Minimum Guesswork with an Unreliable Oracle

Natan Ardimanov*, Ofer Shayevitz, Itzhak Tamo

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We study a guessing game where Alice holds a discrete random variable $X$ , and Bob tries to sequentially guess its value. Before the game begins, Bob can obtain side-information about $X$ by asking an oracle, Carole, any binary question of his choosing. Carole's answer is however unreliable, and is incorrect with probability $\epsilon $. We show that Bob should always ask Carole whether the index of $X$ is odd or even with respect to a descending order of probabilities - this question simultaneously minimizes all the guessing moments for any value of $\epsilon $. In particular, this result settles a conjecture of Burin and Shayevitz. We further consider a more general setup where Bob can ask a multiple-choice $M$ -ary question, and then observe Carole's answer through a noisy channel. When the channel is completely symmetric, i.e., when Carole decides whether to lie regardless of Bob's question and has no preference when she lies, a similar question about the ordered index of $X$ (modulo $M$ ) is optimal. Interestingly however, the problem of testing whether a given question is optimal appears to be generally difficult in other symmetric channels. We provide supporting evidence for this difficulty, by showing that a core property required in our proofs becomes NP-hard to test in the general $M$ -ary case. We establish this hardness result via a reduction from the problem of testing whether a system of modular difference disequations has a solution, which we prove to be NP-hard for $M\geq 3$.

Original languageEnglish
Article number9237980
Pages (from-to)7528-7538
Number of pages11
JournalIEEE Transactions on Information Theory
Volume66
Issue number12
DOIs
StatePublished - Dec 2020

Funding

FundersFunder number
NSF?Binational Science Foundation
National Science Foundation
United States - Israel Binational Science Foundation2015814
Horizon 2020 Framework Programme639573
European Research Council1367/14, 1495/18
United States-Israel Binational Science Foundation
Israel Science Foundation1030/15

    Keywords

    • Information theory
    • combinatorics
    • computational complexity

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