So thanks to everybody for showing up today.
I must say I'm a little intimidated because I'm not used to speaking in the same series with Roger Penrose and Stephen Hawking.
So but I'll do my best. I'm I must say, John, you did a great job in that introduction.
It's a little embarrassing, in a way. I sometimes wonder if I have a sort of Don Quixote tendency.
But, you know, now I'm trying to do something useful with my life. So I'm going to tell you about that part.
And, you know, when you think about the world and what we do as scientists in particular,
well, any kind of science, I would argue we're contributing to the collective awareness.
You know, models that we as a culture and society use to understand our world, to think about the cause,
effect relationships, to understand what our options are and where they will lead us.
And you might think about three levels of awareness that we have.
One starts with understanding the external environment.
This goes all the way back, really, for millennia to people like Ptolemaic, Ptolemy and Archimedes and people like that.
But when Newton solidified the laws of gravitation and mechanics and we now have the ability to,
for example, predict the motions of planets quite well,
and this becomes a practical thing if you think about a problem like avoiding asteroid collisions,
which we know asteroids have hit the earth many times.
They've done enormous damage comparable to what we might expect from a nuclear war.
And so it's a rather serious challenge, actually. And I thought this was something we had looked till I looked into it.
I was going to use this as an example of something we really had nailed.
And then I started Googling around and I found the NASA's Asteroid Grand Challenge,
and they were quite proud of the fact that we've identified 95% of the asteroids that are greater than one kilometre in radius.
One kilometre radius asteroid would really clobber the earth.
So there's still 5% of them that we don't know are out there. And and if you get down to 300 metre asteroids, we've only identified 10%.
Those could be pretty nasty, too. So even with something like that, we're still actually trying to do things.
Interestingly, they're using crowdsourcing and relying on volunteers and so on, because the issue is if we see the asteroid, say,
ten years before it hits the earth, we can predict things pretty well, but we have to actually see it and we're not looking actively.
Another good example would be weather prediction there.
I think that's a success story. You know, we've been actually predicting the weather for more than a century, initially through subjective methods.
Around the turn of the 19th to 20th century, people started to be able to make weather maps and they began just looking at
these weather maps and seeing patterns subjectively and what was happening.
And professional weather forecasters emerged who would have a catalogue of past weather maps and they would look at the weather map for the day.
You'll remember there were computers back then. They go, Oh, this reminds me of 1925.
And they'd flip through their book and find an example of a weather map that matched sort of today's weather pattern.
Go well, the wind's 20 degrees off and it's five degrees more like your finger.
And then they would make a weather forecast that would be published in the newspaper and people would go on picnics and frequently get rained on.
Now, you know, starting in the fifties, in 1951, Norman was looking for peacetime applications of the digital computer,
and he teamed up with the best meteorologists of the day. In particular.
In the end, a guy named Jules Charney led the charge to they worked hard,
mathematically created a simple one dimensional partial differential equation,
which they simulated on the computer with the help of John Van Norman's wife, who actually did the programming to put it on the computer.
They made a forecast for a couple of weather patterns 12 hours ahead.
And the forecast. Each of them took them 24 hours. So there were well below real time at that point.
But, you know, those forecasts came out not too bad. And so they declared victory.
Billions of dollars got pumped into improving physical weather forecasts.
And I was actually around at the tipping point because in 1980 I was an intern at the
National Centre for Atmospheric Research with working with Edward Lorenz of Chaos Fame.
And and I asked him, Ed, you know, how how are you guys doing relative to this objective weather forecasters?
And it was a very you know, he's a very mild mannered guy, said, well.
We're not doing too bad and we're going to start beating them real soon.
And he was right. 1980 was a tipping point.
But no, that took 30 years to just break even with these guys with their libraries of weather maps.
Now, since 1980, weather forecasts are vastly better than they were in 1980, but your picnic is much less likely to get rained on if the weather.
Weather Forecasting Service says that it won't happen. No.
So weather prediction we're pretty good at.
Now, we're also starting to think about our effect on the environment and climate change is a great example of that.
When you think about it, you know, climate change modellers get a lot of flak, but I think they're doing a pretty good job.
They're telling us something that's rather subtle that's going to happen 50 years from now that will have very profound consequences on the earth,
and even giving us pretty quantitative information about what we have to do to keep it from happening.
So now note that climate change grew out of weather prediction.
They were able to build on a very solid foundation. But we are understanding increasingly more and more about our effect on the environment.
But I think the really hard challenge is to understand our collective effect on ourselves as our world becomes increasingly more artificial.
As you know, there are various estimates,
but many people estimate that we spend about 40% of the energy that's used to fix carbon on the Earth is ultimately going to support humans.
So we're placing a huge load on the planet. And, you know, if you just think about your day, you mostly live in a pretty artificial environment.
So somehow we need to start understanding better our collective effect on ourselves.
Which brings us back into social science and the challenge of understanding things in a quantitative way.
In some areas, like traffic forecasting, for example, we've gotten pretty good.
You know, if I'm these days, if I'm driving back from somewhere at rush hour,
I'll pull out my iPhone and I'll check to see what the traffic's doing and it can reroute me if traffic's messed up.
It does a pretty good job of predicting things like that. But now what about the bigger, bigger things like business cycles, for example?
Well, so here is an example of a plot of a business cycle.
And what you can see is various measures. And I'll come back to some in a minute.
Things like GDP and unemployment belts up and down by large factors as a function of time.
And you can see that we have some big events like this is the crisis of 2008, which the Dallas Fed estimated cost the US between six and $15 trillion.
That's quite a lot of money. That's like a half a year of U.S. GDP.
And for the world, the answer is certainly a lot more than that.
At least many months of GDP, probably a half a year, it suggests.
If that hadn't happened, we could have all gone on vacation for six months and been about even with where we are.
Somehow we didn't get to do that. But so this is an expensive thing.
It costs us as a society a huge amount to have something like that.
