The geometry of random tournaments

TL;DR

This work connects combinatorial properties of tournaments to convex geometry by showing that mean score sequences of random tournaments on any graph $G$ coincide with the image of the graphic zonotope $Z_G$ under a projection that encodes edge-win probabilities. The authors provide short, geometric proofs of Moon's characterization and extend Landau's theorem to general graphs via a hyperplane description of $Z_G$ based on the submodular function $\phi$, then refine the results using zonotopal tilings and mixed subdivisions to realize mean score sequences with randomness restricted to a forest. They demonstrate that every lattice point of $Z_G$ is attainable as a mean score (and potentially as a deterministic score) and that the construction extends to multigraphs. The approach offers a unified, geometric framework for dominance relations and score sequences with potential applications beyond complete graphs.

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.

Theorems & Definitions (11)

• Theorem 1: Landau's Theorem
• Definition 2
• Theorem 3: Moon's Theorem
• Theorem 4: Generalized Moon's Theorem
• proof
• Remark
• Theorem 5
• Corollary 6: Generalized Landau's Theorem
• Lemma 7
• proof : Proof of \ref{['T_tree']}