The Entropy of Lies: Playing Twenty Questions with a Liar

"Twenty questions" is a guessing game played by two players: Bob thinks of an integer between 1 and n, and Alice’s goal is to recover it using a minimal number of Yes/No questions. Shannon’s entropy has a natural interpretation in this context. It characterizes the average number of questions used by an optimal strategy in the distributional variant of the game: let μ be a distribution over [n], then the average number of questions used by an optimal strategy that recovers x∼ μ is between H(μ) and H(μ)+1. We consider an extension of this game where at most k questions can be answered falsely. We extend the classical result by showing that an optimal strategy uses roughly H(μ) + k H_2(μ) questions, where H_2(μ) = ∑_x μ(x)log log 1/μ(x). This also generalizes a result by Rivest et al. (1980) for the uniform distribution. Moreover, we design near optimal strategies that only use comparison queries of the form "x ≤ c?" for c ∈ [n]. The usage of comparison queries lends itself naturally to the context of sorting, where we derive sorting algorithms in the presence of adversarial noise.