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You have a disagreement before you. How do you handle it?
Causes of fake disagreements:
Is the disagreement real? The trivial case is an apparent disagreement occuring over a noisy or low information channel. Internet chat is especially liable to fail this way because of the lack of tone, body language, and relative location cues. People can also disagree through the use of differing definitions with corresponding denotations and connotations. Fortunately, when recognized this cause of disagreement rarely produces problems; the topic at issue rarely is the definitions themselves. If there is a game theoretic reason the agents may also give the appearance of disagreement even though they might well agree in private. The agents could also disagree if they are victims of a Man-in-the-middle attack where someone is intercepting and altering the messages passed between the two parties. Finally, the agents could disagree simply because they are in different contexts. Is the sun yellow I ask? Yes, say you. No, say the aliens at Eta Carinae.
Causes of disagreements about predictions:
Assuming the disagreement is real what does that give us? Most commonly the disagreement is about the facts that predicate our actions. To handle these we must first consider our relationship to the other person and how they think (a la superrationality); observations made by others may not be given the same weight we would give those observations if we had made them ourselves. After considering this we must then merge their evidence with our own in a controlled way. With people this gets a bit tricky. Rarely do people give us information we can handle in a cleanly Bayesian way (a la Aumann's agreement theorem). Instead we must merge our explicit evidence sets along with vague abstracted probabilistic intuitions that are half speculation and half partially forgotten memories.