In the distributional Twenty Questions game, Bob chooses a number $x$ from $1$ to $n$ according to a distribution $\mu$, and Alice (who knows $\mu$) attempts to identify $x$ using Yes/No questions, which Bob answers truthfully. Her goal is to minimize the expected number of questions. The optimal strategy for the Twenty Questions game corresponds to a Huffman code for $\mu$, yet this strategy could potentially uses all $2^n$ possible questions. Dagan et al. constructed a set of $1.25^{n+o(n)}$ questions which suffice to construct an optimal strategy for all $\mu$, and showed that this number is optimal (up to sub-exponential factors) for infinitely many $n$. We determine the optimal size of such a set of questions for all $n$ (up to sub-exponential factors), answering an open question of Dagan et al. In addition, we generalize the results of Dagan et al. to the $d$-ary setting, obtaining similar results with $1.25$ replaced by $1 + (d-1)/d^{d/(d-1)}$.

翻译：在分布式二十问游戏中，Bob依据分布μ从1到n中选择一个数x，而知晓μ的Alice试图通过是非题（Bob如实回答）来辨识x。她的目标是最小化问题的期望数量。该游戏的最优策略对应于μ的霍夫曼编码，但此策略理论上可能使用所有2^n个可能的问题。Dagan等人构造了一个包含1.25^{n+o(n)}个问题的集合，足以对所有μ构建最优策略，并证明了对于无穷多个n，该数量（忽略次指数因子）是最优的。我们确定了所有n下这类问题集合的最优大小（忽略次指数因子），回答了Dagan等人提出的一个开放问题。此外，我们将Dagan等人的结果推广至d元设定，以1 + (d-1)/d^{d/(d-1)}替代1.25，获得了类似结论。