And it's not just the rich fat cats that suffer.
It hits the poor the most of all. You know, at this point, unemployment for Greeks under 25 is at 80%.
It's pretty horrendous. It's having a huge toll on Greek society.
You have some very close Greek friends.
They say it's depressing to be in Athens these days because people are just so dispirited about what's going on.
And so here are some examples of what what crises do to unemployment from some arbitrary starting point.
You can see it's not that uncommon for unemployment to swing down to as much as 20% lower than where it was before the crisis started.
So these things hurt. And I would argue I would say there's even a pretty good argument that the Brexit vote might have gone the other
way if we hadn't had the crisis of 2008 and that Donald Trump would not have been nominated for president.
Okay. Now, 2008 wasn't the only crisis.
I already showed you several on this plot here. Financial panics are endemic.
I. This really hit me. It's a long story, so I'm going to give you the short version.
But I happened to be in the Treasury building. I was lobbying for something involving the Office of Financial Research,
which I played a small role in helping start the remarkable grassroots movement of various scientists and
Wall Street practitioners who were shocked by the fact that when Lehman Brothers was about to fail in 2008,
we didn't have a clue who their counterparties were and what effect it would have on the financial system if if we didn't prop them up.
And so much of legislation got the text for legislation got developed by a committee of which I played a small part.
It's about 20 pages. It actually went verbatim into the Dodd-Frank bill and got enacted.
And it's an author's to collect all the data together and allow us to really think systematically about markets.
Now, I was in the Treasury building,
actually lobby the Deputy Secretary of the Treasury to to appoint somebody who had a strong science background to the post.
I was not successful, but that's another story.
But as I was leading, I started to walk out of the building and I noticed through these portraits of the past secretaries of the Treasury.
So I started, you know, looking at them and reading the little inscriptions below.
And it's kind of as I was going backwards in time, boring, boring, boring.
But then I got to the ones that were actually dead when the inscriptions had been written,
and then the inscriptions got much more truthful and much more interesting.
And I noticed that about every other one would say either did a great job of dealing with the
financial panic of the year or did a terrible job of dealing with a financial panic of X year.
And after I got out of there, I actually looked it up and I made a list of these, the financial panics.
As you can see, there's a lot of them. There were routine events back in the old days now.
So and I can't resist throwing the some of these pictures in just for the graphics.
Cartoonists have been good for a while. A particularly important panic was the panic of 1907 that was triggered by events in copper markets.
It looked like the whole economy was on the verge of collapse, and JPMorgan rounded up all the big players on Wall Street,
put them in his office and said and basically gave them orders about what to do to prop up the financial market.
This ended up being hugely controversial because people were understandably suspicious of a guy like Jp morgan.
Did he profit from it? He actually later felt quite, let's say, frustrated because he felt like he'd done the world a good turn.
But this led finally for the US to start a central bank, the Fed in 1913.
And by the way, since then you'll notice if you go back and look at my list panics up until the Fed was
formed came about every little less than eight years on average since the Fed was formed.
Well, we had a doozy, obviously, in 1929 and another big one up here, but they've been somehow bigger but less frequent.
So it seems that it has made a difference, but bigger but less frequent.
They're still happening. We still don't really understand when as they're coming on us, we don't understand they're coming.
And and it's still debated how we should deal with them once we're there.
Witness the big debates about austerity.
Now, given this, you might think just if you go back and do the numbers, if we're talking, say, $50 trillion that a crisis can cost us.
Well, how much research should we be doing to deal with that?
Well, just to take some numbers, the National Science Foundation for Economics and social science in the United States is $30 million a year,
not much the US Federal Reserve and it's like 12 branches.
It spends a total of about $500 million. Some fraction of that goes for research.
And I made a ballpark estimate of how much money gets paid out to academic economists throughout the world.
Maybe it's $1,000,000,000.
So, you know, that's not much when you think that the mistakes that are happening can cost us $50 trillion, you know, $50 trillion.
That's five times ten to the 13 versus ten to the nine.
So we're talking about the order of 10,000 or 50,000 difference.
So even if there's low odds of of making things more successful, it looks like investing more money in economics should be a really good investment.
I'll note just as a reference point, $1,000,000,000 is about what the US budget is for polar research.
Now I'm a great advocate of polar research. I don't want them to cut that one little bit.
But I it does seem like a dichotomy to me.
Now. No, but you ought to ask, of course, would this more funding be effective?
And there are several arguments that you might make to say that it's not maybe the economy is inherently unpredictable.
For example, maybe there's fundamental limits like market efficiency that say it can't happen.
Or maybe social science is a quixotic quest in and of itself.
Maybe people can't be described by law as people like George Soros, who ultimately pays my salary,
have suggested that that's true, though he's still paying my salary, so he must not think it's a complete lost cause.
And now maybe there's another effect, though, which is maybe the culture of economics doesn't scale with more funding.
And I'll note that they are unlike, say, meteorology,
where you can bet every budget cycle they're there in Congress saying we need more money, we need to spur computers,
we need better satellites, we need better measuring instruments,
we need more money because we really we're always pushing the limits of what big science can do.
Economics is not doing that. Now, I just want to note that these arguments are only valid if they're really strong,
because remember, there's that factor of 10,000 or 50,000 there.
So to to think about this question, let's step back now and let's think about the standard template for an economic model.
How do economic models work? So I'm going to teach you economics in about 3 minutes.
A challenging task, but it's only made a little easy and that economic models all follow more or less this template.
So what do economists do? They assume that agents have preferences and beliefs.
The preferences might come in the form of a utility function.
The beliefs might come in in the form of something called rational expectations.
So they they optimise that utility function perfectly.
Then they just assume there's a fixed point equilibrium where agents will maximise their preferences according to their beliefs.
And so then they compute what everybody's actions are.
And then from there you can compute prices and quantities and all the other things that are come out in the economic model.
