  We say that a digraph $D$ is competitive if any pair of vertices has a common
out-neighbor in $D$ and that a graph $G$ is competitively orientable if there
exists a competitive orientation of $G$. The notion of competitive digraphs
arose while studying digraph whose competition graphs are complete. We derive
some useful properties of competitively orientable graphs and show that a
complete graph of order $n$ is competitively orientable if and only if $n \geq
7$. Then we completely characterize a competitively orientable complete
multipartite graph in terms of the sizes of its partite sets. Moreover, we
present a way to build a competitive multipartite tournament in each of
competitively orientable cases.
