Aumann's Agreement Theorem and Its Discontents

 

In quantum physics, there are mathematical equations, such as Schrödinger's equation, upon which nearly every educated person agrees.  Not only do they agree on the math, they also agree on the evidence.  What they disagree upon is how to interpret this math.  Thus we have De Broglie-Bohm "pilot wave" interpretation, the Copenhagen interpretation, the "Many Worlds" interpretation, QBism (quantum Bayesianism), and so on.  In a sense, we might say that for the varying schools of interpretation, they all agree on the math, but they disagree on how to "put it into words," to translate the truth of the mathematical equations into a natural language such as English.  But this is not merely a linguistic phenomenon, a mere dispute over semantics.  It might be better to say that there's a dispute over how to translate the truth of the math into human intuition.  Each of the different interpretations of quantum physics is nestled into a different set of intuitions about the universe, our place in it, and how to understand it.  

Of course, another option is simply to choose, resolutely, not to interpret the math.  (This is similar to, but, I would insist, not the same as, the Copenhagen interpretation.)  I imagine a scientist filling a chalkboard with hundreds of lines of complex mathematical formulae, and deriving one particular equation at the very end, which he proudly shows off to his colleagues.  They ask, "What does it mean?" and he replies, equally proudly, "I don't know."  For instance, one might reasonably reflect upon the fact that humans evolved for survival, mostly likely in the African Savannah, tens or hundreds of thousands of years ago, and there's no reason to think that, under such circumstances, an organism would evolve an intuition for quantum physics.  Thus, a person might conclude, it is mere hubris to suppose that anyone ever can or will develop a real intuitive understanding of how our universe works.  (I actually met a person like this once.  I happened to take a class with a graduate student in physics when I was an undergraduate, and he was very, very intelligent, especially when it came to math, and got good grades, but he once told me that deep down, he never felt that the theoretical physics he was working on had anything to do with reality - to him, it just seemed like a game he was playing with numbers.  I don't know whether this was a deep-seated philosophical conviction on his part, or if he just had a case of imposter syndrome.)

But for many of this, this is not enough.  We want to feel smug.  We want to feel confident.  We want to feel that we really understand physics.  And we have faith that such a thing is possible.  Is it merely our own sense of pride, our striving for honor, that motivates us to feel this way?  I can imagine one of the advocates of the interpretations of quantum physics (Copenhagen, many-worlds, etc., etc.) accosting the person who refuses to interpret it at all: Where's your spirit?  The spirit of inquiry, the spirit of intellectual adventure, the human spirit - how can you just abandon it like that?  How can you be indifferent as to the nature of reality?  If that's your attitude, why bother studying physics at all?  Or science at all?  Why bother using rationality?  

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There's a rough analogy between quantum physics and Aumann's Agreement Theorem.  It's not a perfect analogy, but I hope it will help me clarify my position on the theorem.   I don't dispute the mathematical argument at its center - the math seems solid to me.  But I dispute how to interpret this math, or, better put, I want to demonstrate that more than one interpretation of the math is possible, and I very strenuously insist that, quite often, people take the consequences of this argument far further than the math can support.  People take Aumann's theorem to mean that two people cannot agree to disagree, which is, of course, obviously, patently untrue: people agree to disagree all the time.  It has, in fact, happened to me twice today.  Sadly, this faulty interpretation of Aumann's theorem begins with Robert Aumann himself, who unfortunately and unhelpfully titled his 1976 paper "Agreeing to Disagree".  I suspect that Aumann gave it this title provocatively, and perhaps even a bit tongue-in-cheek.  But ever since, people have been saying that it has been "proven" that is impossible for two people to agree to disagree.  If you push someone on it, they may admit that yes, of course people sometimes agree to disagree, but really they shouldn't - which is interesting.  It shows that there's a hidden moral, normative dimension to what they themselves believe to be a purely logical and mathematical argument.

In reality, Aumann's theorem does not prove that people cannot agree to disagree, nor that they should not.  I would say that Aumann's theorem points out an interesting quirk of the specific conditions of an artificial mathematical model that game theory sometimes uses - conditions that Aumann himself partly helped establish, even in this very proof - but that it has little if any applicability to the real world.

What separates the artificial world of Aumann's theorem from the real world?  There are several possible objections:

Objection 1. We humans are not "rational" in the Bayesian sense (or, for that matter, in many other senses of the word).   And, by the way, the current generation of LLMs are not perfect Bayesian agents, either.  

Objection 1a. Robin Hanson and Tyler Cowen think that Aumann's theorem does not describe human debates very closely, because we tend not to believe that other people are fully rational.  That's also true, and might count as a secondary objection.  But they do not go far enough.  The problem is not merely that we believe that other people aren't fully "rational" (which would be enough to undermine Aumann's theorem).  The problem is that we humans really aren't "rational," in the Bayesian sense.  Even those people who aspire to be rational agents in the Bayesian sense are not.  (And that is a vanishingly small portion of humanity.)  And, by the way, I am not at all convinced that we should be "rational" agents in the Bayesian sense.  I think Bayesian statistical thinking has limited applicability, as I've argued before. 