And so this programs really hit its stride back in the fifties, maybe reached its peak about 1980.
And and then since then, the program in economics has been to modify these assumptions one at a time.
So they, for example, in the old days, they would assume utility and perfect rationality, that everybody would know everything.
They say, no, okay, maybe not everybody knows everything. Maybe there's asymmetric information, maybe there's institutional constraints.
So there are some things that people can't do. Maybe only some of the agents are rational and a few of the other ones are rational.
So they they play with these assumptions one at a time. But the thing that is invariant and they always, almost always do when I say they,
I'm referring to the sort of mainstream, because there are some rebels out there like me that are challenging this.
But but the sticks point assumption is something that's invariant in this.
So I'm going to examine that is the first thing I do in this lecture.
And before I go forward, I just want to comment that there are many kinds of equilibrium.
You know, I was educated as a physicist, so, you know, you immediately learn about forces balancing this trivial kind of equilibrium.
Then you learn about thermodynamic equilibrium. It's a bit more subtle.
Energy flows are balanced in a detailed way. And but in physics, we know what equilibrium is.
We develop a framework outside of equilibrium, and then we can test to see whether equilibrium is an appropriate assumption or not.
When it is appropriate, it's great because it really simplifies everything and you can do the calculations really well.
I've already referred to dynamic equilibrium. That is the notion of a fixed point.
So where things don't change, which underlies almost all math,
economic theory and strategic equilibrium, which is specifically what they're talking about,
where all the agents fully take each other into account and and do that in order to consider their reasoning and in fact, in economic models.
One of the difficulties. Well, it's a good thing and a bad thing.
Things are greatly simplified by assuming equilibrium off the bat, but you're left wondering whether the equilibrium assumption is valid.
The pinnacle of economic theory that this has led us to, or what are called dynamic,
stochastic general equilibrium models and what any Haldane has dubbed a rocking horse economy.
And the reason I say that is, if you think about the analogy to physical equilibrium,
imagine that you have a rocking horse and you bang that rocking horse with a hammer.
So you knock it away from its equilibrium. So now it's rocking and it starts to damp down again.
And then you hit it with a hammer again and it rocks again.
So that's a little bit the assumption.
The assumption is if it weren't for shocks, the economy is in equilibrium, but zuse throw shocks down from on high.
The shocks could be technological shocks because some technology has gotten better.
That could be behavioural shocks because people have changed their attitudes towards things like that.
They become more impatient for some reason, so they're discounting differently than they used to.
The shocks knock the world out of equilibrium, the world moves back towards equilibrium and the shocks hit again.
And these models spend a lot of effort in diagnosing what these shocks are and and and
trying to make some predictions now to see the kind of thing that they're talking about.
Here's another picture of the business cycle. In this case.
We're looking at the logarithm of gross national product and over about half the little more than half a century.
And we're looking at the deviations around the mean trend, which is upward.
So there's already an art in how you do that stuff.
And so so under this view, all these fluctuations are driven by these shocks hitting the world, which is always trying to go back to that mean trend.
No. I want to comment and I'll come back later to say some more about limits to prediction.
But when you make a model like this, you've inherently said that the noise is doing a lot of the work,
because you're saying that the main explanation for knocking for this whole movement in something like a real business cycle model,
I find that a bit of a strange name is that is that these shocks are hitting
and the shocks are something that economists kept on priori cannot predict.
It's always worth remembering that, by the way, whenever somebody talks about noise, noise is typically not fundamental.
I mean, the only fundamental source of noise that we know of is measurement in quantum mechanics.
Everything else just amounts to our ignorance. And not that noise models aren't useful, but they're inherently self-limiting.
So. So now why would you expect that the economy might not be in equilibrium?
Well, first of all, beliefs and outcomes may not be consistent.
And when you dig in to equilibrium theories,
you see that consistency between beliefs and outcomes is a is a deeply important part of what is going on in equilibrium.
And as I've already suggested, the strategy dynamics may not settle into a fixed point.
So to illustrate this, I'm going to begin by thinking about the strategic context where people are reasoning,
and I'm going to use something called a Keynes beauty contest. Now in Keynes's Day, papers would publish.
This is an example of a page of a newspaper from the time they would publish pictures of women.
I admit that's a very sexist thing to do, that I don't think they ever did this with men.
And they would they would then ask people to rate how attractive they thought these women were and people would then rate them.
And then if you voted for the choice that was most popular, your name could be drawn out of a hat and you might win a vacation or something like that.
And but Keynes pointed out that what matters isn't there's no inherent measure of which women are the most beautiful.
The the criterion is, is to guess what other people will think is most beautiful.
And he pointed out that the stock market is very much that kind of thing.
It's not necessarily what makes prices go up isn't what company is the best company, it's what company the people think is the best company.
So that was the context of his doing this. But I'll now give you an example of a different kind.
In that spirit.
You're asked to name a number between zero and 100, and the winner is going to be the one whose guess is two thirds of the average guess.
So I want you to think about this for a few seconds to think what would your guess be?
Now, I've done this in the past by polling people and I've discovered that it's a bit of a dangerous
thing to do because it can be a little embarrassing once I explain what's happening.
So I don't want to embarrass anybody. Also, I can't make my snotty remarks because I really wouldn't want to offend anybody.
So just think about it. Formulate your guess. I want everybody to pick a number, you know, how are you going to do it?
All right. Now, let me show you what about 30,000 Danes did.
Now, this was published in the newspaper, so they had more chance to think about it than you did.
And and you can see that, you know, the guess is there are people who actually guessed 100.
Now, let's go back and think about you wonder a little bit, you know, okay.
I mean, this kind of might explain why people are actually going to vote for Donald Trump.
Okay. Let's suppose all of you but one, just 100.
Okay. If everybody goes to 100, then what would the winning guess be?
What, 66? Right, two thirds of 100.
So obviously, anything greater than 100 is a loser.
