  Given $\eta \in [0, 1]$, a colouring $C$ of $V(G)$ is an $\eta$-majority
colouring if at most $\eta d^+(v)$ out-neighbours of $v$ have colour $C(v)$,
for any $v \in V(G)$. We show that every digraph $G$ equipped with an
assignment of lists $L$, each of size at least $k$, has a $2/k$-majority
$L$-colouring. For even $k$ this is best possible, while for odd $k$ the
constant $2/k$ cannot be replaced by any number less than $2/(k+1)$. This
generalizes a result of Anholcer, Bosek and Grytczuk, who proved the cases
$k=3$ and $k=4$ and gave a weaker result for general $k$.
