Video Title: An Evening With The Philosophical Muser #9: Scott
Aaronson And I Discuss Aumann's Agreement Theorem
Video Author: The Philosophical Muser (Always Seeking The Truth)


James: Good evening, everybody, and welcome to another edition of "An
Evening With The Philosophical Muser." I'm really delighted to be
joined today by Scott Aronson, a theoretical computer scientist and
Professor of Computer Science at the University of Texas in Austin.
Welcome, Scott.


Scott: Thanks, it's great to be here.


James: Yeah, I'm really stoked to have you with us. It's very timely
as well, because you know those Facebook memories that pop up from the
past showing you old posts? Well, one of them popped up this week and
it was a post I'd written in about 2013 I think it was. And it was the
people that I'd really like to have a conversation with who are
currently alive today. I'd made a list of people and it had E.O.
Wilson on it, Martin Scorsese, Paul McCartney, Deidre McCloskey the
economist, David Friedman, Patti Smith, Joni Mitchell, and people like
that. And yourself, Scott Aaronson.  You are...


Scott: I'm truly honored to be on that list. To be on such a list… I
don't deserve it, and I'm happy to be alive, uh, still, 10 years after
you wrote that.


James: Yeah, absolutely. It's a distinguished list and I'm very, very
glad you're on it. I think the only person who hasn't survived the
list is Cormac McCarthy, who died quite recently.


Scott: E.O. Wilson?


James: Oh, did he pass away? Oh gosh, have I missed him?


Scott: Yeah, a couple of years ago.


James: Bless. Okay, well he won't be coming on the show then. They're
really missed.

Okay, so a lot of people won't know exactly why it's so exciting for
me to have you on, but many, many will, and it's about a theorem
called Aumann's Agreement Theorem which I've written about before and
blogged about; and it's in two of my books. And I know that you,
Scott, were into mathematics before you got formally into computer
programming, and I guess that's probably where you came to learn about
the theorem, isn't it? And stumbled upon it.

So I'm really familiar with it and I've spoken about it a lot in my
work but I'd love you to explain it in your words to give the viewers
an understanding of what it is and hear it from your perspective and
why it's such an intriguing theorem.


Scott: Well, Aumann's theorem was proved in 1976 in a now-famous paper
called "Agreeing to Disagree." And, you know, I think it was about a
three or four-page paper and the statement and the proof of this
theorem within it was about one paragraph.

And you know, that is maybe the most recognizable thing for which
Aumann later won the Nobel prize in economics. So, it was surprising
to people. Basically, what it says is that under certain
reasonable-sounding assumptions, two rational people should never be
able to agree to disagree. Okay, so if they have a common, what we
call common, knowledge of each other's opinion which means they know
each other's opinion, they know that they know it, they know that they
know that they know it, and so forth, right? But, by the way, one of
the things that Aumann's paper is famous for is for formalizing the very
concept of common knowledge. He explains why that is different from
merely knowing each other's opinion. They have to not only know, but
know that they know, and know that they know that they know, and so
on, right? If they're in that state, we call it common knowledge. It
dictates that under those conditions, their opinions must be equal.
They must have the same opinion.

Now, there is a key assumption that underlies this, which is called
the Assumption of a Common Prior. The setting that we imagine is that
we have two people. In computer science, we like to call them Alice
and Bob. There is a probability distribution over all the possible
states of the world. To the extent that Alice and Bob might disagree
with each other, it is only because they have different information.

In a complete vacuum, if we imagined Alice and Bob as souls who
haven't been born yet, then in that original condition, they would
assign the same probability to any possible state of affairs. For
example, whether Rishi Sunak winning the election, or, uh, it raining
tomorrow or any event at all, okay? But then they might disagree with
each other because they have been born and lived different lives and
have accumulated different experiences, right? Which have given them
different knowledge.

So, maybe, Alice has checked the weather report and Bob has not, or
Bob knows people in politics and Alice doesn't. And so, because they
have different information, if you ask them to estimate something
like, 'With what probability will it rain tomorrow?' or 'With what
probability will this candidate win the election?' then they might
give you different answers. Alice is using the information that she
has, and Bob is using the information that he has.

But what Bayes' Theorem says is that if their opinion ever becomes
common knowledge - so Alice knows Bob's estimate and Bob knows that
Alice knows his estimate, and Alice knows that Bob knows that Alice
knows, and so on and likewise for Alice's estimate - Bob knows it,
Alice knows that Bob knows it, and so on, their estimates must
actually be the same.

That sort of restores the situation in this hypothetical original
condition before they ever accumulated different knowledge and
different experiences.


James: Yeah, absolutely. So, for any that don't know what common
priors are. Basically, what we're talking about are Bayesian priors,
aren't we? So, In probability and statistics, the Bayesian theorem
describes the probability of an event based on prior knowledge of
conditions that might be related to the event.

So you know, if the risk of X is known to increase with, say, Y, then
rather than taking statistics from the set as a whole, Bayesian
probability looks to assess X more accurately relative to changes in
Y. As you said with the election example.

I remember when I first stumbled upon this theorem, it was almost like
a light bulb moment for me. It's like when someone articulates
something that you've thought about and worked on for a long time.
Bringing it together so elegantly and succinctly was really liberating
for me.

What it did was it gave me a renewed sense of hope. If people can
agree in theory on a relatively short execution time, provided they
are both seeking the truth and without too many impinging factors, it
can help further and advance the conversation. In terms of how we can
make progress as people, it was great to see that theorem formalized.

You've proven a part of it in your paper, 'The Complexity of
Agreement' in 2005. That's more about the execution time, right? And
you're elaborating on it?


Scott: Yes, I can tell you a story. Back in 2003 or so, I was reading
things online and came across the webpage of an economist at George
Mason University, named Robin Hanson. He had an enormous number of
heterodox opinions, even crazy-sounding opinions, then and now. Maybe
some listeners have encountered him, but he had a paper with another
Economist named Tyler Cohen called "Are Our Disagreements Honest?" or
"Are Our Disagreements Rational?".

It explained Aumann's Theorem. That was the first time that I had ever
encountered it. My first reaction on seeing it was that I must have
misunderstood. This can't possibly be right that rational people
should never agree to disagree about anything. So then, my immediate
reaction was to start looking for loopholes in it. Indeed, many people
have done that over the past half-century.

One loophole might be this assumption of a common prior. We're
assuming not only that Alice and Bob are Bayesians, as you said, which
seems pretty reasonable. Being Bayesian could even be taken as one
definition of rationality. It just says that you're updating your
probabilities for different possible events — then you are updating
those probabilities. You know, as you gain new knowledge in the way
that math says you should. Right? That, you know, that's really, uh,
all that being a Bayesian means.

