Aumann`s Agreement Theorem Proof

Aumann`s agreement theorem proof: understanding the mathematics

Aumann’s agreement theorem is a fundamental concept in game theory that was first introduced by Robert Aumann in 1976. The theorem states that, under certain conditions, rational players with common knowledge of each other`s beliefs will eventually converge to a common understanding or “agreement”. In other words, if two or more people start with different beliefs about a particular topic, they will eventually come to a consensus as they share information.

The proof of Aumann’s agreement theorem is a complex mathematical formula that involves probability theory, measure theory, and set theory. It can be difficult to understand for those who are not well-versed in mathematics, but we will try to break it down as simply as possible.

The proof starts by assuming that all players have common knowledge of the rules of the game being played and that they are rational, meaning they make decisions based on logical reasoning. It also assumes that players have a “belief state” or a probability distribution over all possible outcomes of the game.

The next step is to define the concept of “common knowledge”. Common knowledge means that all players know something, and they know that all other players know it, and they know that all other players know that they know it, and so on. In other words, it is not just a matter of each player having the same information, but also knowing that everyone else has the same information.

The key insight in the proof is the use of Bayesian updating. This means that each player updates their beliefs using Bayes’ theorem, which is a mathematical formula that calculates the probability of an event based on prior knowledge of related events. As players share information and update their beliefs, they converge to a common understanding.

The proof also shows that this convergence is almost sure, meaning that it will happen with probability 1. This means that while there is a small chance that players will not reach a consensus, it is highly unlikely.

To summarize, Aumann’s agreement theorem is a powerful concept that explains why rational actors can come to a shared understanding even if they have different starting points. The proof of the theorem is complex and requires a strong mathematical background, but the key insights are the assumptions of rationality, common knowledge, and Bayesian updating. Understanding these concepts can help us make sense of how people make decisions in complex situations and how knowledge is shared and created over time.