Axiomatic and Erd\H{o}s-Moon approaches to tournament rankings

TL;DR

This paper explores combining axiomatic and Erd extendash{}Moon methods for tournament rankings, establishing the minimal proportion of backward arcs achievable under specific axioms, notably proving it is 3/4 for the Copeland axiom.

Contribution

It introduces a combined approach to tournament rankings, defining the Erd extendash{}Moon number for axioms and calculating it for the Copeland axiom as 3/4.

Findings

The Erd extendash{}Moon number for the Copeland axiom is 3/4.

The combined approach minimizes backward arcs among rankings satisfying given axioms.

It extends the understanding of tournament ranking optimization under axiomatic constraints.

Abstract

Tournament ranking is a function that assigns each vertex of a tournament (i.e., a directed graph without loops, in which each pair of different vertexes is connected by exactly one arc) a number called the rank of the vertex. One of approaches to constructing tournament rankings suggests choosing a ranking that satisfies a fixed set of axioms. In another approach, proposed by Erd\H{o}s and Moon, only injective rankings are considered, and among them, one that minimises the number of backward arcs is selected (an arc $x\to y$ is called backward iff the rank of $x$ is less than the rank of $y$). We combine these two approaches as follows: among the rankings that satisfy a fixed set of axioms, we choose one that minimises the number of backward arcs. The Erd\H{o}s-Moon approach naturally leads to the question of how small the proportion of backward arcs can be guaranteed when using injective rankings. Erd\H{o}s and Moon showed that the answer to this question is $1/2$. A similar question arises in our approach: how small the proportion of backward arcs can be guaranteed when using rankings that satisfy a set of axioms $\mathcal{A}$? We call this number the Erd\H{o}s-Moon number of $\mathcal{A}$. We prove that the Erd\H{o}s-Moon number of the Copeland axiom equals $3/4$.