  We consider the setting of prediction with expert advice; a learner makes
predictions by aggregating those of a group of experts. Under this setting, and
for the right choice of loss function and "mixing" algorithm, it is possible
for the learner to achieve a constant regret regardless of the number of
prediction rounds. For example, a constant regret can be achieved for
\emph{mixable} losses using the \emph{aggregating algorithm}. The
\emph{Generalized Aggregating Algorithm} (GAA) is a name for a family of
algorithms parameterized by convex functions on simplices (entropies), which
reduce to the aggregating algorithm when using the \emph{Shannon entropy}
$\operatorname{S}$. For a given entropy $\Phi$, losses for which a constant
regret is possible using the \textsc{GAA} are called $\Phi$-mixable. Which
losses are $\Phi$-mixable was previously left as an open question. We fully
characterize $\Phi$-mixability and answer other open questions posed by
\cite{Reid2015}. We show that the Shannon entropy $\operatorname{S}$ is
fundamental in nature when it comes to mixability; any $\Phi$-mixable loss is
necessarily $\operatorname{S}$-mixable, and the lowest worst-case regret of the
\textsc{GAA} is achieved using the Shannon entropy. Finally, by leveraging the
connection between the \emph{mirror descent algorithm} and the update step of
the GAA, we suggest a new \emph{adaptive} generalized aggregating algorithm and
analyze its performance in terms of the regret bound.
