Searching for knights and spies: a majority/minority game

Abstract: There are n people, each of whom is either a knight or a spy. It is known that at least k knights are present, where n/2 < k < n. Knights always tell the truth. We consider both spies who always lie and spies who answer as they see fit. This paper determines the number of questions required to find a spy or prove that everyone in the room is a knight. We also determine the minimum number of questions needed to find at least one person's identity, or a nominated person's identity, or to find a spy (under the assumption that a spy is present). For spies who always lie, we prove that these searching problems, and the problem of finding a knight, can be solved by a simultaneous optimal strategy. We also give some computational results on the problem of finding all identities when spies always lie, and end by stating some open problems.