The football model, stochastic ordering and martingale transport

Abstract

Tournaments are competitions between a number of teams, the outcome of which determines the relative strength or rank of each team. In many cases, the strength of a team in the tournament is given by a score. Perhaps, the most striking mathematical result on the tournament is Moon's theorem, which provides a necessary and sufficient condition for a feasible score sequence via majorization. To give a probabilistic interpretation of Moon's result, Aldous and Kolesnik introduced the football model, the existence of which gives a short proof of Moon's theorem. However, the existence proof of Aldous and Kolesnik is nonconstructive, leading to the question of a ``canonical'' construction of the football model. The purpose of this paper is to provide explicit constructions of the football model with an additional stochastic ordering constraint, which can be formulated by martingale transport. Two solutions are given: one is by solving an entropy optimization problem via Sinkhorn's algorithm, and the other relies on the idea of shadow couplings. It turns out that both constructions yield the property of strong stochastic transitivity. The nontransitive situations of the football model are also considered.

Figures (1)

• Figure 1: A martingale transport between $\mu=\delta_{0.5}+\delta_{0.7}+\delta_{1.8}$ and $\nu=\delta_0+\delta_1+\delta_2$ is obtained considering $3!$ permutations, two of which are represented on the two columns of this figure. On the left column the permutation $\sigma$ is the identity. $S^{\nu}(\delta_{0.5})=\frac{1}{2}\delta_0+\frac{1}{2} \delta_1$ is obtained as in Remark \ref{['rem:levels']} for the quantile levels of $\nu$ in the interval $[0.5,1.5]$ over the whole interval $[0,3]$ of all orders. We have $\nu^\sigma_1=\nu-S^\nu(\delta_{0.5})=\frac{1}{2} \delta_0+\frac{1}{2}\delta_1+\delta_2$. For the second atom $S^{\nu^\sigma_1}(\delta_{0.7})=\frac{4}{10}\delta_0+\frac{5}{10}\delta_1+\frac{1}{10}\delta_2$. Finally $S^{\nu^\sigma_2}(\delta_{1.8})=\nu^{\sigma}_2=\frac{1}{10}\delta_0+\frac{9}{10}\delta_2$. On the right column where $\sigma$ is different we find $S^\nu(\delta_{0.7})=\frac{3}{10}\delta_{0}+\frac{7}{10}\delta_1$ and $S^{\nu_1^\sigma}(\delta_{1.8})=\frac{2}{10}\delta_1+\frac{8}{10}\delta_2$. Finally $S^{\nu_2^\sigma}(\delta_{0.5})=\frac{7}{10}\delta_0+\frac{1}{10}\delta_1+\frac{2}{10}\delta_2$.

Theorems & Definitions (43)

• Definition 1.1: Generalized tournament matrices and scores
• Definition 1.2: Majorization
• Theorem 1.3: Moon63
• Remark 1.4
• Theorem 1.5: HLPCFMStr65
• proof : Proof of Theorem \ref{['thm:moon']}
• Lemma 2.1
• proof
• Proposition 2.2
• proof