So all those guys that there's a lot of quite a few of them,
either they didn't understand the game or they really weren't thinking, now let's suppose then you go, okay, I'll get 66.
But you go, well, 66 is pretty high because other people aren't total dummies either.
And why would they all get to 100? Well, okay, so maybe they would guess 50.
And now let's take two thirds of that. So that would be about a third.
So you see there's actually a peak there, but then you might go, well, okay,
suppose people get stopped then I'm going to suppose I'm going to reason one more all two time.
That's the typical gas. And so in fact, that's the winner around around just above 20.
So but but note, if you were an economist and you were trained to think about rational expectations, you would have iterated this process, too,
and you would have guessed the Nash equilibrium, which is zero, because that's where you end up if you assume that everybody is really smart.
If we were all smart and we all knew that everybody else was. Mart, we would just go to zero and it would be a fixed point equilibrium.
Everything would be peachy. It would be easy. So my snotty remark is, okay, these people are obviously idiots and these guys are idiots for months.
So, all right. My snotty remark. Okay.
So when is this assumption justified? So I'm going to show you an example of some work we did to test this in the context of game theory.
The players can make one of several possible moves. So a game is something where players make moves.
So it's like the games you're familiar with. At each turn, players receive payments according to the set of moves of all players.
In other words, if on that turn, Martin makes a move and I make a move and a list makes a move,
and we make move number, you know, listen, makes move number one. I make number two.
Martin makes move number three.
We then look up in the payoff matrix what are payment is and each of us get some payoff that may be quite specific to each of us.
And so that payoff matrix defines the game. Then we're going to play that game repeatedly.
So we're going to play it again and again so we have a chance to learn from it.
So and by the way, I think if we played that Keane's beauty contest game repeatedly, we would have seen a progression towards zero.
And we're good, but we're going to make the players learn their strategies, unlike the.
Well, I guess I had excuse me this on my next slide.
There is there is a standard theory by Nash which says that there's always a fixed point in a game like that.
There's always an equilibrium like the zero equilibrium in the beauty contest.
As long as players can choose moves and introduce an element of randomness to keep the other players guessing,
as long as they can do that, there always exists to fix point.
But Nash didn't prove it stable so we don't know that that's behaviourally what people are actually going to do.
And the experiment I just showed you suggest that that's not what people typically do.
But what do they do now? Oh, okay. Let me give you a little bit of intuition.
So the first thought I have on this is simple games should be easier to learn than difficult games.
So I'm reminded of when I was about nine or ten on the playground,
we started playing this game called Tic TAC tOe, which I think in Britain you called nuts and crosses.
So you make a little three by three grid, you put four zero, you try and get a, you know, a diagonal, horizontal or vertical line of connected things.
Whoever does that first wins.
So at some point, I figured out that I could be zero and I could always block the other guy from doing it, so I could always at least get a draw.
And so for about a day, maybe I felt like a real hero because I was just cleaning up.
And then all my friends figured it out and the game became completely boring.
We quit playing it. So that's a good example of tic tac toe is a simple game.
Children learn the Nash equilibrium. Now, what about more complicated games?
Say, take chess. So now I'm going to tell you a little joke.
How did two economists play chess? Well, you know, so there's a random element to chess.
You begin by choosing white or black, and then they choose white or black,
and then they argue about whether white or black wins under the Nash equilibrium of the game.
Now, why is the joke a little silly? Well, because we don't know what the Nash equilibrium of the game is.
It is a deterministic game once you've chosen white or black. No, I have to tell the corollary, the joke, because I'm a physicist,
so it's rude to tell jokes about economists without telling a joke on your own tribe.
So how did two physicists play chess? Well, they get quite drunk and they start moving the pieces around randomly.
And they argue about whether the game is more or less interesting under scaling of the size of the board and the number of pieces.
So you'll see from what I'm about to present that I'm a physicist now.
So what we did is we said we wanted to understand generic behaviour and I did want to make sure
that my comment was registered that there's a we're not the first to think about this issue.
Lots of people have worried about this from the beginning.
In fact, von Neumann himself was quite critical about game theory for its emphasis on Nash equilibrium, and in particular at Oxford.
We have a couple of people who have been pioneers in this Paten, Yang and Vince Crawford.
So other people have thought about this, but we took a different approach and we really wanted to understand what's generically true.
So we constructed games at random.
So remember, there's this payoff matrix that determines how much you get paid under the collection of moves that get made.
We put random numbers into that payoff matrix.
Once we put those random numbers in, we keep the payoff matrix fixed.
And then we have the players. We give the players a learning algorithm and we have them play.
Now our goal is a bit like we want to characterise the behaviour a priori as much as we can.
In other words,
we want to know are there some rules of thumb that might tell you in advance whether a game's likely to converge to an equilibrium or not?
An analogy can be made to the Reynolds number and fluid turbulence,
which is a number the ratio of the shear stress to viscosity nine dimensional number that numbers around one.
Then I pretty much know the fluids laminar that is the dynamics sits at a fixed point.
On the other hand, if that number is 10,000,
then I have a quite I'm virtually certain that the fluid flow is going to be highly turbulent and chaotic.
So we're looking for something a little like that now.
So we actually did something. We didn't just populate these payoff matrices completely at random.
We choose payoffs.
So they were normally distributed and we constrain the playoffs and we began by studying two players and then we'll go to more players later.
So when we have two players, we constrain them so that we have a parameter gamma that involves the relative payoff matrices of the two players.
If game is one, the payoff matrices of the players are identical.
So if I'm playing Martin, his payoff matrix and mine are exactly the same.
If I win, he wins. If I'm a zero, if I win, he might well, he might lose completely uncorrelated.
And if karma's minus one, we have a zero sum game. If I win, he loses.
So that's an important parameter we discovered.
We also chose a learning algorithm in which players learn their strategies using an algorithm called experience weighted attraction.
We chose this. Under this algorithm, players choose their strategies probabilistically.
