非递增谷权重下的最优完全反馈猜牌：Diaconis-Fulman-Holmes 货架洗牌猜想的证明
Optimal complete-feedback card guessing under nonincreasing valley weights: A proof of the Diaconis-Fulman-Holmes shelf-shuffling conjecture
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中文总结 AI 辅助
该论文证明了Diaconis-Fulman-Holmes关于货架洗牌后方向跟踪策略最优的猜想，通过更一般的非递增谷权重定理，利用保持前缀的相邻交换，最大化条件概率与期望正确猜测数。
中文摘要 AI 辅助
设随机排列 $W\in S_n$ 表示 $n$ 张不同牌的顺序。每次猜测后，玩家得知实际牌，目标是最大化正确猜测次数的期望总数。Diaconis、Fulman 和 Holmes 猜想：在单次均匀无偏货架洗牌后，方向跟踪策略是最优的：首先猜测最小的牌，之后在最近一次上升或下降的方向上选择最近的剩余牌。我们证明了一个更一般的定理：如果 \\[ \mathbb P(W=w)=\frac{f(v(w))}{\sum_{\sigma\in S_n}f(v(\sigma))}, \\] 其中 $v(w)$ 是排列的内部谷数，$f$ 是非负且非递增的，且分母为正，那么方向跟踪在每次正概率历史之后最大化正确猜测下一张牌的条件概率。因此，它在所有完全反馈策略中最大化正确猜测次数的期望总数。这里内部谷是满足 $w_{i-1}>w_i<w_{i+1}$ 的位置 $2\le i\le n-1$。结合已知的货架洗牌概率公式，该定理证明了该猜想对任意牌堆大小和货架数量均成立。证明的关键是交换相邻剩余值且保持已揭示前缀：从首选候选开始，交换使谷数不变或增加一，从而将权重的单调性转化为每次历史之后的条件概率排序。
英文摘要
Let a random permutation $W\in S_n$ represent the order of $n$ distinct cards. After each guess, the player learns the actual card, and the objective is to maximize the expected total number of correct guesses. Diaconis, Fulman, and Holmes conjectured that, after a single uniform, unbiased shelf shuffle, the direction-tracking strategy is optimal: first guess the smallest card, and thereafter choose the nearest remaining card in the direction of the most recent rise or fall. We prove a more general theorem: if \[ \mathbb P(W=w)=\frac{f(v(w))}{\sum_{σ\in S_n}f(v(σ))}, \] where $v(w)$ is the number of interior valleys of the permutation, $f$ is nonnegative and nonincreasing, and the denominator is positive, then direction tracking maximizes the conditional probability of correctly guessing the next card after every history of positive probability. It therefore maximizes the expected total number of correct guesses among all complete-feedback strategies. Here an interior valley is a position $2\le i\le n-1$ satisfying $w_{i-1}>w_i<w_{i+1}$. Combined with the known shelf-shuffling probability formula, this theorem proves the conjecture for every deck size and number of shelves. The key to the proof is a swap of adjacent remaining values that preserves the revealed prefix: starting from the preferred candidate, the swap leaves the valley count unchanged or increases it by one, thereby turning monotonicity of the weights into a conditional probability ordering after every history.
发表机构
- School of Data Science, Fudan University（复旦大学数据科学学院）
机构由 AI 辅助整理，请以论文原文为准。
