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The Complexity of Agreement Scott Aaronson #

Summary: The Complexity of Agreement
Scott Aaronson #
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 e#ciently? This paper answers both questions in the a#rmative, 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 su#ces 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


Source: Aaronson, Scott - Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (MIT)


Collections: Physics; Computer Technologies and Information Sciences