Okay, but, uh, then, you know, the set-up of Aumann's Theorem somehow
goes further than that in saying that: first of all, Alice and Bob,
you know, have probabilities for everything. They're never unwilling
to say probabilities. They always have a probability, you know. You
ask them, "Is there life on other planets?" "Is there a God?" They
have a probability for it, everything is a question of probabilities
for them. And crucially, until they gain knowledge, their
probabilities are the same.

And that, that is sort of the metaphysically weird part here, right?
And I thought, "Why should we assume that their probabilities — their
prior probabilities, as they're called — are the same?" But in their
paper, Robin Hansen and Tyler Cohen had an argument for that just
completely rocked my world. And it still does, I think.

Their argument was the following: if you were Alice, then it's not
inevitable that, as you're thinking about it, you would have been
Alice. Maybe you would have been born as Bob instead. So, the very
fact that you are you and not someone else is just another piece of
information. That you know, another piece of evidence that you've
gotten that you should then, uh, conditionalize on, you know, in the
Bayesian way.


James: Yeah, I think you said, didn't you, in your paper though, if
Alice and Bob exchanged everything they knew then, uh...


Scott: That's right, yeah. So I'm coming to that, yes. But you know,
so um, what they said is, you know, the very fact that you are the
person you are and have the DNA that you have, and so forth, these
could all just be treated as knowledge that you've obtained and that
you've conditionalized on, right?

And somehow, you know, the sort of indexical facts, you know, the fact
that you are you, should not change what your opinions are compared to
if you were someone else, except insofar as it has given you different
information, right? Except insofar as it has given you maybe
privileged information that other people don't have, that you can then
use as evidence. Okay, but then, Aumann's theorem comes into play and
says that if your opinions are common knowledge, then the fact you
have different evidence doesn't matter anymore. Then your opinion
should really be the same.

Okay, so then the next thought that I had was that, well, this must,
maybe this principle is true, but clearly, it doesn't describe real
life disagreements. In real life, when you see people disagree with
each other, they don't just exchange a few messages and then gain
common knowledge of each other's opinions, then come to an agreement.
We almost never see that happen. More often, we see two people get
even more polarized as a result of disagreeing with each other.

So, I thought the mismatch between this theorem and reality might have
something to do with complexity. Perhaps it relates to the length of a
conversation required for people to actually attain common knowledge
of each other's opinions and come to agreement. As he said, if two
people exchanged all the knowledge that they had, uploaded their
entire brain's worth of information into each other, they would
effectively become the same person. At that point, it is obvious
they'll have the same opinion because there is nothing anymore to
break the symmetry and give them a different opinion.

However, in real life, that would take an entire lifetime. If I were
to have all the knowledge that you have in order to have the same
opinion as you, I would basically have to have lived your life, right?
And you would have to have lived my life in order to share my
opinions. So, I thought that maybe, that's the loophole. Okay, so I
thought maybe I can prove a theorem, you know, showing that the length
of the conversation has to be very large.

Okay, so now we come to Aumann's Theorem. It didn't talk directly
about a conversation between Alice and Bob, it just sort of assumed
that Alice and Bob are in this condition of common knowledge, however
they got into it, and then it says that therefore their opinions must
be the same. But then later authors, like Ginakoplos and
Polemarchakis, in the 1980s, thought about how would Alice and Bob
actually come to that condition of common knowledge.

So, they imagine the conversation that would go as follows: that Alice
would basically just tell Bob, "I think that there's a 70% chance of
rain tomorrow." And then Bob would say, "Oh well, that's interesting.
Now that you say that, the fact that you've said that causes me to
update my opinion. And now I think that there's a 60% chance of rain
tomorrow."

And Alice says, "Well, now that you've said that, I actually think
there's a 65% chance of rain tomorrow." And then eventually, Bob says,
"Oh, well now I also think it's a 65% chance." Right, and then they
agree, okay. So, basically, what Genocopolis and Paula Marchakos noted
was that if there's only a finite number of possible states of the
world, say like a finite amount of knowledge that Alice and Bob could
have, then that conversation will terminate after a finite amount of
time. It will end with Alice and Bob having the same opinion and with
common knowledge that they have at the same time, as Aumann's theorem
would have predicted.

The fascinating thing about this conversation is that, at no point,
are they ever mentioning evidence. At no point are they ever giving
reasons why they think it will or will not rain. All they're doing is
stating their opinions, and their opinions given each other's
opinions. Yet, that is enough for them to reach that state of common
knowledge.

Coming back to Robin Hanson, he wrote articles in the early 2000s
where he said there is something else that's very strange about that
conversation. If we imagine a conversation that you had with someone
whose opinion disagreed, maybe what happens is that you start out in
two different places and then you get even more polarized as you
argue. But in the best case, you'd say you start out kind of far away
and then maybe you compromise a little bit. You might say, "you say
20, I say 80, and then maybe you have a point. Maybe I'll lower my
estimate to 70, right? And maybe you'll raise yours to 30. And maybe
in that way, we'll sort of gradually inch closer to agreement, okay?"

And what Robin pointed out is that an Aumannian conversation - you
know, this kind where we're Bayesians with common priors and we're
just stating each other's opinions - it doesn't look like that at all.
Okay, in fact, our opinions follow a random walk. Which means that my
own estimate of my future, I should never be able to predict the
direction in which I will disagree with you in the future. Or predict
in which direction my opinion will move.

So like, if I'm at 20 and you're at 80, let's say, right? I should not
be able to predict that as a result of talking to you, I will move
closer to you, I will go up to 30. It should be just as likely from my
standpoint that I'll go down, that talking to you will cause me to go
down to 10 or whatever.

And the reason for that is, if I knew in what direction my opinion was
going to shift, then I should have shifted it already. Okay, so this
is the reason for this random walk property, right? Our opinions, if
someone watched us, would just be jumping around, totally erratically.

Okay, now there's only one type of conversation that I've had in real
life that even approaches that and that's like when I'm arguing at a
blackboard with someone about a math problem, let's say, right? And,
you know, we might be arguing. I might think that a conjecture is true
and the other person might think it's false. Then, five minutes later,
we might be arguing just as vehemently but now the positions have
shifted. Now the positions have switched, right? And now I think it's
false and the other person thinks it's true. That basically never
happens with politics, with religion, with anything like that. But
okay, so okay.

