A \emph{majority coloring} of a digraph is a coloring of its vertices such
that for each vertex v, at most half of the out-neighbors of v has the same
color as v. A digraph D is \emph{majority k-choosable} if for any
assignment of lists of colors of size k to the vertices there is a majority
coloring of D from these lists. We prove that every digraph is majority
4-choosable. This gives a positive answer to a question posed recently by
Kreutzer, Oum, Seymour, van der Zypen, and Wood in \cite{Kreutzer}. We obtain
this result as a consequence of a more general theorem, in which majority
condition is profitably extended. For instance, the theorem implies also that
every digraph has a coloring from arbitrary lists of size three, in which at
most 2/3 of the out-neighbors of any vertex share its color. This solves
another problem posed in \cite{Kreutzer}, and supports an intriguing conjecture
stating that every digraph is majority 3-colorable