The Future of Everything

March 12, 2021

Quantum economics and finance – video series

Filed under: Quantum Economics and Finance — Tags: , — David @ 8:40 pm

This series of short videos introduces the key ideas of quantum economics and finance. The only background assumed is basic linear algebra. The material is based on the book Quantum Economics and Finance: An Applied Mathematics Introduction.

Notes:

QEF01 – Introduction to Quantum Economics and Finance

QEF02 – Quantum Probability and Logic

QEF03 – Basics of Quantum Computing

QEF04 – Quantum Cognition

QEF05 – The Quantum Walk

QEF06 – The Penny Flip Game

QEF07 – The Prisoner’s Dilemma

QEF08 – Quantizing Propensity

QEF09 – Threshold Effects in Quantum Economics

QEF10 – A Quantum Option Pricing Model

QEF11 – The Money Bomb

December 14, 2021

Quantum economics – the story so far

This piece gives a brief summary of my work to date (2016-2021) in quantum economics.

The idea that the financial system could best be represented as a quantum system came to me (dawned on me? evolved?) while working on The Evolution of Money (Columbia University Press, 2016). “Money objects bind the virtual to the real, and abstract number to the fuzzy idea of value, in a way similar to the particle/wave duality in quantum physics,” I offered. “Money serves as a means to quantify value, in the sense of reducing it to a mathematical quantity – but as in quantum measurement, the process is approximate.” Price is best seen as an emergent feature of the financial system. I summarised this theory in two papers for the journal Economic Thought: “A Quantum Theory of Money and Value” and “A Quantum Theory of Money and Value, Part 2: The Uncertainty Principle“.

While I had some background in quantum physics – I studied the topic in undergraduate university, taught a course on mathematical physics one year at UCL, and encountered quantum phenomena first-hand while working on the design of particle accelerators in my early career – my aim in the book (co-authored with Roman Chlupaty) was not to impose quantum ideas onto the economy. My primary research interest was in computational biology and forecasting and I had not touched quantum mechanics in many years. The dual real/virtual nature of money just had an obvious similarity to the dual nature of quantum entities, and in fact I was surprised that I appeared to have been the first to make this connection in a serious way and come up with a quantum theory of money.

I was aware that a number of researchers were working in applying quantum models to cognition and psychology, but it was only after finishing the book that I learned about the area of quantum finance (I also discovered a separate paper on “Quantum economics” by the physicist Asghar Qadir from 1978, which argued that the quantum formalism was well suited to modelling things like economic preferences). The reason I hadn’t come across these works in my research about money was because just like in neoclassical economics there was no discussion of that topic. Nor was there much discussion of quantum phenomena such as entanglement or interference. Instead the emphasis in quantum finance was on using quantum techniques to solve classical problems such as the Black-Scholes option-pricing algorithm, or portfolio optimisation.

My motivation was completely different. In books such as Economyths, and The Money Formula (with Paul Wilmott), I had investigated the drawbacks and limitations of these traditional models – so rather than invent more efficient ways of solving them, I wanted to replace them with something more realistic. Money was the the thing which linked finance and psychology, so a quantum theory of money could be a first step in developing a new approach to economics.

I sketched out the basic idea as an Economic Thought paper “Quantum economics” which served as a blueprint for my 2018 book of the same name. It tied together the quantum theory of money, with ideas from quantum finance, quantum cognition, quantum game theory, and the broader field of quantum social science. The ideas were also summarised in a piece for Aeon magazine – which was when I found out why no one had probably bothered to develop a quantum theory of money. The article was not well received, by economists but especially it seemed by physicists, some of whom went out of their way to trash the idea.

I was not new to having my work come under criticism. Indeed, much of my career has focused on pointing out the drawbacks and limitations of mathematical models, which has frequently brought me into conflict with people who don’t see it that way, starting with my D.Phil. thesis on model error in weather forecasting (see Apollo’s Arrow). My book Economyths also drew howls of outrage from some economists. However quantum economics felt different, and seemed to touch on a range of taboos, in particular from physicists who have long resisted the adoption of quantum ideas by other fields. But quantum mathematics is not owned by physicists, it is simply an alternative version of probability which was first used to model subatomic particles, but also can be used to describe phenomena such as uncertainty, entanglement, and interference which affect mental systems including the economy.