Now, once we understand that the opinions follow a random walk, then
we can take a step further and this is what I did in my 2005 paper. We
can ask, how long would the conversation have to continue before the
two people probably, approximately agree with each other?

Now, the theorem that I proved in 2005 says that the number of
messages we have to exchange before we come to an approximate
agreement is completely independent of how much knowledge we have. It
doesn't matter how long we've lived for, how many books we've read,
how much experience we have. All that matters is, with what
probability do we want to agree with each other? As judged by our
shared prior and within what accuracy do we want to agree with each
other?

And basically, I proved a bound which says that, like if we want to
agree with a probability of, let's say, one minus delta, we'll say.
And if we want to agree, let's say, about some probability which is a
number between zero and one, and we want to agree to within an
accuracy of epsilon, then the number of messages that we have to
exchange - so the length of our conversation - goes like, 1 divided by
delta times epsilon squared. That's it.

So, for example, if we wanted to agree, at least 90 percent of the
time or at least, if we want to go into the conversation expecting
that, at least 90 percent of the time, we will agree to, within at
most 10 let's say, then we might have to exchange a certain number of
messages. The theorem would then say, at most a thousand. In practice,
it would probably be much, much fewer than that.

But it says that, at any rate, it can't be more. So, that showed that
the length of the conversation was not the loophole to Aumann's
theorem that I had thought it was. It's always good to win when math
doesn't just formalize what you thought was obvious, but actually
challenges it and changes your view of something. So, that was what I
wrote a paper about.


James: Okay, so a couple of questions then. Particularly on the random
walk. So, let's imagine a search space - we'll call that state one.
Okay, and we have two agents opening, so let's say that consists of N
bits. You're saying that, and yourself, it was Alice and Bob you said.
So the theorem requires a prior probability distribution over, say,
two possible states. And you're saying that it requires them to
calculate expectations over that distribution and update.



Scott: For now, we're assuming that Alice and Bob do not face any
limitations of computing power, right?  I also proved a theorem that
starts to deal with the case that they're limited in their
computational abilities. Okay, but in the theorem I told you before,
we were only interested in the length of their conversation, right?


James: And you think that the execution time is shorter than Aumann
predicted. Is that well?


Scott: I mean, Aumann just didn't make a prediction about it.  It's
just not the question that he was focused on, right?


James: Okay, so in terms of the random walk. Yes, and you said, Robin
Hanson came up with the idea that it's random.


Scott: I mean, it may have been known to people before him, but he
wrote a paper that made it very explicit.


James: My contention, having heard that, would be that it's probably
not going to follow an exact random walk. It'll probably be, there'll
be some bias to the random walk. For all intents and purposes, it may
have lots of random elements, but well, okay...


Scott: But the question is, you know, if you know that your future
updates to your opinion will be biased in a certain direction, then,
and you're a Bayesian, the question is always why haven't you updated
your opinion in that direction already?

Right, well, one thing that seems to help a lot for people to get
intuition about this is to consider the stock market. If it can be
reliably predicted that a certain stock will go up, then the market
should have already priced that in. The stock should have already gone
up.

This is why if you look at the prices of stocks over time, they
actually do seem to follow a trajectory that looks a lot like a random
walk. Not always, there are all sorts of complicating factors. But if
a broker tries to sell you a stock promising that it's going to go up,
the obvious retort would be, 'Well then, why are you selling it to
me?'

So, you could say that there's actually a version of Aumann's Theorem
about the prices of commodities, which suggests that no one should
ever be engaging in purely speculative trade, right? I mean, if people
have different liquidity needs or whatever; then, you know, these are
all reasons to trade. However, they should never engage in speculative
trade because the very fact that someone wants to sell you something
means that they must know something that you don't.


James: That signifies that it's less valuable than they're saying it
is.  Okay, so if you're both rational, I'll try to illustrate a random
walk, just in case anyone doesn't know what that is, and we'll try to
use that to work out if we think that it is following a random walk
process. So if we picture a random walk being a drunkard leaving a pub
and he has an equal probability of turning left or right, then what
we're saying is, over the course of his journey from the pub, we can't
predict where he might be in N number of steps. That would be a random
walk, right?


Scott: It's even saying more than that. It's saying, you know, if you
want to know the distribution over places that will be, you can
predict that it will follow a bell curve centered around the pub. It
will follow what we call a Gaussian distribution.


James: Absolutely. So let's say, for example, in the case of this
particular drunkard, there are constraints on his walk but he may not
know exactly what those constraints are. There are constraints acting
on him and let's try to analogize that with the psychology of the
agents involved. For example, let's say that Alison and Bob are
engaged in a discussion about politics. It's true that they don't
necessarily know why they believe everything they believe. It's also
true, as you say, that had there been indications prior to the
discussion, there's a good chance that they might well have believed
something else anyway. As you say, if they already thought it, why
haven't they already changed their mind?

I think this goes to the heart of where I was picking up on Aumann's
Agreement Theorem in my books. There are a lot of factors driving the
disagreements, and the way that, as you say, we often find humans
diverging when they should, with more rational persuasion, be
converging.

The fact that there are psychological biases, tribal affiliations, and
all sorts of unconscious things acting on it, isn't it more likely
that it's following a slightly biased, random walk?


Scott: Well, in real life, we know that there are countless ways that
we deviate from rationality. Now there's almost like an embarrassment
of riches in terms of asking, "Where could we be going wrong?", such
that we deviate from the prediction of Aumann's Theorem. The question
is more: which one is it? Of our many failings, which are the most
relevant ones, right? That cause us to deviate from this.

I mean now, it's important to say we have these intuitions, for
example, that in real life maybe people could come to agreement, could
come to understand each other's perspective. But for that to happen,
it's very important that they share their experiences, their evidence,
and explain where they're each coming from. And then, maybe there's a
chance.

What's so surprising about Aumann's Theorem and this
Geanakoplos-Polemarchakis conversation protocol is that it says none
of that is needed at all. All that is needed is for them to reach a
state of common knowledge of what each other's opinions are. So, that
is putting a point on how different this sort of idealized Bayesian
rationality is from whatever it is that we're actually doing.

But it's important to say though, there's another possible loophole
that we haven't touched on. If you are perfectly rational, you're a
Bayesian and so forth, but you're also perfectly honest. However, you
have reason to believe that I am either irrational or dishonest. So,
I'm not telling you my true opinions or my true opinions are just
completely off the wall, I can't even do the Bayesian calculations
correctly, right? Or, I have a completely absurd prior, right? I just
have, you know, my prior says that the probability is almost a hundred
percent that the world is being run by lizardmen from space. And
basically, nothing that I learn can budge me from that fixed opinion.

