Knights, spies, games and ballot sequences

TL;DR

This paper solves the Knights and Spies Problem by determining the minimum questions needed to identify all individuals' roles, considering truthful knights and potentially lying spies, with implications for related game analysis and probabilistic methods.

Contribution

It introduces a solution to the problem, analyzing question strategies and the role of game theory and probability, and presents open questions for future research.

Findings

Minimum questions required for guaranteed identification

Analysis of a related two-player game

Exploration of probabilistic aspects

Abstract

This paper presents a solution to the Knights and Spies Problem: In a room there are n people, each labelled with a unique number between 1 and n. A person may either be a knight or a spy. Knights always tell the truth, while spies may either lie or tell the truth, as they see fit. Each person in the room knows the identity of everyone else. Apart from this, all that is known is that strictly more knights than spies are present. Asking only questions of the form: `Person i, what is the identity of person j?', what is the least number of questions that will guarantee to find the true identities of all n people? The analysis of a related two-player game is critical to the proof. Some probabilistic aspects are also explored. The paper ends by presenting three open questions concerned with generalisations of the problem.