OP-00751 Solution claimed

Fourth-order diffusion approximation for three-player ruin

Three players start with positive integer capitals $A, B, C$. At each step a pair of surviving players is chosen uniformly at random, a fair coin is tossed, and the loser pays the winner one unit; a player whose capital reaches zero is removed. Let $P_{A,B,C}(\sigma)$ be the probability of the elimination order $\sigma \in S_3$, with the final winner listed last, and let $N = A + B + C$. The limit $$P^{\mathrm{BM}}_{A,B,C}(\sigma) = \lim_{m \to \infty} P_{mA,mB,mC}(\sigma)$$ exists by the Brownian diffusion limit, and Diaconis and Ethier express it through the exit distribution of planar Brownian motion from an equilateral triangle. Denisov and Wachtel proved that, for the elimination order $321$, the difference between the discrete probability and its Brownian counterpart is $O(N^{-3})$.

Problem. For every fixed $A, B, C$ and $\sigma$, is $$\left| P_{nA,nB,nC}(\sigma) - P^{\mathrm{BM}}_{A,B,C}(\sigma) \right| = O\bigl((nN)^{-4}\bigr) \qquad (n \to \infty)?$$ The implied constant may depend on the fixed capital proportions and on the elimination order.

Source: Diaconis and Ethier (2022), Conjecture 4.2(b).

Imported from github.com/MColbrook/AIM.

Classification

Primary: PR — Probability
Keywords: AIM Open Applied Problems, diffusion approximation, gambler's ruin, harmonic measure

References

@article{diaconis2022,
  author        = {Diaconis, Persi and Ethier, Stewart N.},
  title         = {Gambler's ruin and the {ICM}},
  journal       = {Statist. Sci.},
  volume        = {37},
  number        = {3},
  year          = {2022},
  pages         = {289--305},
  doi           = {10.1214/21-STS826},
  eprint        = {2011.07610},
  archivePrefix = {arXiv},
}
@article{denisov2024,
  author        = {Denisov, Denis and Wachtel, Vitali},
  title         = {Harmonic measure in a multidimensional gambler's problem},
  journal       = {Ann. Appl. Probab.},
  volume        = {34},
  number        = {5},
  year          = {2024},
  pages         = {4387--4407},
  doi           = {10.1214/24-AAP2069},
  eprint        = {2212.11526},
  archivePrefix = {arXiv},
}
@misc{aim-gambler-scaling,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Fourth-order diffusion approximation for three-player ruin}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/615-gambler-scaling.md},
  note         = {Entry 615 at commit a153f46, 2026-10-13},
}

Resolution claims

Fourth-order diffusion approximation for three-player gambler's ruin v1 GitHub ↗

Kaiwen Jin

For three-player fair gambler's ruin, we prove that the probability of each elimination order differs from its Brownian scaling limit by $O((nN)^{-4})$ when fixed positive capitals with total $N$ are multiplied by $n$. This is Conjecture 4.2(b) of Diaconis and Ethier. The error is antisymmetric when the two losing players are exchanged. On the resulting half triangle, the Brownian harmonic …

Full resolution · not verified · fidelity not established · posted on GitHub 13 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: AIM @ a153f46, paper/gambler_scaling.pdf.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00751, "Fourth-order diffusion approximation for three-player ruin", clopen.solutions.