If I am irrational, or dishonest in any of those ways, well then, you
should devalue my opinion accordingly. And even if you think that
there's a chance that I am irrational or dishonest, or if it's not
common knowledge. So, I don't know, I don't know that I'm rational. I
think that you might be doing things differently because you suspect
that I am irrational.

Yeah, well, this is what happens. So, sort of in the background of
this utopian picture of people who all agree with each other merely
because they all know each other's opinions, is the idea that we have
a common prior. And as a special case, it includes common knowledge of
each other's honesty and rationality, that we're all executing the
protocol correctly. And that is not our world, to put it mildly.

You know, maybe only very, very occasionally - as with scientists
arguing at a blackboard, and even then not always right. But, only
very, very occasionally might we approach that ideal. You can have a
discussion, particularly on religious or political subjects, where you
don't even get off the starting base really because the levels of
distrust are so prominent that, however much you tried to employ
rational persuasion, you're probably never even going to get past the
starting block.


James: But, I wanted to talk about something else as well, in relation
to that, which is complexity. In my book, I mentioned the tribal
affiliations, the confirmation bias, and the fact that we often do not
have sufficient information on which two people could both come
together and converge on an opinion about something. One of the other
elements I talked about in that was the level of complexity. I wanted
to just ask you about that in relation to your paper.

Let me paint a scenario. If you have two mathematicians for example at
the blackboard, and yes, maths is probably one of the subjects
whereby, as long as the agents are honest and looking to converge upon
a theory together, then it's one of the least emotive subjects in
terms of getting two people to agree. So, if you both want to
formulate a theorem mathematically and let's say it's a relatively
simple equation, and let's say the mathematicians are only of college
level. We all agree on what the ground rules are, right? I mean, there
are a few other domains that are like that; maybe chess or certain
sports. You know, it is said that the sports page in the newspaper is
the one you have to be least concerned about being lied to or biased.
There are certain fields where there is an underlying ground truth. We
might not know it at any given time, but we all agree that it's there.
Once we see it, we all have to agree about it.

So let's take that the college level mathematicians are going to agree
on this quite straightforwardly and with fairly short execution time.
Now, let's take something really hot, like a big political dispute. In
our country, the UK, it's Labour and the Conservatives. In yours, it's
Republicans and Democrats. You could certainly find people who are
very hard-wired culturally and socially to those beliefs that are
unlikely to change their minds.

In terms of the level of complexity, one of the things I said in my
book is that it's probably going to be a barrier in some regards
because big subjects like religion, politics, and socio-cultural
elements that are attached to those are going to be a bit of a
minefield for the agents. To gather everything that's required and put
enough computational processing power into their thinking to capture a
lot of complex things, particularly around metaphysical elements to do
with religion and God and stuff. In your paper, yes, I think you said
that one of your conclusions - and you said it was a surprising
conclusion - was that complexity wasn't a major barrier to agreement.
So, is that something that you could just elaborate on in terms of why
you think it isn't? Are we talking about a different sort of
complexity here?


Scott: Yeah, sure. Let's unpack this because you've been using the
word 'complexity' right, but if by 'complexity' you just mean the
amount of knowledge that there is to learn in a given subject - the
amount of information, let's say the number of bits that Alice and Bob
may have learned, that may have caused them to have a different
opinion - or even the computation that they have to do to calculate
what their opinion should be given those bits, my results give strong
evidence that none of that is the limiting factor at all that ought to
prevent them from coming to an agreement.

Right, okay. For example, the length of the conversation that they
have to engage in is completely independent of how much knowledge they
have - that measure of the complexity of the topic. However, listening
to your question, you seem to have been using the word 'complexity'
possibly in a different sense, right? Which is, how difficult is it to
even cash out our disagreement into something empirical at all?

So, the whole framework of Aumann's Theorem and common knowledge and
so forth has presumed that the thing we are disagreeing about is some
underlying truth of the state of the world. And it's the kind of thing
where, as we get more evidence, we update our knowledge of the state
of the world. This seems clearly appropriate to questions like, 'Is it
going to rain tomorrow?' Right now, we're in a state of partial
information about that. We can see the clouds, we can check the
humidity measurements, and so forth. Tomorrow, we will have much more
information about that question.

Likewise, with the question of who is going to win the election or
even scientific questions like, 'What is the mass of the Higgs boson?'
or 'Is there intelligent life on other planets?' That's something
where, in the future, as we explore space more, we will get more
information about that. But a lot of the disagreements that people
have in real life—they're really about which tribe is better. Who
deserves to win in this legal conflict, in this war, in this political
conflict? Who is essentially on the side of good, and who is
essentially on the bad side?

The trouble with that kind of question, however, is that no matter
what the facts are, you can always reinterpret them in light of your
pre-existing beliefs. That sort of tells you who is basically
good—despite the occasional mistakes—and who is basically bad, despite
occasionally getting some factual question right by accident. Still,
that doesn't affect their fundamental badness.

It's not even clear how you cast those questions into the Bayesian
framework at all. I interact a lot with the rationality community,
which revolves around websites like Less Wrong and the writings of
this guy, Eliezer Yudkowsky. What they are constantly trying to do is
to get people to bet on their beliefs.

Instead of just saying something abstract like, 'I believe that
climate change is a huge issue', right? Well, okay, you know, what
specifically do you believe? With what probability do you think there
will be hurricanes in 10 years causing as much destructive power as we
have seen? And you know, give me your over and under right, think
about it more actuarially.

So, they're constantly doing that. The whole idea is that once you
force people to cash out their ideological beliefs into actual
predictions for what might happen in the future, then we have some
chance of getting people to agree. They might realize things are more
uncertain than they thought or it might force them to actually do
their research.

When they make a prediction, they might find out their prediction was
wrong and then they have to concede to the bet. That might force them
to update their belief. Robin Hanson, who I mentioned before as the
person who introduced me to Aumann's Theorem through his paper about
it, is perhaps best known for inventing the idea of prediction
markets.

He believes that for any sort of questions about the future, like who
will win the election or will this economic policy lead to lower
inflation or not, right? Will there, you know, will this foreign
policy, if pursued, lead to a war between these countries or not? He
thinks that these questions should all just be traded on a stock
market essentially, right? And then, you know, if like, there is a
stock that pays a dollar if these two countries go to war. And let's
say now it's trading at 60 cents, then we can learn from that, that
the best consensus right now from all the best experts trading on that
market is that there is a 60% chance that event will happen.

