Twenty (Short) 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 main result gives a set Q of 1.25^n+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 1.25^n−o(n) allowed questions are needed, for infinitely many n. When allowing a small slack of r questions for identifying x over the optimal strategy, we show that a set of roughly (rn)^Θ(1/r) allowed questions is necessary and sufficient.