While writing the book I developed in parallel an online mathematical appendix which presented some key results from quantum cognition, finance, and game theory (an early version was translated into Russian). Because my aim was to develop a theory of quantum economics, I also started applying quantum methods to some key economic problems, including supply and demand, option pricing, and the debt relationship which underlies the creation of money. This online appendix later grew into my technical book Quantum Economics and Finance: An Applied Mathematics Introduction, first published in 2020 and now in its second edition.

For supply and demand, my idea was to model the buyer and seller in terms of a propensity function, which describes a probabilistic propensity to transact as a function of price. A simple choice is to describe the propensity function as a normal distribution. The joint propensity function is the product of the buyer and seller functions. The next step is to use the concept of entropic force to derive an expression for the forces which describes the tendency for each party to move the price closer to their preferred price point. The joint force is just the sum of the forces for the buyer and seller. However there is a contradiction because the probability distribution does not match that produced by an oscillation. To resolve this, we quantize the force to obtain a quantum harmonic oscillator whose ground state matches the joint propensity function. This model, which sounds elaborate but is actually quite minimal in terms of parameters, applies to economic transactions in general, so has numerous applications, including the stock market. The paper “A quantum model of supply and demand” was published in the journal Physica A in 2020.

Typical propensity functions for buyer (to the left), seller (right), and joint (shaded).

The question of how to price options is one of the oldest problems in finance. The modern method dates back to a 1900 thesis by Bachelier and is based on the concept of a random walk. For the quantum version, the logical place to start was with the quantum version of this which is a quantum walk. Instead of assuming that the log price will follow a normal distribution with a standard deviation that grows with the square-root of time, the model has two peaks which speed away from each other linearly in time. It therefore captures the psychological stance of an investor who has a bullish or bearish view on the asset (e.g. price might grow by 10 percent each year), but balances that with the possibility that the opposite might happen in order to obtain a fair price for the option. When coupled with the quantum model of supply and demand, the algorithm can be used to predict option price and volume. “A quantum walk model of financial options” was published in Wilmott magazine in 2021, and the theory was reported on the same year by the Economist in an article “A quantum walk down Wall Street“.

Probability distribution for a quantum walk (solid) versus random walk (dashed).

Finally a main question in quantum economics is the interaction between mind and money which underlies the debt relationship, and also the creation of money objects in the first place. Both of these topics are traditionally neglected in mainstream economics. In quantum economics it is easy to show that the debt relationship can be modelled as a simple circuit with two qubits, representing the debtor and creditor, entangled by a C-NOT gate which represents the loan contract. Interestingly, it turns out that the same circuit can be used to represent the decision-making process within the mind of a single person, where there is an interplay between a subjective context and the final decision. In quantum cognition, this is usually modelled as a two-stage process; however it can also be modelled using two entangled qubits, in which the context and the decision are separated out, as in the debt model. This result was published in a 2021 Frontiers in Artificial Intelligence paper, co-authored with Monireh Houshmand, called “Quantum propensity in economics“. A related paper published in Quantum Reports, that discusses applications including mortgage default, is “The color of money: threshold effects in quantum economics“. 

Two-qubit entanglement circuit for debt contract (A is debtor, B is creditor), or quantum cognition (A is context, B is decision).

For a full list of my research in quantum economics and finance, including links to these and other papers, please see the page Quantum Economics Resources. These findings and others are also presented in my technical book Quantum Economics and Finance: An Applied Mathematics Introduction, and for a general audience in Money, Magic, and How to Dismantle a Financial Bomb: Quantum Economics for the Real World (available 02/2022). The work continues! – if readers are interested in getting involved, please drop me a line here or through LinkedIn.

