Technical Report MSC-2018-01

Title: Twenty Questions Game Using Restricted Sets of Questions
Authors: Yuval Dagan
Supervisors: Yuval Filmus
PDFCurrently accessibly only within the Technion network
Abstract: A basic combinatorial interpretation of Shannon's entropy function is via the ``20 questions'' game. This cooperative game is played by two players, Alice and Bob: Alice picks a distribution $\pi$ over the numbers $\{1,\ldots,n\}$, and announces it to Bob. She then chooses a number $x$ according to $\pi$, and Bob attempts to identify $x$ using as few Yes/No queries as possible, on average.

An optimal strategy for the ``20 questions'' game is given by a Huffman code for $\pi$: Bob's questions reveal the codeword for $x$ bit by bit. This strategy finds $x$ using fewer than $H(\pi)+1$ questions on average. However, the questions asked by Bob could be arbitrary. In this document, we investigate the following question: \emph{Are there restricted sets of questions that match the performance of Huffman codes, either exactly or approximately?}

Our first main result shows that for every distribution $\pi$, Bob has a strategy that uses only questions of the form ``$x < c$?'' and ``$x = c$?'', and uncovers $x$ using at most $H(\pi)+1$ questions on average, matching the performance of Huffman codes in this sense. We also give a natural set of $O(rn^{1/r})$ questions that achieve a performance of at most $H(\pi)+r$, and show that $\Omega(rn^{1/r})$ questions are required to achieve such a guarantee.

Our second main result gives a set $\mathcal{Q}$ of $1.25^{n+o(n)}$ questions such that for every distribution $\pi$, Bob can implement an \emph{optimal} strategy for $\pi$ using only questions from $\mathcal{Q}$. We also show that $1.25^{n-o(n)}$ questions are needed, for infinitely many $n$.

If we allow a small slack of $r$ over the optimal strategy, then roughly $(rn)^{\Theta(1/r)}$ questions are necessary and sufficient.

CopyrightThe above paper is copyright by the Technion, Author(s), or others. Please contact the author(s) for more information

Remark: Any link to this technical report should be to this page (, rather than to the URL of the PDF files directly. The latter URLs may change without notice.

To the list of the MSC technical reports of 2018
To the main CS technical reports page

Computer science department, Technion