 
Summary: The Complexity of Agreement
Scott Aaronson
Abstract
A celebrated 1976 theorem of Aumann asserts that honest, rational Bayesian agents
with common priors will never "agree to disagree": if their opinions about any topic
are common knowledge, then those opinions must be equal. Economists have writ
ten numerous papers examining the assumptions behind this theorem. 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 first show that, for two agents with a common prior to agree within about the
expectation of a [0, 1] variable with high probability over their prior, it suffices for them
to exchange order 1/2 bits. This bound is completely independent of the number of
bits n of relevant knowledge that the agents have. We then extend the bound to three
or more agents; and we give an example where the economists' "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 after exchanging order 1/2 messages,
and that can be simulated by agents with limited computational resources. By this we