November 11, 2021

Ten reasons to (not) be quantum

Filed under: Economics, Quantum, Quantum Economics and Finance — Tags: , — David @ 11:23 pm

While the use of quantum models is becoming more popular in the social sciences including economics, it is still the case that when many people, especially those with a training in physics, hear the expressions “quantum economics” or “quantum finance” they immediately reach for some off-the-shelf arguments about why it must be nonsense (or some smelling salts). Here is a compilation of the usual ones, along with responses.

  1. Quantum mechanics was developed for subatomic particles, so it should not be applied to human systems. As one website claimed, “It’s only when you look at the tiniest quantum particles – atoms, electrons, photons and the like – that you see intriguing things like superposition and entanglement.”

Response: Bohr’s idea of superposition and complementarity was borrowed from psychology, as when we hold conflicting ideas in our heads at the same time, and the concepts of mental interference or entanglement are not so obscure. Also, many ideas from quantum mechanics such as the Hilbert space were invented independently by mathematicians. And calculus was developed for tracking the motion of celestial bodies but we don’t ban its application to other things.

  1. Quantum is too hard for non-physicists to understand. According to the physicist Sean Carroll, “No theory in the history of science has been more misused and abused by cranks and charlatans – and misunderstood by people struggling in good faith with difficult ideas – than quantum mechanics.”

Response: There is often a confusion between quantum probability, which is a mathematical tool, and quantum physics, which is about subatomic particles. Yes, the physics of subatomic particles is complicated – so are things like classical fluid dynamics. But quantum probability is just the next-simplest type of probability after the classical one. And the misuse of mathematical models which has created the most societal problems is the classical methods used in economics. As a side note, most people involved in quantum economics and quantum finance are physicists or (like me) mathematicians.

  1. Quantum economics is physics envy, or an attempt to “appropriate the high prestige of physics” as one physicist put it.

Response: Mainstream economics is directly inspired by, and based on, concepts from classical mechanistic science. There is nothing inherently wrong with using the same mathematical tools for different areas, what is strange is when the tools used don’t change or adapt. As John Cleese said: “people like psychologists and biologists have still got physics envy but it’s envy of Newtonian physics and they haven’t really noticed what’s been happening the last 115 years.”

  1. Quantum is flaky, pretentious hype or woo. Cue nerd jokes about “quantum healing” or “quantum astrology”. In his description of what he called the Intellectual-Yet-Idiot, Nassim Taleb included anyone who “Has mentioned quantum mechanics at least twice in the past five years in conversations that had nothing to do with physics.”

Response: Quantum is a mathematical toolbox – it might come across as flaky or pretentious for a person to talk about it in the wrong context, but not to use it in their work.

  1. Entanglement is unique to special physical systems which can maintain quantum coherence. A physicist explained that “you can never violate a Bell inequality using systems like dice, dollars, or bank accounts. There is simply no way, and certainly no experiment has ever done so. (Maybe one or two ‘crackpot’ people claim otherwise, but they are not to be trusted.)” One science journalist told me that “Dollars don’t become quantum mechanically entangled. If they did, we’d be building quantum computers out of money.”

Response: In mathematical terms, entanglement is a straightforward property of Hilbert spaces, and we can use it to model social and financial systems. The Bell test is not a definition of entanglement, it is a way of teasing out a particular form of entanglement for subatomic particles. It is true that we can’t build quantum computers out of money, but nor can we build classical computers – does that mean money is not classical?

  1. Quantum is too complicated and mathematical – we need simpler models and less math. Variants: The economy cannot be reduced to equations, people are not subatomic particles.

Response: The need for simple models is a theme of many of my books, however what counts is things like the number of parameters in a model. Quantum probability is more complicated than classical probability, but it is the simplest way to capture phenomena such as superposition, interference, and entanglement, which characterise many key mental and financial processes (for example, the quantum walk model for pricing options or the two-qubit model for quantum decisions are not complicated). People are not subatomic particles, but nor are they classical particles, which doesn’t stop economists from using classical models, or talking about physics-like forces of supply and demand (they are just assumed to be at equilibrium). And while it is true that human behaviour cannot be reduced to equations of any sort, we use equations all the time to simulate the economy. Again, many of my books, such as Apollo’s Arrow, or Truth or Beauty, have criticised the overreach of mathematical models, but that is a separate issue and applies as much to classical models.