He thinks that prediction markets would be a way to force people to
bet on their beliefs. And in that way, the real world could better
approach this Bayesian or Omanian rationality. No, you know, I should
say there are now lots of prediction markets that are actually in use.
They tend to get in the news whenever there's an upcoming election.

But in the U.S, prediction markets are severely hampered by regulatory
barriers. They're considered to be gambling, and they're regulated as
if they were online gambling. And some people think well, if people
are allowed to trade stocks, bonds, and futures, right? And, you know,
in companies which sort of indirectly reflect these questions, our
beliefs about what's going to happen in the world, then why shouldn't
we also be allowed to trade in futures that are just directly about
what is going to happen, right? That might be a way of trying to
tackle this complexity problem. When people have these complicated
packages of ideological beliefs, trying to break them down into
individual sub-questions that can be factually resolved, and which you
could then literally bet on or have prediction markets on.


James: I'm sorry, because we know from anecdote and statistical,
empirical evidence - don't we? - that people, when some of the big
problems in society with wrong decision-making are when the decisions
that are made are divorced from the stake of the individual and the
context. Absolutely, you see how people behave politically in terms of
how they want the economy to be run and all the different incentives
behind that. You'll see that people converge upon the smartest
economics often when the stakes are most applicable to their own
situation.

I mean, there's even a, I don't know if you've heard of this, but
there's an economist called John Harsanyi, a Hungarian economist, who
came up with the amnesia theory. He won the Nobel Prize with John
Nash. Absolutely, and based on, well he was, and his rule's veil of
ignorance was based on that, wasn't it? The idea being that, I mean, I
think Will's theories of Justice came to grief a little bit. But how
Sani was the founder of that idea, that if we had, as you mentioned at
the beginning, this sort of pre-birth conception of what might be
right or wrong, and you divorced your own self-interest from that,
then you might be able to construct a moral framework that would be
consensually agreed upon. If you didn't know, if you'd forgotten where
you were in that society, you're absolutely right.

I'm used, when I'm using complexity, I'm using it in the sort of
whether it's empirically intractable or not. How much information is
converged upon the individual and how much sifting there is to do in
order to get to the heart of the matter. So yeah, I think from that
perspective, it's one of the barriers.

Another thing I wanted to mention, which you touched upon as well, is
that it obviously can't be applied to subjective things or things that
are a matter of taste. Aumann's Agreement Theorem is anchored in the
idea of truth being a strength and a factor that sits above the thing
that's being discussed, isn't it?


Scott: Well, I mean, if we're going to be debating like, 'Is chocolate
ice cream better than vanilla?' then at some point we need to define
how we will settle that bet. What are the terms? What if, after all
knowledge has been exchanged, I just still prefer chocolate and you
still prefer vanilla, right? Then you know, and then we would just
have to say well, the truth is that we had different preferences. So,
part of the presupposition here is that you know we are talking about
the same world. We are asking about the truth or falsehood of
something in a world that we can both observe.

And to me, to apply any of this to political disputes, there's an even
more basic presupposition that we care about what is the truth. Like
if someone is saying "Our tax cut is going to turbo charge the
economy," then it has to matter. If you cut taxes, do you actually see
the economy improve?

If there's no feedback mechanism, if you're just saying that as sort
of a slogan and you can just repeat a failed policy any number of
times and it doesn't affect the outcome, well then obviously you have
no hope of reaching agreement there.


James: And also, because I think there are a lot more objective
answers than we often care to think.

I think if you imagine analyzing the world through an omniscient mind,
let's say God gave us access to his mind, whereby when we talk about a
truth or an objective fact, that is that which would be true,
irrespective of human opinion about it. Now, I know, obviously, some
of the things in that set of facts will be conditioned by human
opinion and of course, human behavior. But there are a lot of things
that I think we classify as subjective or determined by preference.
These things are preferential, but they're also based on far more
objectivity than I think we often realize. In other words, if we did
have access to all the facts, then there would be, for example, a
right way to manage an economy. Or there would be a right way to
establish the rule of law that's better for the individual in
question. There probably would be a correct level of tax for whatever
incentive was in place. And then there would be a correct level or,
well, it's hard to say.


Scott: I think that some political disagreements really do come down
to differing views on trade-offs that different interest groups judge
differently.


James: Right, yeah, I agree.


Scott: Some people believe it's more important to uphold individual
responsibility, and others think it's crucial to have a social safety
net. And then, you know, that might never be resolved by any amount of
evidence. However, there are many other disagreements, as mentioned,
that might seem like they're about values, but in reality, they're
just different predictions.

We all want a society that's prospering, where people live long lives,
enjoying good healthcare, and they're happy, and they're not
committing suicide, and they're not dying of drug overdoses, and
they're not in prison.  Like, we all want that outcome, and we're just
disagreeing about which policies are going to lead to that outcome.

Yeah, and for those questions, it seems like yes, prediction markets,
you know, could help more generally. Forcing people to bet on their
beliefs, to sort of state their beliefs as empirical predictions and
test them, that could help enormously.


James: Yeah, absolutely. And this goes back to what we're saying about
having a stake in it because, you know, you notice with, um, say a lot
of young people in our country, and probably yours too, a lot of young
people tend to be quite left-wing economically because they don't have
much of a stake in the economy at that age.

And then when they start to have more responsibility, when they become
business owners or when they have a job, when they actually start
paying taxes, people tend to move slightly more economically right as
they get older.

And I think, you know, the other thing that's worth mentioning now in
terms of human behavior is if you took two sets of, um, two groups
together: the amount of things that humans agree on and the amount of
things that they disagree on, I think that's another factor which we
have to mention. Because actually, the set of things we agree on is
way larger than the set of things we disagree on. Yet it's like the
Freudian narcissism of small differences; it'The things that we
disagree on make the headlines. It's the things that we disagree on
that make up most of our discussions and conversations right now.


Scott: Now, of course, the left-winger might say, "Well, you know,
it's not surprising that people come to right-wing opinions if they
weren't born in this poor community or this oppressed or marginalized
community, right?" And it's sort of not epistemically valid to just
form your opinions based on the accident that you weren't born holding
the short end of the stick.

So yeah, I mean, I think that it's a really hard problem to try to see
everyone's perspective. There will be questions that are just
trade-offs of values that maybe cannot be resolved. But then we can
separate out questions that are just empirical predictions. Like, you
predict that communism will lead to a utopian society for everyone?
Well okay, that's a prediction that, in some sense, we've tested.


