Aumann's agreement theorem
Encyclopedia
Aumann
's agreement theorem says that two people acting rationally
(in a certain precise sense) and with common knowledge
of each other's beliefs cannot agree to disagree
. More specifically, if two people are genuine Bayesian rationalists with common priors
, and if they each have common knowledge
of their individual posteriors
, then their posteriors must be equal.
A question arises whether such an agreement can be reached in a reasonable time and, from a mathematical perspective, whether this can be done efficiently. Scott Aaronson
has shown that this is indeed the case.
Of course, the assumption of common priors is a rather strong one and may not hold in practice. However, Robin Hanson
has presented an argument that Bayesians who agree about the processes that gave rise to their priors (e.g., genetic and environmental influences) should, if they adhere to a certain pre-rationality condition, have common priors.
Robert Aumann
Robert John Aumann is an Israeli-American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel...
's agreement theorem says that two people acting rationally
Rational choice theory
Rational choice theory, also known as choice theory or rational action theory, is a framework for understanding and often formally modeling social and economic behavior. It is the main theoretical paradigm in the currently-dominant school of microeconomics...
(in a certain precise sense) and with common knowledge
Common knowledge (logic)
Common knowledge is a special kind of knowledge for a group of agents. There is common knowledge of p in a group of agents G when all the agents in G know p, they all know that they know p, they all know that they all know that they know p, and so on ad infinitum.The concept was first introduced in...
of each other's beliefs cannot agree to disagree
Agree to disagree
The term "agree to disagree" or "agreeing to disagree" is a phrase in English referring to the resolution of a conflict whereby all parties tolerate but do not accept the opposing position. It generally occurs when all sides recognise that further conflict would be unnecessary, ineffective or...
. More specifically, if two people are genuine Bayesian rationalists with common priors
Prior probability
In Bayesian statistical inference, a prior probability distribution, often called simply the prior, of an uncertain quantity p is the probability distribution that would express one's uncertainty about p before the "data"...
, and if they each have common knowledge
Common knowledge (logic)
Common knowledge is a special kind of knowledge for a group of agents. There is common knowledge of p in a group of agents G when all the agents in G know p, they all know that they know p, they all know that they all know that they know p, and so on ad infinitum.The concept was first introduced in...
of their individual posteriors
Posterior probability
In Bayesian statistics, the posterior probability of a random event or an uncertain proposition is the conditional probability that is assigned after the relevant evidence is taken into account...
, then their posteriors must be equal.
A question arises whether such an agreement can be reached in a reasonable time and, from a mathematical perspective, whether this can be done efficiently. Scott Aaronson
Scott Aaronson
Scott Joel Aaronson is a theoretical computer scientist and faculty member in the Electrical Engineering and Computer Science department at the Massachusetts Institute of Technology.-Education:...
has shown that this is indeed the case.
Of course, the assumption of common priors is a rather strong one and may not hold in practice. However, Robin Hanson
Robin Hanson
Robin D. Hanson is an associate professor of economics at George Mason University and a research associate at the Future of Humanity Institute of Oxford University. He is known as an expert on idea futures, markets and was involved in the creation of the Foresight Exchange and DARPA's FutureMAP...
has presented an argument that Bayesians who agree about the processes that gave rise to their priors (e.g., genetic and environmental influences) should, if they adhere to a certain pre-rationality condition, have common priors.