  In this paper, we compute competition indices and periods of multipartite
tournaments. We first show that the competition period of an acyclic digraph
$D$ is one and $\zeta(D) +1$ is a sharp upper bound of the competition index of
$D$ where $\zeta(D)$ is the sink elimination index of $D$. Then we prove that,
especially, for an acyclic $k$-partite tournament $D$, the competition index of
$D$ is $\zeta(D)$ or $\zeta(D) +1$ for an integer $k \ge 3$. By developing
useful tools to create infinitely many directed walks in a certain regular
pattern from given directed walks, we show that the competition period of a
multipartite tournament with sinks and directed cycles is at most three. We
also prove that the competition index of a primitive digraph does not exceed
its exponent.