So if I was saying rock, paper, scissors, I would have a certain probability of playing rock,
certain probability of paper, certain probability of playing scissors on any given turn.
Those probabilities can change dynamically at time and time,
but on every turn I'm going to roll some dice that are going to be balanced according to those probabilities,
to determine what how likely I'm going to select something.
And we chose this because some economists had done experiments and shown that it's
a pretty good model for how people actually behave in games in a laboratory.
It was also analytically tractable for us to study it.
Now, this this learning algorithm has a parameter alpha that determines memory because the
probability of mimicking a given move is dependent on how well it's been working in the past.
If Alpha Zero, the past can be infinitely long as I just accumulate more and more data.
And something that happened years ago is equivalent to something that that happened the last turn.
If Alpha is greater than zero, then I pay more attention to things that happened the last turn than things that happened in the past.
Now, what did we find? And I should say one more comment that's important.
Being physicists, we chose the limit where there's lots of possible moves.
We looked at really complicated games.
Now we did that because that allowed us to go into the thermodynamic limit where we can actually compute stuff using cool analytic techniques.
So that my collaborator, Tobi Scala, is an expert in.
So what did we find? What we found is what happens really depends on the parameters.
This parameter alpha that determines memory of the strategy is important and the parameter.
Gamma is also important. And there's a big region here where you just converge to a unique fixed point that's close to a Nash equilibrium.
So that's very much like what the economists are typically assuming.
This is the zone where traditional economics is okay.
But then we found this chaotic zone over here that I'll tell you more about in a minute.
And then there was so this happens, by the way, when Gamma is less than zero.
So the games are competitive in the sense that if I win, Martin tends to lose.
And so we saw these chaotic cases here where there was no convergence to a fixed point.
When Gamma is greater than zero, we saw multiple fixed points, which in the large state spaces we were looking at where each player had 50 moves.
There could be hundreds of these fix points. And which one you converge to depends on exactly where you start.
So there's multiple basins of attraction in the dynamical system that corresponds to the learning dynamics.
Now, what's going on in these chaotic cases when the cases so we saw lots of things.
So we saw, for example, now here I'm going to pick three out of the 50 strategies or I should call them three out of pure strategies.
This is another term or three out of the 50 possible moves.
So this is the probability of making that move as a function of time.
And what you can see in this case that corresponds to these parameters, there's just an oscillation.
So the players are just going around around in a loop. History is repeating itself.
And in the state space where you take a couple of the moves and you just project things out of that axis,
you see an attractor whose dimension is essentially one, and they just go around on this loop.
In other cases, you see things like this where you see what we're able to do to tell for using various techniques is a low dimensional,
strange attractor. And in other cases, the dimension of that attractor gets bigger and bigger.
It's still a strange attractor, but the dimension gets bigger and the behaviour gets more and more random.
Here it starts to just look like a ball of yarn. It's to all intents and purposes, a random process.
And though there are some interesting things going on here, if you look at one of the strategies here,
the strategy is sometimes played with the probability of approaching one.
And notice this is a logarithmic scale. And another points, the probability of the strategy being used can be ten to the minus,
20 virtually never used, and then it can come back and become very popular again.
So there's an automatic generation of fads of different strategies being used in this world, which has a certain realism.
Now you can ask what happens when you go to more players?
What changes if instead of two player games you have multiplayer games?
Now let me first make the remark that we were able to analytically calculate this blue line to very high approximation.
So we were quite proud of that. And it takes 20 pages of hard math to do that, but it matched pretty well with it with what we did in the simulations.
Now, what about when we have more than two players?
Well, when you have more than two players, by the time you get up to, say,
ten players, it becomes really hard to simulate this for more than two moves.
So we can't even go into that high move zone again.
So here we really had to rely on our theory. We tested our theory as best we could.
We pushed the theory out there, and we showed that the area of the chaotic zone increases is the square root of the number of players.
And so, in fact, what happens is, as you add more players, this boundary just translates itself over like this.
And chaos takes more and more of the parameter space, as does this complicated region.
And in the limit where you have lots of players, it's you never see the classic Nash fixed equilibrium.
Now, if you're an economist,
this should make you a little nervous because this is suggesting that when there's a lot of players in a complicated game,
there may be it may be the generic case.
I would argue you have to justify that you're going to use a model that just has a fixed point, because what it means,
if you assume a fixed point in something complicated, like I would say the stock market is a good example.
You're you're approximating this big ball of yarn with this complicated dynamics of the strategies by a point sitting in the middle of this.
Now it's a point with some noise. And as I said, this is for many practical purposes, a random process.
But I think there can be key differences, like as I actually showed in one of my earlier papers,
you can sometimes take advantage of this when there's chaos to make a prediction about the endogenous dynamics.
And and I think it can be insightful as to the type of the model.
And by the way, as a side effect we saw here, you see the change in the total payoff as a function of time.
You see these periods where things are pretty quiet.
You see periods where things are quite noisy.
This is very reminiscent of what you see if you look at returns in the stock market, something called clustered volatility.
This was a surprise, not something we expected to see.
Now I have to add some caveats and I'm worried about running out of time here.
But what happens with other learning algorithms? Well,
we've explored some like we've explored some versions of the kind of level K learning that you you saw people using in that beauty contest example.
We've, you know, you can you can ask whether the ensemble of games we're choosing is representative.
You can ask about games with just a few actions, like two by two games.
Marco Pagano, my graduate student here in math, has been looking at that,
has actually made a a classification of what happens under all two by two games and showed that while chaos is now much less likely,
there is still a zone where chaos happens even in two by two games.
No. So how do you cope with this kind of nonlinear non-equilibrium behaviour?
If you throw out those six point models, what do you do instead?
So I would argue that you have to you have to go to simulation,
because if you want to make any kind of realistic models, you end up with nonlinear equations.
And nonlinear equations are typically not solvable via any other means than simulation.
