Communication, consensus and order. Who wants to speak first?

Parikh and Krasucki [R. Parikh, P. Krasucki, Communication, consensus and knowledge, J. Econ. Theory 52 (1990) 178–189] show that, in a group of rational agents, communication of the value of a function f leads to a consensus on the value of f, provided some conditions on the communication protocol and the function f hold. In this article, we address the issue of the influence of the protocol on the outcome of the communication process, when agents value information positively. We show that, if it is common knowledge in a group of agents that some of them disagree on two protocols, then the consensus value of f must be the same for both protocols.