The geometry of random tournaments

TL;DR

This paper provides geometric proofs for classical theorems characterizing score sequences in random tournaments, using zonotopes from convex geometry, and explores realizability of mean score sequences.

Contribution

It introduces short, natural proofs of Landau's and Moon's theorems using zonotopes, applicable to any graph, and discusses realizability by tournaments with mixed randomness.

Findings

Zonotope-based proofs of Landau's and Moon's theorems.

Characterization of mean score sequences via convex geometry.

Existence of tournaments realizing mean scores with partial randomness.

Abstract

A tournament is an orientation of a graph. Each edge represents a match, directed towards the winner. The score sequence lists the number of wins by each team. Landau (1953) characterized score sequences of the complete graph. Moon (1963) showed that the same conditions are necessary and sufficient for mean score sequences of random tournaments. We present short and natural proofs of these results that work for any graph using zonotopes from convex geometry. A zonotope is a linear image of a cube. Moon's Theorem follows by identifying elements of the cube with distributions and the linear map as the expectation operator. Our proof of Landau's Theorem combines zonotopal tilings with the theory of mixed subdivisions. We also show that any mean score sequence can be realized by a tournament that is random within a subforest, and deterministic otherwise.