Probability Space & Aumann Agreement

I think there's another, more fundamental reason why Aumann agreement doesn't matter in practice. It requires each party to assume the other is completely rational and honest.

Acting as if the other party is rational is good for promoting calm and reasonable discussion. Seriously considering the possibility that the other party is rational is certainly valuable. But assuming that the other party is in fact totally rational is just silly. We know we're talking to other flawed human beings, and either or both of us might just be totally off base, even if we're hanging around on a rationality discussion board.

Sure all by itself this first paper doesn't seem very relevant for real disagreements, but there is a whole literature beyond this first paper, which weakens the assumptions required for similar results. Keep reading.

One question on your objections: how would you characterize the state of two human rationalist wannabes who have failed to reach agreement? Would you say that their disagreement is common knowledge, or instead are they uncertain if they have a disagreement?

ISTM that people usually find themselves rather certain that they are in disagreement and that this is common knowledge. Aumann's theorem seems to forbid this even if we assume that the calculations are intractable.

The rational way to characterize the situation, if in fact intractability is a practical objection, would be that each party says he is unsure of what his opinion should be, because the information is too complex for him to make a decision. If circumstances force him to adopt a belief to act on, maybe it is rational for the two to choose different actions, but they should admit that they do not really have good grounds to assume that their choice is better than the other person's. Hence they really are not certain that they are in disagreement, in accordance with the theorem. Again this is in striking contrast to actual human behavior even among wannabes.

Should people really adopt the "common knowledge" terminology? Surely that terminology is highly misleading and is responsible for many misunderstandings.

If people take common English words and give them an esoteric technical meaning that differs dramatically from a literal reading, then shouldn't they at least capitalise them?

I too found my understanding changed dramatically when I looked into Aumann's original paper. Basically, the result has a misleading billing - and those citing the result rarely seemed to bother explaining much about the actual result or its significance.

I also found myself wondering why people remained puzzled about the high observed levels of disagreement. It seems obvious to me that people are poor approximations of truth-seeking agents - and instead promote their own interests. If you understand that, then the existence of many real-world disagreements is explained: people disagree in order to manipulate the opinions and actions of others for their own benefit.

Sorry, I think I got a bit confused about the "meet" operation, mind clarifying?

is (I^J)(w) equal to the intersection of I(w) and J(w) (which seems to be the implied way it works based on the overall description here) or something else? (Since the definition of meet you gave involved unions rather than intersections, and some sort of merging operation)

Thanks.

EDIT: whoops. am stupid today. Meant to say intersection, not disjunction

I'm not sure I understand how $\Omega$ represents the set of world histories. If world histories were to live anywhere, they'd live in the sigma algebra — as collections of events, per the definition. If not, and every element of $\Omega$ truly is a world history, then how can $F$ represent "information about which events occur in which possible world-histories", when each $f \in F$ is made up of atoms from $\Omega$, that is, when every element in $F$ is a collection of world histories? One of these definitions ought to be recast, I believe. It might be most sensible to make $\Omega$ the set of all possible events across all possible histories, that way you can largely keep your other definitions as-is

Interesting that the problems with Aumann's theorem were pointed out ten years ago, but belief in it continues to be prevalent.

Diagrams would be wonderful, anyone up to drawing them?

I think that I understand this proof now. Does the following dialogue capture it?

AGENT 1: My observations establish that our world is in the world-set S. However, as far as I can tell, any world in S could be our world.

AGENT 2: My observations establish that our world is in the world-set T. However, as far as I can tell, any world in T could be our world.

TOGETHER: So now we both know that our world is in the world-set S ∩ T—though, as far as we can tell, any world in S ∩ T could be our world. Therefore, since we share the same priors, we both arrive at the same value when we compute P(E | S ∩ T), the probability that a given event E occurred in our world.

ETA: janos's comment indicates that I'm missing something, but I don't have the time this second to think it through. Sounds like the set that they ultimately condition on isn't S ∩ T but rather a subset of it.

ETA2: Well, I couldn't resist thinking about it, even though I couldn't spare the time :). The upshot is that I don't understand janos's comment, and I agree with Psy-Kosh. As stated, for example, in this paper:

The meet [of partitions π1 and π2] has as blocks [i.e., elements] all nonempty intersections of a block from π1 with a block from π2.

From this it follows that the element of I∧J containing w is precisely I(w) ∩ J(w). So, unless I'm missing something, my dialogue above completely captures the proof in the OP.

ETA3: It turns out that both possible ways of orienting the partial order relation are in common use. Everything that I've seen discussing the theory of set partitions puts refinements lower in the lattice. This was the convention that I was using above. But, as Vladimir Nesov points out, it's natural to use the opposite convention when talking about epistemic agents, and this is the usage in Wei Dai's post. The clash between these conventions was a large part of the cause of my confusion. At any rate, under the convention that Wei Dai is using, the element of I∧J containing w is not in general I(w) ∩ J(w).

Efforts to illuminate Aumann's disagreement result do seem rather rare - thanks for your efforts here.

It appears to me that reducing this to an equation is totally irrelevant, in that it obscures the premises of the argument, and an argument is only as good as the reliability of the premises. Moreover, the theorem appears faulty based on inductive logic, in that the premises can be true and the conclusion false. I'm really interested in why this thought process is wrong.

While I see your point, I wouldn't say that the agreement issue is over rated at all.

There are many disagreements that don't change at all over arbitrarily many iterations, which sure don't look right given AAT. Even if the beliefs don't converge exactly, I don't think its too much to ask for some motion towards convergence.

I think the more important parts are the parts that talk about predicting disagreements

The main problem I have always had with this is that the reference set is "actual world history" when in fact that is the exact thing that observers are trying to decipher.

We all realize that there is in fact an "actual world history" however if it was known then this wouldn't be an issue. Using it as a reference set then, seems spurious in all practicality.

The most obvious way to achieve it is for the two agents to simply tell each other I(w) and J(w), after which they share a new, common information partition.

I think that summation is a good way to interpret the problem I addressed in as practical a manner as is currently available; I would note however that most people arbitrarily weight observational inference, so there is a skewing of the data.

The sad part about the whole thing is that both or all observers exchanging information may be the same deviation away from w such that their combined probabilities of l(w)are further away from w than either individually.