  A $d$-distinguishing vertex (arc) labeling of a digraph is a vertex (arc)
labeling using $d$ labels that is not preserved by any nontrivial automorphism.
Let $\rho(T)$ ($\rho'(T)$) be the minimum size of a label class in a
2-distinguishing vertex (arc) labeling of a tournament $T$. Gluck's Theorem
implies that $\rho(T) \le \lfloor n/2 \rfloor$ for any tournament $T$ of order
$n$. In this paper we construct a family of tournaments $\cal H$ such that
$\rho(T) \ge \lfloor n/2 \rfloor$ for any order $n$ tournament in $\cal H$.
Additionally, we prove that $\rho'(T) \le \lfloor 7n/36 \rfloor + 3$ for any
tournament $T$ of order $n$ and $\rho'(T) \ge \lceil n/6 \rceil$ when $T \in
{\cal H}$ and has order $n$. These results answer some open questions stated by
Boutin.
