Generalized Kings and Single-Elimination Winners in Random Tournaments

TL;DR

This paper characterizes the probability thresholds for the emergence of generalized kings in random tournaments and analyzes conditions under which all alternatives can win in single-elimination tournaments.

Contribution

It provides an almost complete characterization of thresholds for k-kings in two random models and bounds the probability for all alternatives to win in single-elimination tournaments.

Findings

Thresholds change mainly between k=2 and k=3.

All k-kings appear with high probability in certain regimes.

Almost complete characterization of k-kings in random models.

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.