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)+kH2(µ) questions, where H2(µ) = ^P_x µ(x) log log _µ(¹x). 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.