# Aumann's agreement theorem

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In game theory, Aumann's agreement theorem is a theorem which demonstrates that rational agents with common knowledge of each other's beliefs cannot agree to disagree. It was first formulated in the 1976 paper titled "Agreeing to Disagree" by Robert Aumann, after whom the theorem is named.

Game theory is the study of mathematical models of strategic interaction among rational decision-makers. It has applications in all fields of social science, as well as in logic, systems science, and computer science. Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers. In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.

In economics, game theory, decision theory, and artificial intelligence, a rational agent is an agent that has clear preferences, models uncertainty via expected values of variables or functions of variables, and always chooses to perform the action with the optimal expected outcome for itself from among all feasible actions. A rational agent can be anything that makes decisions, typically a person, firm, machine, or software.

## Explanation

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 posterior probabilities, then their posteriors must be equal.  This theorem holds even if the people's individual posteriors are based on different observed information about the world. Simply knowing that another agent observed some information and came to their respective conclusion will force each to revise their beliefs, resulting eventually in total agreement on the correct posterior. Thus, two rational Bayesian agents with the same priors and who know each other's posteriors will have to agree.

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. The basic premise of rational choice theory is that aggregate social behavior results from the behavior of individual actors, each of whom is making their individual decisions. The theory also focuses on the determinants of the individual choices. Rational choice theory then assumes that an individual has preferences among the available choice alternatives that allow them to state which option they prefer. These preferences are assumed to be complete and transitive. The rational agent is assumed to take account of available information, probabilities of events, and potential costs and benefits in determining preferences, and to act consistently in choosing the self-determined best choice of action.

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.

"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(s). It generally occurs when all sides recognise that further conflict would be unnecessary, ineffective or otherwise undesirable. They may also remain on amicable terms while continuing to disagree about the unresolved issues.

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. Scott Joel Aaronson is an American theoretical computer scientist and David J. Bruton Jr. Centennial Professor of Computer Science at the University of Texas at Austin. His primary areas of research are quantum computing and computational complexity theory. Robin Dale 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 and markets, and he was involved in the creation of the Foresight Institute's Foresight Exchange and DARPA’s FutureMAP project. He invented market scoring rules like LMSR used by prediction markets such as Consensus Point, and has conducted research on signalling.

Studying the same issue from a different perspective, a research paper by Ziv Hellman considers what happens if priors are not common. The paper presents a way to measure how distant priors are from being common. If this distance is ε then, under common knowledge, disagreement on events is always bounded from above by ε. When ε goes to zero, Aumann's original agreement theorem is recapitulated.  In a 2013 paper, Joseph Halpern and Willemien Kets argued that "players can agree to disagree in the presence of ambiguity, even if there is a common prior, but that allowing for ambiguity is more restrictive than assuming heterogeneous priors." 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. He also holds a visiting position at Stony Brook University, and is one of the founding members of the Stony Brook Center for Game Theory. Joseph Yehuda Halpern is a professor of computer science at Cornell University. Most of his research is on reasoning about knowledge and uncertainty.

## Related Research Articles

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1. Aumann, Robert J. (1976). "Agreeing to Disagree" (PDF). The Annals of Statistics. 4 (6): 1236–1239. doi:10.1214/aos/1176343654. ISSN   0090-5364. JSTOR   2958591.
2. Aaronson, Scott (2005). The complexity of agreement (PDF). Proceedings of ACM STOC. pp. 634–643. doi:10.1145/1060590.1060686. ISBN   978-1-58113-960-0 . Retrieved 2010-08-09.
3. Hanson, Robin (2006). "Uncommon Priors Require Origin Disputes". Theory and Decision. 61 (4): 319–328. CiteSeerX  . doi:10.1007/s11238-006-9004-4.
4. Hellman, Ziv (2013). "Almost Common Priors". International Journal of Game Theory. 42 (2): 399–410. doi:10.1007/s00182-012-0347-5.
5. Halpern, Joseph; Willemien Kets (2013-10-28). "Ambiguous Language and Consensus" (PDF). Retrieved 2014-01-13.