James: Absolutely, a social or political question can be tested, and
now we have an answer.


Scott: That answer was bought at the cost of hundreds of millions of
deaths, but now we have it.


James: It shows you how powerful these tribal affiliations are,
because, you know, we've probably had this too in America with, I
think of Bernie Sanders, but we've had this resurgence of socialism in
the UK in the last five years with a guy called Jeremy Corbyn, who
you've probably heard of. Yes, yeah, sure. And no one, no one who
remembers when socialism had any influence in this country last, which
was the 1970s, no one sensible and rational would want to go back to
that.  As you say, the world is a natural landscape for testing
empirical predictions, right?


Scott: These comparisons are hard because in the US, many people will
say, "Well, what about Sweden and Norway? All these countries have
socialized health care and they're doing much better than we are. They
have much better outcomes, right?"

So, I think once you submit yourself to the rigor of empirical
evidence, you have to agree that it could go either way. It could go
in a direction that you don't want. I'm someone who doesn't know if I
should be honored or hurt to be attacked both by left-wing and
right-wing people constantly on my blog. But, I am trying to get the
right answers. I don't always succeed, but I at least acknowledge that
there is no ideological camp that has a monopoly on the right answers.
I feel like understanding that is a prerequisite to making headway.


James: Yes, I think I'm similar, we can discuss this further in a minute.

It's quite intriguing but, I believe, truth is such an alien concept
to very many people. As Plato and Mark Twain articulated, there's a
probability of suffering if you don't partake in these advantageous
relationships and instead attempt to uncover the truth in a
non-partisan way. It makes people feel uncomfortable. Recognizing this
is important as it's apparent that you will never be perceived as a
reliable ally for their tribe, mainly because you might be persuaded
by an argument made by the opposition.

Einstein had a great term to describe the mind's ability to sift ideas
and stimulate creativity; he referred to it as "combinatory play".
This concept involves opening up one mental channel and exploring
another, thus bringing together diverse things to create new ideas and
thought processes based on these relationships. He elicited that the
results emerging from combinatory play are very difficult to induce
deliberately, artificially, or even precipitously. They must be
welcomed and arrived at and greeted when they come upon us. And I
think, yeah, you know, that's really interesting. Because one, I have
a friend who's a physicist and a mathematician. He and I are very
similar and I think you and I are similar in this way too. You and him
would be similar in the way that we're not that interested, it seems,
from what I've learned of you so far. We love the truth, and we value
the truth more than we love being part of a particular group. We're
more feline than canine in that regard.


Scott: I try to be that way, you know. I might fall short of it, right?


James: Yeah, and I've noticed with him, he's nothing like me
personality-wise. I'm very extroverted, I'm very sociable. He's very
introverted and he hides away in his room and he does lots of
equations and stuff and doesn't mind socializing so much. But what I
found between the two of us, and this is one of the reasons why my
wife and I have such a great relationship, is because she's very
unpartisan as well. She's really open to the truth. We have a great
relationship because of that.

And I find that with this friend of mine, we pretty much agree on
everything. Now obviously, sometimes, that's not healthy either. But
the reason why I think the only things we have as bones of contention,
I find, are matters of degree. It's like, you know, I'm slightly more
libertarian and market friendly than him. But he appreciates the
market. He's a very highly cautious person, so he has issues with
having too much freedom and things like that. Fundamentally, we agree
on almost everything because we're both looking for the truth. We have
great discussions, and I find that the people I disagree with most are
those who I believe are not fundamental truth-seekers. They're living
a life of convenience of belief and convenience of viewpoint, which
happens to tailor to what they want their life to be like.

My relationship with him is a prime example, and if you and I were
friends I'm sure we'd have a similar type of relationship. It
simulates, in a sense, the logical calculation steps whereby we are
both quite dispassionately involved in the truth. We are a bit like
computers in that sense, trying to converge on the truth.


Scott: I mean, there clearly are people who care more about the truth
than others. Now, the danger, or rather the paradox, emerges. As soon
as you form a group that claims to be the ones who care about the
truth, and those "other people" only care about what is convenient for
them to believe, you've reproduced the in-group and out-group dynamics
that you were trying to avoid.

I think you see this happen repeatedly in history. People will realize
that the establishment is systematically wrong about something or many
things and they step away. They'll say, "We're going to take a stand
and create a group that cares about the truth.  Build a new club of
just the truth seekers, the people who care about the truth, right?"
And then, once they've decided that we have a monopoly on
truth-seeking and these other people are not even trying to get it
right, they just reproduce the 'us versus them' dynamics. And they're
blind too, when actually the other side has gotten it right.


James: Yeah, you have to be careful of that. Absolutely, yeah, you do.


Scott: I mean, this is the central thing you have to be careful of.


James: But equally, if you're open to the truth, you love the truth.
Because actually, this is the other thing about truth: truth underpins
every other quality. Every other quality has propositions related to
it whereby the truthful proposition is the most valuable. That
includes love, justice - truth is the thing that underpins them all.

So actually, this hypothetical group, yes, you don't want to be so
hallowed that you end up in a sort of pantheon of truth seekers where
you dismiss the rest as non-truthy. I'm not saying that. But actually,
it's a club that in order to join, you don't mind being shown to be
wrong as well. Because the love of truth triumphs and it trumps the
so-called embarrassment and discomfort that some people have.

You know, I don't know what you think, but whenever people show me a
better way of thinking about the world, or they show me somewhere
where I was, uh, not thinking about it as well as I could be, I'm
actually exhilarated. Because if I debate with somebody and I come
away still believing what I believe, then as far as I'm concerned,
I've consolidated my position.

That happens a lot with religious discussion, um, where it's a lot
more to do with ethics and, um, value judgments. But equally, if
somebody says to me, "Actually James, what you've said, here's a
better way of looking at it," I think, "Great, I've got a new one!
I've learned something new, I've got a new way of interpreting the
world. I can add that to my storehouse of knowledge and I'm a better
person for it." And I think that's a healthy way to be because,
actually, we're all subservient to the power and quality of the truth,
whether we like to admit it or not.


Scott: Yeah, now you know, I agree and I worry about this enormously
in the context of social media, especially sites like Twitter. My
experience has consistently been that you can sometimes approach the
Aumannian ideal in a relatively small conversation among people who
have different perspectives but who basically trust each other to be
speaking honestly and are trying to understand each other.

