On interactive knowledge with bounded communication

The effect of upper bounds on message delivery times in a computer network upon the dynamics of knowledge gain is investigated. Recent work has identified centipedes and brooms-causal structures that combine message chains with time bound information-as necessary conditions for knowledge gain and common knowledge gain, respectively. This paper shows that, under the full-information protocol, these structures are both necessary and sufficient for such epistemic gain. We then apply this analysis to gain insights into the relation between "everyone knows" and common knowledge. We prove a tight threshold on the depth k, beyond which E ^k _G (everyone in G knows nested to depth k) collapses into C _G (common knowledge), when this knowledge concerns the occurrence of a spontaneous event. The threshold depends on the size of the group G of agents, as well as the time that has elapsed since the event of interest occurred. The existence of such a threshold is not guaranteed for all protocols, which is demonstrated here by presenting a counterexample in which no such threshold exists.