  1. Quantum is a forced analogy or a metaphor. As economist Paul Samuelson once wrote, “There is really nothing more pathetic than to have an economist or a retired engineer try to force analogies between the concepts of physics and the concepts of economics … and when an economist makes reference to a Heisenberg Principle of [quantum] indeterminacy in the social world, at best this must be regarded as a figure of speech or a play on words, rather than a valid application of the relations of quantum mechanics.”

Response: Quantum probability is a mathematical tool, which is not the same as an analogy or metaphor. The purpose of a metaphor is usually to describe something which is abstract and complicated in terms of something that is more concrete, so it would make more sense to go the other way and use human behaviour as a metaphor to help describe subatomic behaviour.

  1. The brain has not been shown to rely directly on quantum processes.

Response: Quantum effects appear to be exploited by biological systems in a number of processes (see quantum biology) but whether they are used in the brain or not makes no difference to economics. The argument is not that the economy inherits quantum properties from subatomic interactions in the brain, but that it can be modelled as a quantum system in its own right. For example, a debt contract can be expressed using a quantum circuit in a way which captures effects such as uncertainty, subjective context, power relationships, and so on.

  1. Markets are not quantum because there is no uncertainty. For example, something like a bank account, or an order book for a stock market, has clearly posted amounts and prices. One person compared her bank account to Schrödinger’s cat: “I am a PhD physicist, so for me the word quantum that gets thrown around is a bit ridiculous … So think about your bank account, it might be empty until you open it, so are you telling me that this is uh quantum finance or quantum economics okay you can have a million in your account or you can have zero we don’t know?”

Response: While it may be true that bank accounts are not like Schrödinger’s cat, I will let The Economist answer that one, from an article called “Schrödinger’s markets” in the print edition: “on a closer look finance bears a striking resemblance to the quantum world. A beam of light might seem continuous, but is in fact a stream of discrete packets of energy called photons. Cash flows come in similarly distinct chunks. Like the position of a particle, the true price of an asset is unknowable without making a measurement – a transaction – that in turn changes it. In both fields uncertainty, or risk, is best understood not as a peripheral source of error, but as the fundamental feature of the system.”

As computer scientist Scott Aaronson notes, quantum methods are adapted to handle “information and probabilities and observables, and how they relate to each other.” Since the financial system seem a pretty good example of information, probabilities, and observables (in this case through transactions) it seems like a suitable approach. Much of the confusion comes down to the fact that quantum economics is not quantum physics applied to the economy, but rather quantum mathematics applied to the economy (see figure below). Quantum mathematics should be viewed as a mathematical toolbox that can be applied to either physical or social systems where appropriate.

The idea of quantum economics is not that physics can be directly applied to social behaviour as shown here …
… but instead that quantum mathematics can be applied to both physical and social systems. Figures from: Quantum Economics and Finance: An Applied Mathematics Introduction

The above nine reasons for rejecting a quantum approach, which are the ones most commonly produced, are very superficial and are easily dismissed with a little reflection. (Skeptics sometimes prefer to say that they are “not convinced” without giving a specific reason, but my aim is not to convince people of anything, it is to lay out the facts as I see them and let others do their own research and come to their own conclusions.) Also, arguing against these reasons, as I have done above, will in my experience have absolutely no effect. One reason is that getting the quantum approach seems to involve something of an aha moment where it suddenly clicks into place. The other reason though is that they are not the real reason. So why is it that no one even tried to apply quantum methods to the economy until about a century after they were invented? This points to:

    10. Quantum economics touches on a range of taboo topics.

These taboos will be the subject of a future article, but for the full picture read Money, Magic, and How to Dismantle a Financial Bomb: Quantum Economics for the Real World (available 02/2022). Finally, given the numerous reasons to not take a quantum approach, I should point out that there also many reasons why the opposite is true, and the economy is amenable to a quantum treatment! In particular, quantum is the best framework for expressing in mathematical terms the complex interactions between mind and money. To see why, the best place to start is again with the books. For a mathematical treatment, see Quantum Economics and Finance: An Applied Mathematics Introduction.

