How do we define knowledge, and, crucially for cryptography, ignorance? In this chapter we lay the basis for future security proofs by formalizing the notion of knowledge of a quantum party, such as the memory of an eavesdropper, about a classical piece of information, such as a secret key. For this we introduce an appropriate measure of conditional entropy, the min-entropy, and introduce important tools to bound it using guessing games.

A reading of the literature on cognitive hierarchies leaves the impression that a subject’s type is predetermined before she comes into the lab so that the distribution of types is exogenous and immutable across games. In this chapter we view the choice of a person’s cognitive level as endogenous and explain it by focusing on subject’s ’expectations about the cognitive levels endogenously chosen by others. We run a set of experiments using the two-thirds guessing game where subjects receive public advice offered by a set of advisors. We discover that certain types of public advice, those that are commonly interpreted as meaningful, are capable of shifting the distribution of observed cognitive types, indicating that the distribution is endogenous.