Thank you for the clarification: Aumann’s theorem does not assume that the people have the same information. They just know each other’s posteriors. After reading the original paper, I understand that the concensus comes about iteratively in the following way: they know each other’s conclusions (posteriors). If they have different conclusions, then they must infer that the other has different information, and they modify their posteriors based on this different, unknown information to some extent. They then recompare their posteriors. If they’re still different, they conclude that the other’s evidence must have been stronger than they estimated, and they recalculate. So without actually sharing the information, they deduce the net result of the information by mutually comparing the posteriors.
In Aumann’s original paper, the statement of the theorem doesn’t involve any assumption that the two parties have performed any sort of iterative procedure. In informal explanations of why the result makes sense, such iterative procedures are usually described. I think this illustrates the point that the innocuous-sounding description of what the two parties are supposed to know (“their posteriors are common knowledge”) conceals more than meets the eye: to get a situation where anything like it is true, you need to assume that they’ve been through some higher-quality information exchange procedure.
The proof looks at an ordering of posteriors p1, p2, etc, that result from subsequent levels of knowledge of knowledge of the other’s posteriors. However, these are shown to be equal, so in a sense all of the iterations happens in some way simultaneously. -- Actually, I looked at it again and I’m not so sure this is true. It’s how I understand it.
Thank you for the clarification: Aumann’s theorem does not assume that the people have the same information. They just know each other’s posteriors. After reading the original paper, I understand that the concensus comes about iteratively in the following way: they know each other’s conclusions (posteriors). If they have different conclusions, then they must infer that the other has different information, and they modify their posteriors based on this different, unknown information to some extent. They then recompare their posteriors. If they’re still different, they conclude that the other’s evidence must have been stronger than they estimated, and they recalculate. So without actually sharing the information, they deduce the net result of the information by mutually comparing the posteriors.
In Aumann’s original paper, the statement of the theorem doesn’t involve any assumption that the two parties have performed any sort of iterative procedure. In informal explanations of why the result makes sense, such iterative procedures are usually described. I think this illustrates the point that the innocuous-sounding description of what the two parties are supposed to know (“their posteriors are common knowledge”) conceals more than meets the eye: to get a situation where anything like it is true, you need to assume that they’ve been through some higher-quality information exchange procedure.
The proof looks at an ordering of posteriors p1, p2, etc, that result from subsequent levels of knowledge of knowledge of the other’s posteriors. However, these are shown to be equal, so in a sense all of the iterations happens in some way simultaneously. -- Actually, I looked at it again and I’m not so sure this is true. It’s how I understand it.