And just as a note, computation has revolutionised the physical and natural sciences.
I think it accounts for most of the progress we've made over the last 50 or 60 years.
And this was really driven home to me, this concept, the importance of nonlinear dynamics or nonlinear equations in general.
When when I was a graduate student, Stan Hull came to the physics department at UC Santa Cruz,
where I was studying and gave a lecture in which he began by apologising for using the term nonlinear.
And he said, I'm apologising because calling an equation nonlinear is calling is like calling an animal a non elephant.
Because most equations are nonlinear and we've been blinded to that fact by the by.
Because up until now Ulam was one of the pi.
Up until recently, all we could solve were linear equations.
So we were like the mathematician sitting under the lamppost looking for our keys,
because that was the only place there was any light, even though we knew we've dropped the key somewhere else.
And so now I was actually one of the first people to use computers to look at nonlinear.
Nonlinear systems was the very first example of simulating something on a computer.
In 1945 was the Fermi past ultimate experiment in which they put masses with
coupled springs on the computer and they expected just to see ergodic behaviour.
They were just trying to confirm something that was standard in statistical mechanics.
They thought the energy would be equally partitioned among the modes of oscillation and instead
they saw solitary waves propagating back and forth the opposite of what they predicted.
And so no social science, I would argue, has not benefited as physical and natural science have.
I think there's two reasons for that. One is the elementary processes are not well understood, so it can be harder to put things on the computer.
And secondly, there is different cultural attitudes. That has to do with, I think, symmetry breaking in the history of economics.
But I'll maybe come back and say a little bit about that at the end. Now, I want to give you another example from my work of how you can deal with.
Out of equilibrium behaviour that relates to something called leverage cycles and leverage cycles,
as we discovered in 2008, are one of the most important examples of the dynamics of systemic risk.
They were foreseen before 2008 by people like Hyman Minsky who wrote about them in words.
He didn't have a mathematical model, and basic story is that during calm times, people get complacent.
They compete for profits. Leverage creeps up.
I'll explain what leverage is in a minute. But people take more risks.
And then as leverage gets higher and higher, eventually some negative shocks are going to hit and they're going to cause a crash.
JANJUA Two couples who I've collaborated with on many papers also had a theory of leverage cycles.
He actually coined the term involving heterogeneous investors that optimists use more leverage.
Bad news comes along that wipes out the optimists.
And because the optimists were baked into prices before then, there's a big downward spike because they get removed from the picture.
Now I'm going to show an even simpler model than either of these two.
That was developed with Christoph Almond's, who was my graduate student here and is now a postdoc at the LSC.
So it involves leverage. Now, what is that? That involves buying assets with borrowed money.
And the leverage is a number that's the ratio, the value of those risky assets to how much money you actually put down.
So if you buy a house and you make a down payment, that's equal to 20% of the price of the house.
That implies a 5 to 1 leverage because the value of the house is at the moment where you purchase it 5 to 1, the ratio is 5 to 1.
Now, of course, after that, the value of the House may change and your leverage may go up and down.
And it turns out that's very important.
Our model is a model that it's targeted at explaining rather ambitiously the great moderation and the subsequent crisis.
What happened historically is that stock market prices went up over a period of a decade or so.
Stock market volatility went down and leverage for assets of US broker dealers,
speculative traders went up and actually skyrocketed in 2008, then crash down sent volatility really high.
The stock market came down with it. So can we explain this kind of behaviour from a simple model?
Now I have to say before I present this model, we begin by making a rather complicated model.
We had a bunch of banks and a bunch of assets and the banks were managing things and initially we didn't have any risk control.
And we said, what happens when we throw in Basel two the regulation that banks are supposed to follow?
And we saw the whole economy going up and down and this big oscillation and we said, Holy [INAUDIBLE], what's going on?
And we began simplifying the model and we simplified it down to the model I'm going to show you here.
So when we strip things down to the essence, all we needed, it turned out, were two agents, a bank,
which you can think of as any agent that using leverage and a fundamentalist who is
an agent who isn't and who buys undervalued assets and sells overvalued assets.
We assume there's one risky asset plus cash and we make four assumptions.
We assume the bank uses an exponential moving average of historical volatility to estimate the expected volatility going forward.
So what does that mean? That the bank if you go back to the kind of picture I showed you a moment ago,
the banks who are leveraging are looking at this time series and they're using a historical average of what this has
done over the last couple of years or so in order to decide how volatile things are likely to be going forward.
That's actually a very reasonable thing to do because volatility is pretty persistent.
And so the path of the future is in general, like the past.
There can be spikes, as we see here, but most of the time it's a pretty good rule.
And I stand on that assumption because I can just tell you, you survey what people do.
They do that when I was a prediction company,
the firm where Norman Packard and I formed after we didn't get rich playing roulette, we started an investment management firm.
They don't throw you out for winning. And we did pretty well.
And we did this, too, like everybody else. Basel two risk management.
This is a standard thing. Actually this came in while we were a prediction company in the late nineties.
You something called value at risk where you set the leverage target.
I'll come back and explain what that is. We use a standard assumption about price formation supply equals demand,
so we find prices that way and we assume this fundamentalist buys undervalued assets and vice versa.
So the diagram of the model looks like this. We have a fund, we have a bank.
The fund detects a mispricing, we just assume a fixed valuation.
It updates the portfolio. Wait, there's a little shock that keeps knocking things out of equilibrium.
So we're using a little bit. The same trick is that dynamic, stochastic general equilibrium model.
But with the important thing, we don't enforce strategic equilibrium in this model.
And. The bank similarly updates its risk estimate in force as a leverage constraint, and we form prices.
And now let me mention a key fact, which is that for a passive investor with leverage greater than one like the bank.
If prices drop, leverage goes up and vice versa.
Now, so why is that? If you look at the formula for leverage with prices drop, it affects the risky assets.
