Twenty questions game using simple questions

Yuval Dagan (M.Sc. Thesis Seminar)

Tuesday, 14.03.2017, 13:00

Taub 601

Advisor: Prof. Yuval Filmus

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 paper, we investigate the following question: 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.