Twenty (simple) questions

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 Π over the numbers {1,…,n}, and announces it to Bob. She then chooses a number x according to Π, 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 Π: Bob's questions reveal the codeword for x bit by bit. This strategy finds x using fewer than H(Π)+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 Π, Bob has a strategy that uses only questions of the form "x < c?" and "x = c?", and uncovers x using at most H(Π)+1 questions on average, matching the performance of Huffman codes in this sense. We also give a natural set of O(rn1/r) questions that achieve a performance of at most H(Π)+r, and show that Ωrn1/r) questions are required to achieve such a guarantee. Our second main result gives a set Q of 1.25n+o(n) questions such that for every distribution Π, Bob can implement an optimal strategy for Π using only questions from Q. We also show that 1.25n-o(n) questions are needed, for infinitely many n. If we allow a small slack of r over the optimal strategy, then roughly (rn)Θ(1/r) questions are necessary and sufficient.