Gambler’s Ruin and the ICM

Abstract

Consider gambler's ruin with three players, 1, 2, and 3, having initial capitals $A$, $B$, and $C$. 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 the game continues with the remaining two. Let $\sigma\in S_3$ be the elimination order (e.g., $\sigma=132$ means player 1 is eliminated first, player 3 is eliminated second, and player 2 is left with $A+B+C$). We seek approximations (and exact formulas) for the probabilities $P_{A,B,C}(\sigma)$. 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 players' elimination order probabilities.