Objection 2. We do not always share the same priors.  This objection has been discussed extensively, but I think it is a minor, secondary concern, compared with Objection 1.

Objection 3. We do not have "common knowledge" in the game theoretical sense of this term.

Objection 4. Even if rational agents, for Aumann-style reasons, are led to agree on matters of fact, they might still disagree in other ways - normative issues, such as aesthetics, morality, metaphysics, priorities of values, emotional stances, etc., etc., and there is no clear Aumann-style argument to reconcile these kinds of arguments.  For instance, if two perfect Bayesian agents were fans of two opposing football teams, after arguing over the facts and the evidence, they might eventually come to agree on whether a specific player on Team A completed a pass into an endzone.  But they would still likely disagree on whether it was a good thing or a bad thing that the pass was completed - if they finally agreed that the pass was complete, the Team A fan would most likely be happy, and the Team B fan sad.  Thus, two fully Bayesian agents may still agree to disagree on a wide variety of issues, political, moral, religious, and so on, even if they are compelled to agree on facts.

Objection 5. Even within the realm of material facts, there may be measurements that differ relative to an observer: for instance, if you are traveling at near the speed of light and measure the diameter of a planet, whereas I am at rest relative to the planet's surface, if our measurements do not match, we will, in a certain manner of speaking, have to agree to disagree.  Likewise, if two particles are emitted from the same process of radioactive decay, and, lightyears away, one of them goes past me and the other goes past you, and I measure my particle's angular momentum and you don't, we may have to agree to disagree on certain properties of your particle, in a certain sense.  We know about these kinds of observer effects - who's to say that there aren't other kinds of observer effects that we haven't even begun to guess?  Perhaps there are ways that two observers could get accurate, yet conflicting, results that cannot be so easily reconciled.  There's nothing in Aumann's agreement theorem to prove that that is impossible.

Objection 6. Although fundamentalist believers in Aumann's agreement theorem may be correct that eventually two fully rational Bayesian agents will have to agree on matters of fact, nonetheless the process by which these two opposing beliefs converge may take a very long time.  In the meantime, until they have reached convergence, these rational agents will have to agree to disagree.

And so on.  I'm sure that many, many more objections are possible, if we sat down and thought about it. 

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All of that having been said, there's another way of looking at all of this.  I have to admit that, when I articulate objections to Aumann's agreement theorem, there's a part of me that feels like "Phew!" and this leads me to question my own motives here.  Is this an example of motivated reasoning?  I must admit that if it turned out that what I'll call Aumannian Fundamentalists (people who, to my mind, interpret Aumann's argument as implying a lot more about the real world than I do) turned out to be right, I would be deeply disappointed.  Which leads me to delve deeper into my feelings.  Why do I feel this way?  

If the Aumannian Fundamentalists were correct, this would force me to reevaluate a lot of the principles that I hold dear.  After all, I'm a red-blooded American boy, and in a way you might say that one of the founding principles of the United States of America is that people can agree to disagree.  Legally, the First Amendment of our Constitution guarantees that the Federal Government cannot compel you or me to conform to some set societal standard.  We have freedom of religion in this country, as well as freedom of speech, freedom of the press, and so on.  More than that, we have what used to be called "freedom of conscience."  This is a concept that is not limited to the United States: it is officially enshrined in the European Convention on Human Rights, and the Universal Declaration of Human Rights.  And it is much older than America's founding: the concept of "freedom of conscience" emerged during the Protestant Reformation, after Michael Servetus was burned at the stake for heresy in 1553, when writers like Sebastian Costellio wrote pamphlets denouncing this execution.  It's not a major exaggeration to say that this principle has undergirded western civilization, and fostered a culture of diversity of ideas and opinions and convictions, expressed in literature and art and music and movies and fashion and all kinds of forms of expression and ways of life that makes all of our lives so much richer and more exciting and interesting than they might otherwise be.  And Aumann's agreement theorem, in its fundamentalist interpretation, seems to risk taking all of that away from us.  People like me get nervous when considering Aumann's theorem because assenting to it feels like surrendering your capacity for individual, independent, critical thought to some kind of Borg-like hive mind.  In short, for many people, myself included, Aumann's agreement theorem, at least at an unconscious level, represents a collectivist threat to bourgeois values.  We intuit the frightening prospect of a "group-think", something like what Emile Durkheim called "collective consciousness."

But I think this intuition, however sincere, is misguided.  Not so much because we don't have anything to fear from conformity - we do - but because both capitalism and communism are, each in their own way, collectivist and conformist.