Prediction with expert evaluators' advice

We introduce a new protocol for prediction with expert advice in which each expert evaluates the learner's and his own performance using a loss function that may be different from the loss functions used by the other experts. The learner's goal is to perform better or not much worse than each expert, as evaluated by that expert, for all experts simultaneously. If the loss functions used by the experts are all proper scoring rules and all mixable, we show that the defensive forecasting algorithm enjoys the same performance guarantee as that attainable in the standard setting and known to be optimal.