Abstract
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.
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