  The area of judicious partitioning considers the general family of
partitioning problems in which one seeks to optimize several parameters
simultaneously, and these problems have been widely studied in various
combinatorial contexts. In this paper, we study essentially the most
fundamental judicious partitioning problem for directed graphs, which naturally
extends the classical Max Cut problem to this setting: we seek bipartitions in
which many edges cross in each direction. It is easy to see that a minimum
outdegree condition is required in order for the problem to be nontrivial, and
we prove that every directed graph with M edges and minimum outdegree at least
two admits a bipartition in which at least (1/6 + o(1))M edges cross in each
direction. We also prove that if the minimum outdegree is at least three, then
the constant can be increased to 1/5. If the minimum outdegree tends to
infinity with N, then the constant increases to 1/4. All of these constants are
best-possible, and provide asymptotic answers to a question of Alex Scott.