It affects the equity, which is the assets minus the liabilities.
It affects these assets by the same amount. But because you're subtracting off liabilities in the denominator, this effect is bigger.
And so you see an amplification. So so when when prices drop, leverage goes up.
Now, if you're under a if you have to control your leverage, this becomes destabilising because now your leverage is higher than it should be.
So you're above your target. So you have to sell assets to match your target.
So notice your risk control is now destabilising the market.
So although it's a sensible thing for you to do as an individual for the world as a whole, if if everybody is doing this,
their price impact is destabilising thing because they're buying into rising markets and selling in into falling markets.
And in fact, when we originally tried to make this model, we tried to make it with just banks and it just blew up because it's just destabilising.
We had to put the fundamentalists in there even to stabilise things.
Now just to explain value at risk, that is a.k.a. Basel two,
value at risk is an and a little rule that mathematically looks like this whereby when volatility increases, your leverage decreases.
And according to Basel two, it would follow this blue line where this B parameter here is minus a half and there's some limit, but up to the limit.
If you set the limit to zero, you can see it's just your leverage is just going as one over the standard deviation of volatility.
Very sensible thing to do from a macroprudential point of view, because if the world's riskier, you should take less leverage and take less risk.
Supported and endorsed by the establishment.
By the way, this dash green line would correspond to constant leverage.
Now, what happens when we run this model? Well, in one regime, when we make the banking sector small, we see random oscillations.
And I should mention this model is now a dynamical system. It's got it's about six dimensional.
And it's it's stochastic in general.
So here we're running it and you see that things are kind of randomly varying around some price, but not much is happening.
The leverage is fluctuating a bit, but not much is happening. Now, suppose we make the banking sector bigger.
And what does that mean? Well, when the banking sector is small, it means we don't give them much money or we don't let them take very much leverage.
But if we let them do either one of those things more strongly, then we see something happening like this.
This is that oscillation I already mentioned, where prices are slowly building and then crashing and slowly building and then crashing.
And leverage is similarly slowly building and crashing. Now, interestingly, we can turn the noise completely off if we turn the noise off.
Now we go to a fixed point in the small banking sector version, things just get flat.
But for the large banking sector, we see an oscillation.
And in fact, this oscillation we can show is a chaotic oscillation where things go around some loop
in a space that involve price and who owns the asset and what the perceived risk is.
And you can see there's some kind of complicated behaviour over in this part.
But if you slice this thing, you'll see the characteristic fractal structure that you expect from a low dimensional, strange attractor.
And by the way, there's a hyperbolic fixed point for those of you that have taken dynamical systems and know what that means.
That's driven by this place where the leverage of is going to infinity, where things get really unstable in this model.
Now let me just discuss a little bit what happens as you very parameters.
There's two key parameters. One is alpha, which is controlling the risk of the individual agents.
And B, which is controlling the risk policy that we let them follow because we're interested in, you know, what what are the alternatives?
And the the face base looks like this. So Basel two corresponds to -0.5 for this parameter B, that makes the risk control more or less pro-cyclical.
It's called and as long and this this is corresponds to varying alpha but I plot leverage instead because that really simplifies the plot.
And what we see is there's a region here where things are nice and stable. This is where the fixed point equilibrium works.
There's a region up here where things oscillate and the boundary between those involves a hop bifurcation.
Things happen very suddenly. It's a flip bifurcation.
Things happen very suddenly. You suddenly go from a fixed point to a finite amplitude oscillation.
And as a regulator, you really better understand this because obviously the consequences could be serious.
Now, does this explain the great moderation in the crisis?
Well, I wouldn't want to argue that it is the explanation.
But on the other hand, it could have caused it.
And I forgot to say something essential back here, which is that when you look at this oscillation here, the time axis is not arbitrary.
We've calibrated this model. These are measured in years. So these these run ups automatically come out with a period of around 15 years.
And the levels of leverage that trigger this to happen are quite plausible for what happens in the market.
So it is we get a reasonably good match with what's actually going on.
And the model very clearly says the leverage was too high leading up in the period leading up to the crisis.
So it could have caused the great moderation in the crisis. And and therefore, the housing bubble may have just been the spark that made this happen.
Now, how can we do things more realistically?
Well, we've been working on making what are called agent based models.
What I just showed you was a very simple example of an agent based model in general.
If you want to make things more realistic, you've got to do better than that.
You're not going to get an accurate model of the economy with a with a model that just has two agents and one asset.
We're not that lucky.
I argue that to surmount the fundamental problem of macroeconomics, which is that if you think about the business cycle I showed you,
there's only about 15 well observed business cycles in the history of civilisation.
The ones in the distant past are probably not completely relevant to what's going on now.
So we have very limited data to use to calibrate models.
We have to, I would argue, make good fundamental models.
We can't just use statistical models because the course of dimensionality blocks us.
And and and we've got to use micro data to validate them.
Now I'm out of time, so I'm not going to say much about this. But just to say, if you look at something like climate models,
it's been key that they're not just looking at time series like global mean temperature and global carbon dioxide.
They're looking at details about the weather patterns they predict.
And they can show they can do things like join Tierra del Fuego together with Antarctica.
There are simulations. The world suddenly gets vastly warmer than it is now because they block the Antarctic current.
And you can look back in the in the record and see that that's in fact how it was done.
Now, my vision of the economics of the future would be a simulation of the economies of the major countries of the world.
We get the micro big data inputs directly off the Internet.
We might even couple this to other social models. We'd have teams focusing on each components on households, firms, banks,
etc. and they would be monitoring, using the Internet whether their models are really being followed.
I'm going to skip over this because I want to wrap up pretty soon.
Now, how well do I think we can do that?
Well, we're not going to do that perfectly, because there are things like market efficiency, market efficiencies, a self-organising process.
I'll say a bit in a moment, it limits the predictability of tradable assets.
I've already alluded to the fact that I think in the real world we see something
that's very close to turbulence in financial assets and in the economy.
