting algorithm and the Aggregating Algorithm can be adapted to the new setting and give the same performance guarantees as in the traditional setting. Also the paper outlines an application of the Defensive Forecasting algorithm to a setting with multiple loss functions