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TR04-061 | 30th June 2004 00:00
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#### The Complexity of Agreement

**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 written 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

epsilon about the expectation of a [0,1] variable with high probability

over their prior, it suffices for them to exchange order 1/epsilon^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 epsilon after exchanging order

1/epsilon^2 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/epsilon^6 but not

in n.