Absolute continuity of stationary Elo ratings
Fix an integer $N \ge 2$, a vector of true skills $\rho \in \mathbb{R}^N$ with $\sum_i \rho_i = 0$, and constants $c, K > 0$ with $Kc < 1$; let $$Z_N = \Bigl\{x \in \mathbb{R}^N : \sum_i x_i = 0\Bigr\}, \qquad b(u) = \tanh(cu).$$ The binary-score logistic Elo chain on $Z_N$ moves as follows. At each step an ordered pair $(i, j)$ of distinct players is chosen uniformly, independently of the past; given the pair, a fresh score $S \in \{-1, 1\}$ is drawn with $$\mathbb{P}(S = 1 \mid i, j) = \frac{1 + b(\rho_i - \rho_j)}{2},$$ and only the coordinates $i$ and $j$ are updated: $$X_i' = X_i + K\{S - b(X_i - X_j)\}, \qquad X_j' = X_j - K\{S - b(X_i - X_j)\}.$$ Cortez and Tossounian proved that this chain has a unique invariant probability measure $\pi_{N,\rho,c,K}$ on $Z_N$, that the chain converges to it exponentially fast in the $2$-Wasserstein distance, and that $\pi_{N,\rho,c,K}$ has full support and a finite exponential moment. When the scores have densities on $[-1, 1]$ instead, they note that the invariant law clearly has a density.
Problem. Is $\pi_{N,\rho,c,K}$ absolutely continuous with respect to $(N-1)$-dimensional Lebesgue measure on $Z_N$ for every such choice of $N$, $\rho$, $c$ and $K$?
Source: Cortez and Tossounian (2026), Section 6, "Smoothness of π", who note that the existence of a density is not obvious for binary scores. Stated by AIM — Open Applied Problems for the logistic binary-score chain, for every admissible choice of parameters.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- ST
- Keywords
- AIM Open Applied Problems, Elo rating system, invariant measures, Markov chains
References
@article{cortez2026,
author = {Cortez, Roberto and Tossounian, Hagop},
title = {Convergence and stationary distribution of {E}lo rating systems},
journal = {Ann. Appl. Probab.},
volume = {36},
number = {4},
year = {2026},
pages = {3395--3418},
doi = {10.1214/26-AAP2306},
eprint = {2410.09180},
archivePrefix = {arXiv},
}
@misc{aim-elo-density,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Absolute continuity of stationary Elo ratings}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/637-elo-density.md},
note = {Entry 637 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We give a computer-assisted counterexample to the universal absolute-continuity assertion in AIM Problem 637. The admissible parameters are two equally skilled players, logistic scale $c=1/2$, and update size $K=9/10$. The stationary law is singular continuous, has full support on the zero-sum line, and has measure Hausdorff dimension at most $10\log 2/7<1$. The central finite certificate verifies an explicit drift inequality …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, siavash-sadeghi-637/aim637_counterexample.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00679, "Absolute continuity of stationary Elo ratings", clopen.solutions.