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August 2022 Gambler’s Ruin and the ICM
Persi Diaconis, Stewart N. Ethier
Author Affiliations +
Statist. Sci. 37(3): 289-305 (August 2022). DOI: 10.1214/21-STS826


Consider gambler’s ruin with three players, 1, 2, and 3, having initial capitals A, B, and C units. At each round a pair of players is chosen (uniformly at random) and a fair coin flip is made resulting in the transfer of one unit between these two players. Eventually, one of the players is eliminated and play continues with the remaining two. Let σS3 be the elimination order (e.g., σ=132 means player 1 is eliminated first and player 3 is eliminated second, leaving player 2 with A+B+C units).

We seek approximations (and exact formulas) for the elimination order probabilities PA,B,C(σ). Exact, as well as arbitrarily precise, computation of these probabilities is possible when N:=A+B+C is not too large. Linear interpolation can then give reasonable approximations for large N. One frequently used approximation, the independent chip model (ICM), is shown to be inadequate. A regression adjustment is proposed, which seems to give good approximations to the elimination order probabilities.

Funding Statement

The first author was supported by NSF Grant DMS-1954042.
The second author was supported by Simons Foundation Grant 429675.


We thank Laurent Saloff-Coste, Laurent Miclo, Lexing Ying, Gene Kim, Sangchul Lee, Sourav Chatterjee, Guanyang Wang, Thomas Bruss, Pat Fitzsimmons, Bruce Hajek, Denis Denisov, Steve Stigler, Bernard Bru, Mason Malmuth, and Tom Ferguson for their help. We are particularly indebted to Chris Ferguson for emphasizing the utility and mathematical depth of the problem.


Download Citation

Persi Diaconis. Stewart N. Ethier. "Gambler’s Ruin and the ICM." Statist. Sci. 37 (3) 289 - 305, August 2022.


Published: August 2022
First available in Project Euclid: 21 June 2022

MathSciNet: MR4444368
zbMATH: 07569962
Digital Object Identifier: 10.1214/21-STS826

Keywords: Gambler’s ruin problem , independent chip model (ICM) , linear interpolation , Linear regression , Plackett–Luce model , tower problem

Rights: Copyright © 2022 Institute of Mathematical Statistics

Vol.37 • No. 3 • August 2022
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