数千名玩家独立筹码模型名次概率的近似精确计算
Near-Exact Computation of Independent Chip Model Placement Probabilities for Thousands of Players
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中文总结 AI 辅助
提出 DE-ICM 算法，在 O(M n^2) 时间内精确计算数千名玩家的 ICM 名次概率，数值精度接近双精度，且速度极快。
中文摘要 AI 辅助
独立筹码模型（ICM）将扑克锦标赛中剩余玩家的筹码堆转换为名次概率和奖金权益。它是锦标赛求解器中的标准模型，但其定义考虑了所有可能的完赛顺序，对于大规模参赛者而言，精确计算一直被认为是难以处理的。同样的数学也出现在其他领域；例如，ICM 是一种以筹码为权重的 Plackett-Luce 排名模型。我们提出了 DE-ICM，一种确定性算法，在 O(M n^2) 时间内计算所有 n 名玩家在所有获奖名次上的名次概率，其中正交节点数 M 对于可接受的输入和目标精度是固定的。其数值方案允许目标设置为接近双精度算术的精度。该算法评估了一个经典的积分表示，其中以玩家的指数时钟为条件，领先的玩家数量服从泊松二项分布。在单个网格上的双指数正交计算每个积分；其截断误差具有闭式界限，而控制离散化误差的步长由场大小决定。然后，数值稳定的反卷积从单个共享乘积中恢复每个玩家的留一系数，时间复杂度为 O(n)。在最多 4000 名玩家的精确可解实例上，权益的最差观测相对误差为 4.0 × 10^-14，名次概率的最差绝对误差为 1.1 × 10^-14。1000 名玩家的完整名次矩阵在单个 CPU 核心上仅需 0.2 秒。
英文摘要
The Independent Chip Model (ICM) converts the chip stacks of the players remaining in a poker tournament into finishing-place probabilities and prize equities. It is the standard model in tournament solvers, but its definition considers all possible finishing orders, and exact computation has been regarded as intractable for large fields. The same mathematics appears in other fields; for instance, the ICM is a Plackett-Luce ranking model with stacks as weights. We present DE-ICM, a deterministic algorithm that computes the placement probabilities of all $n$ players for all paid places in $O(M n^{2})$ time, where the number $M$ of quadrature nodes is fixed for the admissible inputs and the target accuracy. Its numerical scheme allows the target to be set close to the accuracy of double-precision arithmetic. The algorithm evaluates a classical integral representation in which, conditional on a player's exponential clock, the number of players ahead is Poisson-binomial. A double-exponential quadrature on a single grid evaluates every integral; its truncation errors have closed-form bounds, and the step size, which controls the discretization error, follows from the field size. A numerically stable deconvolution then recovers each player's leave-one-out coefficients in $O(n)$ time from one shared product. On exactly solvable instances with up to 4,000 players, the worst observed relative error of an equity is $4.0 \times 10^{-14}$ and the worst absolute error of a placement probability $1.1 \times 10^{-14}$. The full placement matrix for 1,000 players takes 0.2 seconds on one CPU core.
发表机构
- GTO Wizard
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