The epistemic gossip problem

In the gossip problem information (‘secrets’) must be shared among a certain number of agents using the minimum number of calls. We extend the gossip problem to arbitrary epistemic depths. For example, we may require not only that all agents know all secrets but also that all agents know that all agents know all secrets. We give optimal protocols for various versions of this epistemic gossip problem, depending on the graph of communication links, in the case of two-way communication, one-way communication and parallel communication. We show, among other things, that increasing epistemic depth from 1 (all agents know all secrets) to 2 (so that all agents know that all agents know all secrets) does not double the required number of calls but increases this number by $3∕2$ (for a complete graph). We also show that the following counter-intuitive result generalises to the epistemic gossip problem: asymptotically the same number of calls are required whether calls are two-way or one-way.