Collaborative Discovery: A study of Guru-follower dynamics

A model is suggested for abstracting the interaction between a guru and her followers. A guru has mental powers of insight that surpass those of her followers, but they (i) might not believe in her powers and (ii) have limited attention span, so may terminate interaction with her before reaching full “enlightenment”. The main question that interests us is to understand when are gurus likely to retain their flock. This question affects not only spiritual leaders, but also automated processes competing to retain users in a congested information world. Our model assumes followers interact with the guru while keeping in mind a single “retention parameter” that measures the strength of their belief in her predictive power, and the guru’s objective is to reinforce and maximize this parameter through “informative” and “correct” predictions. We make three contributions. First, we define a natural class of retentive scoring rules to model the way followers update their retention parameter and thus evaluate gurus they interact with. These rules are shown to be tightly connected to truth-eliciting “proper scoring rules” studied in Decision Theory since the 1950’s [McCarthy, PNAS 1956]. Second, we focus on the intrinsic properties of phenomena that are amenable to collaborative discovery with a guru. Assuming follower types (or “identities”) are sampled from a distribution D, the retention complexity of D is the minimal initial retention value (or “strength of faith”) that a follower must have before approaching the guru, in order for the guru to retain that follower throughout the collaborative discovery, during which the follower “discovers” his true “identity”. Third and last, we take a modest first step towards relating retention complexity to other established computational complexity measures, namely, dual distance and query complexity, when D is a uniform distribution over a linear space. We show that (i) the retention complexity of Walsh-Hadamard codes is constant and (ii) that of random Low Density Parity Check (LDPC) codes is, with high probability, linear in the code’s blocklength; intriguingly, for these two interesting families of linear codes, retention complexity is roughly equal to query complexity as locally testable codes.