  The R\'{e}nyi $\alpha$-entropy $H_{\alpha}$ of complete antisymmetric
directed graphs (i.e., tournaments) is explored. We optimize $H_{\alpha}$ when
$\alpha = 2$ and $3$, and find that as $\alpha$ increases $H_{\alpha}$'s
sensitivity to what we refer to as `regularity' increases as well. A regular
tournament on $n$ vertices is one with each vertex having out-degree
$\frac{n-1}{2}$, but there is a lot of diversity in terms of structure among
the regular tournaments; for example, a regular tournament may be such that
each vertex's out-set induces a regular tournament (a doubly-regular
tournament) or a transitive tournament (a rotational tournament). As $\alpha$
increases, on the set of regular tournaments, $H_{\alpha}$ has maximum value on
doubly regular tournaments and minimum value on rotational tournaments. The
more `regular', the higher the entropy. We show, however, that $H_2$ and $H_3$
are maximized, among all tournaments on any number of vertices by any regular
tournament. We also provide a calculation that is equivalent to the von Neumann
entropy, but may be applied to any directed or undirected graph and shows that
the von Neumann entropy is a measure of how quickly a random walk on the graph
or directed graph settles.