You can do that even about the most controversial subjects. For
example, on my blog eight years ago, there was a discussion on
feminism and dating. Essentially, the question was - 'who has a harder
life, nerdy young men or nerdy young women?' These were absolutely
explosive questions and they got discussed in the comment section of
my blog. Men, women, people from all different perspectives were
really trying to understand each other. I feel like we were actually
making progress.

But then, once the conversation got on Twitter, once it entered the
wider social media world, it was met with swift condemnation. Anybody
who said something that differed from the accepted narrative was
ridiculed, embarrassed, and condemned as much as possible.


James: I was going to ask you about this actually. It feels like you
can have reasoned discussions, you can make progress on these really
difficult issues when you're among people who are acting in good
faith. But, social media puts a spanner in the works.


Scott: Social media transforms dialogue into performance for an
audience of hundreds of thousands of people, which completely changes
the dynamics.


James: Yeah, and if you're a public figure, then you've got so much at
stake, your reputation and your career. I mean, imagine someone like
Richard Dawkins. How is that man going to have a wholly balanced view
of whether God exists, given how much he has invested in it? But yeah,
no, if you are a public figure, like Dawkins is, let's say, it's very
hard.


Scott: It becomes very hard to even think out loud, right? To sort of
think in the Aumannian way where you're acknowledging when you made a
mistake about something.

Dawkins does a far better job than most of acknowledging when he makes
a mistake about something. Right. But you know, there are tens of
thousands of people who are going to leap on anything he's said. So,
that is a dynamic that if you are a truth seeker, you have to
constantly be on guard against.


James: Yeah, and I wonder what the effects of because you and I are
probably the right age where we can remember a pre-internet age and
the emergence of it to the point where you can now find out anything
you want to by a search engine. The young people who have been brought
up with that way of life, and no different, it's very natural to them.
I think I wrote a blog a while back, and it was to do with whether
we've reached peak stupid"—a term I didn't invent. It's always
dangerous to make a pronouncement that we've reached peak stupidity.
There are always new depths to plumb, I know.

It's a curious thing that in the last 15 years or so, never have we,
as a global society, had so much information poured onto us at any one
time. Remember, of course, we've evolved minds for the Savannah, and
we haven't evolved minds over 200,000 years that have equipped us to
deal with this kind of information overload. We can't avoid
outsourcing our thinking to other agents.

At the risk of sounding hubristic, I think it's less of a seduction
for people like ourselves because we know that there's not enough of a
payoff to join that kind of mentality. I'm sure we do it too. However,
for young people who are fighting their life battles and don't really
know what they believe about a lot of things, they're getting drawn
into this world where the Overton window has shifted so much. Now,
even what seems like a normal question—like "what gender am I?"—is a
question that young people, even at primary school age, are now asking
themselves. Whereas 25-30 years ago, it was a marginal consideration
for a few people.

I think the internet, yes. I wrote a blog post called "The Internet Is
Making Us Smarter But More Divided". So, it is making us more
knowledgeable, but we're not doing great things so much for the
information we have. Because there's so much of it, right? I mean,
there's a theory that you know, there have always been people who
believe that, uh, the government is controlled by Satanist child
molesters or, you know, whatever ridiculous thing, right? But those
people sort of never had a voice before, right? They were always sort
of confined to the margins and disenfranchised.


Scott: And the internet, you know, like, I am old enough to remember
the 90s when the big promise was that it was going to democratize
information and give everyone a voice. And, it turns out that it's
delivered on that promise, right? But, you know, what many people
wanted to say, once they had a voice, was absolutely horrifying,
right? And we're now just, in the multi-decade process of working
through the consequences of that.


James: Yeah, absolutely.  Um, and one thing you touched on a moment
ago, which I wanted to ask you about as well, is you talked about the
idea of a symposium of Truth Seekers. I remember in the paper, you
asked a question: rather than having Alice and Bob, what about if it's
Alice, Bob and another person? So, how does your theorem respond to
predictions about when other agents are involved? Does that increase
the probability or decrease it, do you think?


Scott: Ah, okay, good. So, now we've gone from the loftiest societal
questions back to a straightforward technical question, right?

But yes, I did study the generalization of the setting to, let's say,
N players, or N agents rather, rather than just two. And what I proved
was that it might take more messages exchanged, because now
information has to travel from any person to any other person.

And now, if I had let's say, two disconnected subcultures that never
talk to each other at all, well then, of course, you might never
equilibrate between them. You might never reach agreement between
them. So, the key question is, if you look at the graph of who was
talking to whom, is that graph connected? Meaning, can information
propagate from any person in the group to any other person in the
group?

As long as there's enough information propagation between any pair of
people in the group, then the agreement will happen in a number of
steps that scales with the number of people, and also with how
certainly and how accurately you need them to agree.


James: Do you have any predictions on the time differential then? Do
the execution steps increase or decrease with the next? I know there's
more information.


Scott: I mean, this is again in the setting where we're imagining that
all of the people have common priors. Yes, having knowledge of each
other's honesty and rationality. And, once you've granted those
assumptions, right, then it's really just the question of whether
they're all speaking to each other in a graph that is sufficiently
well-connected.

And then the time that they'll need to reach agreement now depends
both on how accurately and how certainly you need them to agree.
That's that one over delta times epsilon squared thing that I
mentioned before. But now also dependent on the length of the chains
of communication, like how long it takes a message to propagate from
any given person to any other.


James: So, if we imagine, say, 'let's say a scene of the crime'
scenario, where you have an amount of evidence and two detectives on
the scene. Let's imagine two scenarios where in scenario A, we have a
third detective joins the investigation and doesn't have any more
information but has increased contributions in terms of reason and
analysis. And then a second scenario where the third detective comes
into the investigation but has new information to impart. Then, I know
they're not logic machines like computers, but what are the likely
scenarios in terms of differences in execution time do you think in
those two scenarios? I know there's a lot of contingent factors but...


Scott: That's a very interesting question that I didn't prove a
theorem about. I would want to think about it more. Okay, but one
thing that I'll say right now is that the setup we're assuming here
doesn't really account for the contributions made by someone who just
has more powers of analysis of already known facts.

The crazy thing about Bayesians is, in the basic framework of Bayesian
reasoning, there are just the facts you know and the facts you don't
know. And there's the probabilities over the possible states of the
world. As you gain more facts, that causes you to update your
probabilities. But, it's sort of not part of this framework that you
have to think more in order to learn more of the facts. You're just
assumed to already know all of the logical consequences of what you
already believe.

