Everyone Knows that Everyone Knows

A gossip protocol is a procedure for sharing secrets in a network. The basic action in a gossip protocol is a telephone call wherein the caller and the callee exchange all the secrets they know. An agent who knows all secrets is an expert. The usual termination condition is that all agents are experts. Instead, we explore some protocols wherein the termination condition is that all agents know that all agents are experts. We call such agents super experts. Additionally, we model that agents who already know that all agents are experts, do not make and do not answer calls. We also model that such protocols are common knowledge among the agents. We investigate conditions under which such gossip protocols terminate, both in the synchronous case, where there is a global clock, and in the asynchronous case, where there is not. We show that a protocol with missed calls can terminate faster than the same protocol without missed calls.