How to Share Knowledge by Gossiping

Given n agents each of which has a secret (a fact not known to anybody else), the classical version of the gossip problem is to achieve shared knowledge of all secrets in a minimal number of phone calls. There exist protocols achieving shared knowledge in 2(n−2) calls: when the protocol terminates everybody knows all the secrets. We generalize that problem and focus on higher-order shared knowledge: how many calls does it take to obtain that everybody knows that everybody knows all secrets? More generally, how many calls does it take to obtain shared knowledge of order k? This requires not only the communication of secrets, but also the communication of knowledge about secrets. We give a protocol that works in (k+1)(n−2) steps and prove that it is correct: it achieves shared knowledge of level k. The proof is presented in a dynamic epistemic logic that is based on the observability of propositional variables by agents.