Optimal sets of questions for Twenty Questions
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913270593486848 |
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| author | Filmus, Yuval Mehalel, Idan |
| author_facet | Filmus, Yuval Mehalel, Idan |
| contents | In the distributional Twenty Questions game, Bob chooses a number $x$ from $1$ to $n$ according to a distribution $μ$, and Alice (who knows $μ$) 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 $μ$, 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 $μ$, 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 $1 + (d-1)/d^{d/(d-1)}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2106_01737 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Optimal sets of questions for Twenty Questions Filmus, Yuval Mehalel, Idan Discrete Mathematics Combinatorics In the distributional Twenty Questions game, Bob chooses a number $x$ from $1$ to $n$ according to a distribution $μ$, and Alice (who knows $μ$) 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 $μ$, 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 $μ$, 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 $1 + (d-1)/d^{d/(d-1)}$. |
| title | Optimal sets of questions for Twenty Questions |
| topic | Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2106.01737 |