# Coxeter tournaments

## TL;DR

This paper introduces a Coxeter group-based framework for analyzing tournaments, extending classical results with geometric, probabilistic, and algorithmic proofs, and proposes a Coxeter analogue of the Bradley--Terry model for paired comparisons.

## Contribution

It develops a Coxeter group perspective on tournaments, providing new proofs of Moon's theorem and introducing a Coxeter version of the Bradley--Terry model.

## Key findings

- Established a Coxeter version of Moon's theorem.
- Provided geometric, probabilistic, and algorithmic proofs.
- Introduced a Coxeter analogue of the Bradley--Terry model.

## Abstract

We describe the Coxeter permutahedra, recently studied by Ardila, Castillo, Eur and Postnikov, in terms of random Coxeter tournaments, which involve cooperative and solitaire games, as well as the usual competitive games in graph tournaments. In this way, we establish a Coxeter version of Moon's theorem on random tournaments. We present a geometric proof by the Mirsky--Thompson generalized Birkhoff's theorem, a probabilistic proof by Strassen's coupling theorem, and an algorithmic proof by a Coxeter analogue of the Havel--Hakimi algorithm. These proofs have interpretations in terms of players choosing competitors/collaborators with respect to relative weakness/strength. We also introduce a natural Coxeter analogue of the Bradley--Terry model, from the statistical theory of paired comparisons.