Gambler’s Ruin and the ICM

Abstract

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 $\mathit{\sigma }\in {\mathit{S}}_3$ be the elimination order (e.g., $\mathit{\sigma }=132$ means player 1 is eliminated first and player 3 is eliminated second, leaving player 2 with $\mathit{A}+\mathit{B}+\mathit{C}$ units).

We seek approximations (and exact formulas) for the elimination order probabilities ${\mathit{P}}_{\mathit{A},\mathit{B},\mathit{C}}(\mathit{\sigma })$. Exact, as well as arbitrarily precise, computation of these probabilities is possible when $\mathit{N}:=\mathit{A}+\mathit{B}+\mathit{C}$ 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.