And in fact, there's a beautiful paper in nature from about 26 just pointing out that as you vary this,
if you do a match up of the statistics of fluid turbulence to those of the financial system, the match up is quite accurate.
I don't think we're going to be able to make detailed forecasts about the future of the economy.
But I do think there's a lot of stuff we can predict,
and that is there's a lot of information we can compress by making better models that we can use to formulate policies.
It's back to this question of social science versus physics.
You know, unlike atoms, people can't think. That does put.
So people can think, sorry. Sometimes I think they can't.
But yes, people can think. What scientific limits does this impose?
Well, it does impose some limits. It's you know, economics is a [INAUDIBLE] of a lot harder than physics.
I've done both, so I can vouch for that. But that doesn't mean everything's impossible.
That doesn't mean you have to just throw up your hands. And I would point to the consistency of psychological experiments.
There's a lot of psychological experiments.
If you put people in a certain situation, you can make a pretty good prediction that 70% of them will do one thing and 30% will do another thing,
and it holds up fairly consistently through time and so on with some variations, cultural variation and so on.
I think big data empowers us to do incredible things that we couldn't do before.
I already mentioned the example of traffic forecasting, which is a sort of trivial example.
You know, it's a bit frustrating if you look at how we go about running the models that are actually run to understand economies,
they're run through a methodology based on national accounting that was developed in the 1950s and has evolved only little since then.
It involves surveys. It involves boiling everything down into aggregate data.
It involves the numbers coming out often a year after they were actually the events were actually happening.
I just think we can do infinitely better than that. Now, I want to say a bit about market efficiency and then conclude, you know,
the hypothesis of market efficiency says that markets are inherently unpredictable.
Now, when I heard about that, my immediate reaction is, okay, I've got to go try and predict markets.
And, you know, and I think we and many others have since then showed that actually markets are somewhat predictable,
predictable enough to make some somebody but even, you know, they're good firms like ours.
We were right, maybe on 55% of our bets and wrong on 45% of them.
So it's still pretty noisy. So I don't want to say that this is easy.
The world is actually close to being market efficient. The question is, I argue, how much difference does it make?
How much difference do these inefficiencies make? Well, they make an important difference.
They put a lot of money in my pocket. Okay.
But but in general, for society, is this something we ought to care about?
And I argue that we should, because there's a catch.
You can see automatically markets can't be perfectly efficient because market efficiency requires arbitrage,
cause you've got people out there to exploit them, but there's no inefficiencies to exploit the law, go home.
And so the market will be inefficient again. So there must always be some residual inefficiencies.
And so the way I would put this is markets are efficient to first order, but necessarily inefficient at second order.
And I think this has important consequences. I wrote a paper back in 2002 with a market ecology hypothesis,
and the hypothesis is that we need to think about what the strategies that agents are following,
what are these inefficiencies that they're feeding off of? How do these different kinds of agents interact with each other?
And and and what happens? And I did a lot of stuff in that paper where I basically mapped the financial system into the theoretical framework
that's used in ecology and showed that you got all the same things that ecologists have worried about in,
in financial systems.
And I'm working with illicit coinage in health and others to try and use these ideas to actually get better models for understanding financial risk.
Because one of the key hypotheses here, the last one is that imbalances in the market,
food web can drive crises when market ecology is get out of balance.
Strange things can happen.
And I would argue this is what happened in the crisis of 2008, when subprime mortgages and strange new derivatives got way out of hand.
Now, there's a lot of challenges to doing this kind of thing, to making realistic models.
There's a group of us here who are working on this, most of whom are in the Mathematical Institute.
We're trying to create the key building blocks. We're trying to take advantage of the collective,
collective wisdom that's been accumulated by the small cadre of people that are working on out of equilibrium
agent based models to understand how to do simulations in economics make them really simple and easy to use.
We have an open source project. Any of you can track what we're doing.
Join it. All the codes available as it's being created.
If you have programming skills, please, and your interest in this kind of thing, please contact us.
And we'll sign you up. We're trying to create a library of standard models that anybody who wants to join can build
on to deal with the problem that these kind of models actually are time intensive to create.
And if you have to create everything from scratch, it's hard to make to accumulate progress because a typical graduate student will spend several
years just building writing software to get to the point where they can do something useful.
And we're focusing on three use cases at the moment housing markets, insurance industry and bank stress testing.
Now, to conclude, I mean, I apologise.
I haven't said anything in this lecture about long range evolution, about growth,
innovation, the things that actually cause long term structural change in the economy.
There just isn't enough time to talk about it here. Let me just say that's another component of the work I do.
We've shown together with others like Francois Lafond, that actually a lot of these trends are surprisingly forecast able.
There are laws like Moore's Law and Rights Law that make that true.
So I think there is and this is another unexplored area in economics.
Another key application that we desperately need are better economic models to steer us toward a sustainable economy.
Current rates of growth are simply unsustainable. How are we ever going to do this without a huge crisis?
I would argue our existing models are hopelessly flawed. And I think actually this is a mathematics lecture.
I think what we're doing, I think the ecology of the way mathematics done is done, is evolving.
I will stand who was one of the pioneers of that?
As a graduate student, we were pioneers. We started doing what we called experimental economics.
Sorry, experimental mathematics. We put equations on computers.
We played around with them. We tried to map out the phenomenology to serve as a kind of scout for more focussed theory.
And I think that's increasingly the way things are being done and will be done in the future.
I think that will happen in economics as well.
And simulation needs to play a key role in between data on one side and sharper small models on the other side.
We need to be moving more towards an economic big science because while I'm a strong supporter of small science,
you can't do everything with just small science. Imagine if we were trying to deal with global warming without large scale global circulation models.
We just wouldn't be able to do it. Doing this is going to require a culture shift and it's a big job.
And I appreciate you being patient and listening to me.
I'm 10 minutes over time. Thank you.