Twenty (Short) Questions

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 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 \(\mathcal{Q}\) of 1.25n+o(n) questions such that for every distribution π, Bob can implement an optimal strategy for π using only questions from \(\mathcal{Q}\). We also show that 1.25no(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.

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Correspondence to Yuval Filmus.

Additional information

The research was funded by ISF grant 1337/16.

Taub Fellow — supported by the Taub Foundations. The research was funded by ISF grant 1337/16.

Research supported by the National Science Foundation under agreement No. CCF-1412958 and by the Simons Foundations.

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Dagan, Y., Filmus, Y., Gabizon, A. et al. Twenty (Short) Questions. Combinatorica 39, 597–626 (2019). https://doi.org/10.1007/s00493-018-3803-4

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Mathematics Subject Classification (2010)

  • 94A50
  • 90B40
  • 68E05