OPTIMAL SETS OF QUESTIONS FOR TWENTY QUESTIONS

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 2ⁿ possible questions. Dagan et al. constructed a set of 1.25ⁿ+o(n) questions which suffice to construct an optimal strategy for all μ, and showed that this number is optimal (up to subexponential factors) for infinitely many n. We determine the optimal size of such a set of questions for all n (up to subexponential 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