  We study online aggregation of the predictions of experts, and first show new
second-order regret bounds in the standard setting, which are obtained via a
version of the Prod algorithm (and also a version of the polynomially weighted
average algorithm) with multiple learning rates. These bounds are in terms of
excess losses, the differences between the instantaneous losses suffered by the
algorithm and the ones of a given expert. We then demonstrate the interest of
these bounds in the context of experts that report their confidences as a
number in the interval [0,1] using a generic reduction to the standard setting.
We conclude by two other applications in the standard setting, which improve
the known bounds in case of small excess losses and show a bounded regret
against i.i.d. sequences of losses.
