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Bound the maximum Elo rating without projection

Establish that, for n > 2 players under the uncapped Elo process (i.e., Elo_M with M = +∞) driven by the Bradley–Terry–Luce win probabilities and zero-sum ratings with small step-size η, the maximum absolute Elo rating remains small with high probability for a large (polynomial-in-n) number of steps.

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Background

To enable analysis, the paper introduces a capped Elo update with an orthogonal projection step that keeps ratings in [-M, M]n and preserves the zero-sum property. In practice, Elo implementations typically omit this projection, and the authors’ simulations suggest that ratings remain bounded when η is sufficiently small.

The authors provide heuristic arguments for the two-player case and discuss why extending these arguments to n > 2 players is technically challenging: the maximum rating is not a supermartingale, and atypical configurations may occur although they seem unlikely. Formal control of the maximum rating is needed to justify removing the projection and aligning the theory with practice.

References

Unfortunately, we weren't able to make this argument formal for n > 2 players. We leave proving that indeed, with high probability, the maximum rating remains small for a large number of steps as an open problem.

— An Analysis of Elo Rating Systems via Markov Chains (2406.05869 - Olesker-Taylor et al., 9 Jun 2024) in Appendix, Experiments: Further Discussion and Figures (end)