Optimal sets of questions for Twenty Questions

TL;DR

This paper determines the minimal size of question sets needed for optimal Twenty Questions strategies across all distributions, generalizing previous results and solving an open problem in the field.

Contribution

It establishes the exact asymptotic size of question sets for all distributions and extends the results to the d-ary case, answering an open question.

Findings

Optimal question set size is determined up to sub-exponential factors.

Generalization of results to the d-ary Twenty Questions setting.

Provides tight bounds matching previous constructions.

Abstract

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 $1 + (d-1)/d^{d/(d-1)}$.