Now, of course, as applied to real life, that's completely ludicrous.
If we all knew the logical consequences of everything that was known
to us then we would all be perfect chess players. We could all break
any encryption system. And there would be no need for mathematicians.
We would just instantly see the truth of Fermat's Last Theorem. We
wouldn't have to work hard for any of this. And so, there's actually a
lot of research over the past couple decades trying to extend the
Bayesian framework to, you know, what you could call boundedly
rational agents. Yes, agents who don't necessarily know all the
logical consequences of their beliefs and who have to invest a lot of
time and thought to figure them out.

But then it becomes much more complicated, right? That's a very
difficult undertaking. So in your setting where you have two
detectives and then a third detective comes up, if I modeled that in
this Bayesian setup, I would just say each one has some set of
beliefs. If the third one has no relevant information about the crime,
they still have some hunches, they still have some beliefs about it.
That's still going to affect the other two detectives when they have
this conversation.

But I can imagine that depending on details, it could go either way.


James: Yeah, and I think that's why we have to revert back to the
point earlier, which is that actually getting people to agree isn't
the fundamental here because what it's getting them to converge to
what is the truth. I mean, two left-wing extremists might agree on
communism, but it doesn't mean that it's a healthy agreement.

And you know, we should probably close soon because...


Scott: I don't know.  We just have to get people to agree on the
truth?  Well, people, I guess, have been saying that since Plato,
right? But then, there's sort of a secret presupposition that we just
have to get everyone to agree with me, to agree with the truth that I
know, right? I would rather say we have to get everyone to agree on a
truth-seeking epistemology.


James: Absolutely.

And that's a great way for me to ask my final question because the
context of my book is a series of letters that an older person has
written. You know that whole thing about what I wish I could say this
to my younger self?


Scott: Yes, yeah, absolutely.


James: The context of one of the references to Aumann's agreement
theorem in this book, is whereby an older person is writing to the
younger person saying, 'You know what a brilliant theorem this is.'
It's used as a vehicle for encouragement. When I discovered it in
2012, what excited me about it wasn't just that it's a beautiful and
elegant theorem. I was excited about sharing that with people because
I loved its beautiful simplicity. I also love that it's algorithmic
and its ability to influence and encourage people to, as you say, join
the party and see the adventure and the benefits of truth-seeking.

Because the thing about truth-seeking, which I often say to encourage
people, is that you are the only version of your particular self
that's ever existed. You are a unique person, and the only way to
fulfill your potential - and for that self, that's you, to make the
biggest impact you can possibly make - is by seeking the truth. The
greatest benefits lie in learning about your skills and talents, and
figuring out how you can leave an impression and live a full life.
This is best achieved by seeking the truth. By doing so, you discover
things about yourself that align with the blessings that you can
bestow on each other.

So my final question for you is, having known it for so long and given
how much pleasure it clearly gives us, and thank you so much for your
contributions to the theorem yourself, how can we use it as a force
for good in the world? Is it just a case of telling more people about
it, getting them excited about it and about the love of truth seeking
too?


Scott: I mean, to ask for a theorem to fix what is broken in the world
is a very high burden to place on any theorem.

Absolutely, I would say it's good for people to realize that they are
not in sole possession of the truth. They can learn from others, they
might be mistaken, they should update their beliefs, they should try
to predict what will happen. If they have a theory, it should pay
rent.  It should help them be less surprised by what's happening in
the world. And if they're constantly surprised, then maybe the problem
is with their theoretical framework and they should seek a new and
better framework. The way to do that is by discussion and criticism.
It's not name-calling, it's not ganging up, it's not sneering. It's
trying to respond to the best arguments that the other side made, not
the worst arguments.

Now, you don't need to know Aumann's theorem in order to be on board
with any of that.  And there are even people who do know Aumann's
theorem and who are not on board with this, who just fail at it for
very common human reasons.

But I think that what the theorem does is it sort of paints a picture
of what the ideal reasoners look like. We may not reach that ideal,
but we can at least try to approach it. We can ask, are our
institutions or are our social media platforms set up in such a way as
to push us closer to that ideal or further away from it? And if
they're pushing us further away from it, then maybe we need to change
them.


James: Yeah, absolutely, and that's an excellent point as well.
Because the point of having a journey and making progress is that you
have standards. If we aim high with those standards, we're much more
likely to get to a higher place ourselves. The standard mustn't be too
low for the adequacy of the task.

You mentioned the concept of looking at the other side. There's a
famous term, straw manning. I love the opposite of that, steel man.
The Economist, Brian Caplan, came up with the ideological Turing test,
which is basically a way to understand the opposition's view and argue
from their perspective. This can actually strengthen your position and
help you understand what you're arguing against.  If you give the
opposing viewpoint the benefit of the doubt and the best chance in
your own reasoning, you can't lose. Why would you want to do anything
other than that?


Scott: One of the things I've enjoyed most at scientific conferences
are debates. Let's say it's string theory pro and con, or the
interpretation of quantum mechanics— are there multiple universes or
just one?  But they sometimes have people debate the opposite of the
position that they're known for in their books and blog posts. Maybe
you have a coin flip to decide who has to argue which position, right?


James: Yeah, I like that.


Scott: I think you wouldn't want that all the time, but I think that
at least sometimes you ought to have that — steel manning the opposing
side. It's a very important epistemic exercise.


James: Yeah, absolutely. I think I read an example where, if you like,
there were four people on each side. Let's say they're Democrats and
Republicans, and each has to come forward and argue for the
opposition. You have to guess which one out of the four was actually
the Republican — say, arguing for Democrats. And I think that's right.


Scott: I mean, you don't want the debate club mentality where people
just say, 'All that matters is my arguing skill, and I can argue for
either side equally because it doesn't matter, right?  There's no
truth; there's just better or worse debaters.'  Like, no, that's not
the point at all. There is a truth that you're trying to reach, but in
order to reach that truth, it's an excellent exercise to be able to
steelman the opposing case.


James: Absolutely, and that underpins the beauty of Aumann's Theorem,
that there is a truth to converge on. And yeah, I think that's a great
place to end. So, hey Scott, thank you so much for joining me. I've
really enjoyed this discussion, and I hope the viewers will too. I'm
sure they'll find lots of interesting things about it to ponder. So
yeah, I really appreciate you being my guest on the show.


Scott: Thank you for having me.


James: Thank you so much. Okay, all right. Um, thank you. That was
Scott Aaronson.   I just want to say that below in the comments is my
link to my blog. Please do have a look and you'll see many different
arguments and interesting topics, hopefully.   So, thank you for
tuning in, and good night.