  Tournaments can be used to model a variety of practical scenarios including
sports competitions and elections. A natural notion of strength of alternatives
in a tournament is a generalized king: an alternative is said to be a $k$-king
if it can reach every other alternative in the tournament via a directed path
of length at most $k$. In this paper, we provide an almost complete
characterization of the probability threshold such that all, a large number, or
a small number of alternatives are $k$-kings with high probability in two
random models. We show that, perhaps surprisingly, all changes in the threshold
occur in the range of constant $k$, with the biggest change being between $k=2$
and $k=3$. In addition, we establish an asymptotically tight bound on the
probability threshold for which all alternatives are likely able to win a
single-elimination tournament under some bracket.
