论文标题
赌徒的废墟和ICM
Gambler's Ruin and the ICM
论文作者
论文摘要
考虑一下赌徒的废墟，其中有三名球员，1、2和3，具有初始首都$ a $，$ b $和$ c $单位。在每个回合中，选择了一对玩家（随机均匀），并进行公平的硬币翻转，从而在这两个玩家之间转移一个单位。最终，其中一名球员被淘汰，剩下的两个球员仍在继续比赛。令$σ\在S_3 $中为消除订单（例如，$σ= 132 $表示播放器1首先消除，并且播放器3被淘汰第二，而Player 2则为$ a+a+b+c $单位）。 我们寻求近似值（和确切的公式）来消除订单概率$ p_ {a，b，c}（σ）$。当$ n：= a+b+c $不太大时，确切的概念以及任意精确的计算是可能的。然后，线性插值可以为大$ n $提供合理的近似值。一种经常使用的近似值，即独立的芯片模型（ICM），被证明是不足的。提出了回归调整，这似乎可以使消除订单概率具有良好的近似值。
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 $σ\in S_3$ 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 $P_{A,B,C}(σ)$. 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.
