The complexity of agreement

A celebrated 1976 theorem of Aumann asserts that Bayesian agents with common priors can never "agree to disagree": if their opinions about any topic are common knowledge, then those opinions must be equal. But two key questions went unaddressed: first, can the agents reach agreement after a conversation of reasonable length? Second, can the computations needed for that conversation be performed efficiently? This paper answers both questions in the affirmative, thereby strengthening Aumann's original conclusion.We show that for two agents with a common prior to agree within ε about the expectation of a [0,1] variable with high probability over their prior, it suffices for them to exchange O(1/ε2) bits. This bound is completely independent of the number of bits n of relevant knowledge that the agents have. We also extend the bound to three or more agents; and we give an example where the "standard protocol" (which consists of repeatedly announcing one's current expectation) nearly saturates the bound, while a new "attenuated protocol" does better. Finally, we give a protocol that would cause two Bayesians to agree within ε after exchanging O(1/ε2) messages, and that can be simulated by agents with limited computational resources. By this we mean that, after examining the agents' knowledge and a transcript of their conversation, no one would be able to distinguish the agents from perfect Bayesians. The time used by the simulation procedure is exponential in 1/ε6 but not in n.