Agreeing to Disagree?

Before Thanksgiving, I flippantly made the point to the class that "agreeing to disagree is not an equilibrium." This is actually a pretty classic result in game theory, and was originally "proven" in 1976 by Robert Aumann and has become known as "Aumann's agreement theorem." A shorter and slightly more intuitive (read: less topological mathematics) can be found here. 

Here is the basic argument: Two people start with the same "priors" (i.e. would believe the same thing if given the same information), but then receive different information (and therefore have different posterior beliefs). Assume that these beliefs are not represented by single points, but rather by probability distributions taking values over a range of possible "truths."[1] Also assume that the information sets of the agents are "common knowledge" (i.e. each agent knows the other’s opinion and knows that the other agent knows his opinion). If this is the case, then honest agents who exchange information (each had different information at the start) can never disagree.

There is some criticism of the result with respect to the "common knowledge" assumption, but there have been some extensions of the theorem that relax this assumption slightly and still retain the basic result. 
Another interesting point, by Robin Hanson and Tyler Cowen, is that most disagreements are likely to be dishonest because people are deceived into thinking that they are “better arguers” than others. This means that people are not in fact truth-seekers in arguments. They conclude with tips for being more truth-seeking in arguments. They include:

1.     Do not assume that you are more rational than others.

2.     Adopt a moderate opinion (of course, now we can disagree about what constitutes a “moderate” opinion!).

3.     Become aware of signs of self-deception. Signs include: self-interest, emotionality, informality, difficulty articulating arguments, stubbornness, and ignorance of cognitive biases.

I would also add that the greater the degree to which someone is arguing on the basis of conviction rather than fact, the higher the likelihood that they are self deceived. But I suppose that’s an argument for another day.

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[1] It is important to represent the posterior beliefs as probability distributions because the whole point of the theorem is that the agents are honest truth seekers (which implies that they do not believe themselves to know "truth"). More on honesty later.