The entropy of lies: playing twenty questions with a liar

Abstract: 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 $\mu$ be a distribution over $[n]$, then the average number of questions used by an optimal strategy that recovers $x\sim \mu$ is between $H(\mu)$ and $H(\mu)+1$. 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 $H(\mu) + k H_2(\mu)$ questions, where $H_2(\mu) = \sum_x \mu(x)\log\log\frac{1}{\mu(x)}$. This also generalizes a result by Rivest et al. for the uniform distribution. Moreover, we design near optimal strategies that only use comparison queries of the form$x \leq c$?' for $c\in[n]$. The usage of comparison queries lends itself naturally to the context of sorting, where we derive sorting algorithms in the presence of adversarial noise.