Random Knockout Tournaments

TL;DR

This paper analyzes the probabilities of winning in a randomized knockout tournament with probabilistic match outcomes, providing bounds for the best player's chances under various configurations.

Contribution

It introduces bounds on player win probabilities in random knockout tournaments with probabilistic match outcomes, including special case analyses.

Findings

Derived lower bounds for the best player's win probability.

Established upper and lower bounds for all players.

Analyzed the impact of tournament format variations on player success.

Abstract

We consider a random knockout tournament among players $1, \ldots, n$, in which each match involves two players. The match format is specified by the number of matches played in each round, where the constitution of the matches in a round is random. Supposing that there are numbers $v_1, \ldots, v_n$ such that a match between $i$ and $j$ will be won by $i$ with probability $\frac{v_i}{v_i+v_j}$, we obtain a lower bound on the tournament win probability for the best player, as well as upper and lower bounds for all the players. We also obtain additional bounds by considering the best and worst formats for player $1$ in the special case $v_1 > v_2 = v_3 = \cdots = v_n.$