Extreme Score Distributions in Countable-Outcome Round-Robin Tournaments of Equally Strong Players
Yaakov Malinovsky
TL;DR
This work analyzes extreme-score behavior in round-robin tournaments with countable outcome sets under equal player strength. It introduces a general model $M_{[0,1]}$ and shows that exceedance counts above a high threshold converge in distribution to $\mathrm{Poisson}(e^{-t})$ at a rate $O((\log\log n)^2/\log n)$, then translates these results into explicit limiting distributions for the upper order statistics $s_{(n-j)}^{\star}(n)$ with exact tail-cdf expressions $\lim_{n\to\infty} \mathbb{P}\left(\frac{s_{(n-j)}^{\star}(n)}{ }\le x_n(t)\right)=e^{-e^{-t}}\sum_{k=0}^{j}\frac{e^{-t k}}{k!}$. The approach hinges on negative association of the score vector and rigorous Poisson-approximation bounds, extending prior chess-round-robin findings to a broader countable outcome framework and providing explicit convergence rates. These results yield precise rare-event characterizations for large tournaments and have implications for ranking and assessment in paired-comparison models.
Abstract
We consider a general class of round-robin tournament models of equally strong players. In these models, each of the $n$ players competes against every other player exactly once. For each match between two players, the outcome is a value from a countable subset of the unit interval, and the scores of the two players in a match sum to one. The final score of each player is defined as the sum of the scores obtained in matches against all other players. We study the distribution of extreme scores, including the maximum, second maximum, and lower-order extremes. Since the exact distribution is computationally intractable even for small values of $n$, we derive asymptotic results as the number of players $n$ tends to infinity, including limiting distributions, and rates of convergence.