Generalized Kings and Single-Elimination Winners in Random Tournaments

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.