May 17, 2021

Shifting the Economyths

Filed under: Economics, Quantum, Talks — David @ 3:29 pm

This is the text for my contribution to the online conference Beyond the False Dichotomy: Shifting the Narrative

I wrote Economyths a little over ten years ago, and in part the book was my response to the financial crisis. The thesis was that mainstream economics is based on a number of what I called economyths which were defined as beliefs or stories that have shaped economic thought. There were ten in total but to give you an idea here I will just mention four:

One said that the economy is made up of independent individuals. This is the idea that people are like the classical picture of a self-contained atom, and there is “no such thing as society” as Thatcher said.

Another is the idea that the economy is stable and self-correcting. This is the main story of economics from Adam Smith’s invisible hand, to modern equilibrium models.

A third is the idea that the economy is rational and efficient. The assumption that people make rational decisions to optimise utility is connected to the ideas of stability and efficiency. Corporations are defined to be an incarnation of rational economic man.

And then consistent with these is the next economyth which is that the economy is balanced and symmetrical. The issues of power and distribution are viewed as “soft” and peripheral to the subject, and therefore tend to be ignored. As one economist said in an interview, “economists are not good at what’s fair, right?”

Economists have always insisted that their theories are far more complex than these economyths would suggest, but the reality is that these assumptions have been extremely influential and in particular form the basis of the mathematical models used to simulate the economy and make policy recommendations.

In 2017 I did a revised version of the book which included an additional economyth, which was the idea that the economy boils down to barter. This is the myth which in many ways justifies the others, because it means that money isn’t important. Adam Smith for example focused on the “real” economy of labour and commodities and saw money as a kind of veil or a distraction. Economists since then have treated money as just a metric or an inert medium of exchange, and ignored its confounding properties – its dual real/virtual nature, its ability to entangle people through debt, its inherent instability, its tendency to cluster and create inequality, and its psychoactive effects on the human mind.

The drawbacks of omitting things like money and banks from the model became evident after the financial crisis of 2007, when it turned out – as one central banker explained ten years later – that “In the prevalent macro models, the financial sector was absent, considered to have a remote effect on the real economic activity.” And today, the continuing problems of financial instability, social inequality, and environmental degradation can all be largely traced to the money system, which is unstable, unfair, and is reliant on continuous growth to pay off debt.

One reason I called these ideas economyths is because, like myths, their legacy goes back a long way. We could start with post-war economists like Milton Friedman; or go back further to the neoclassical economists who first tried to establish economics as a kind of social physics in the late nineteenth century; or to Adam Smith, who didn’t try to quantify economics but was inspired by Newton. But I would argue that we can actually go back much further. My reason for saying this is that the ten economyths were based directly on a list of opposites, divided into good and evil, from the philosopher Pythagoras, who believed that the universe was based on number, and who lived at a time when coin money was changing the world of commerce. As he said, “number is all” and money is a way of putting numbers on the world.

In order to understand the narrative appeal of this model, it is interesting to compare it with another model which was very influential for a long time, and also reflected the Pythagorean ideals of symmetry and mathematical elegance, namely the Greek model of the cosmos. This model incorporated two main assumptions. The first was that the celestial bodies moved in circles, which were considered the most perfect and symmetrical of forms. The other assumption was that the circles were centered on the Earth. In Aristotle’s version, the planets and stars were thought to be encased in crystalline spheres which rotated around us at different speeds. The fact that planets did not follow perfect circles around the Earth, but sometimes tended to loop back on themselves, was handled by adding epicycles – circles around circles.

This geocentric model was complemented by a theory of physics. According to Aristotle, all matter consisted of the five elements Earth, Water, Air, Fire, and Ether which was reserved for the heavens. His theory was less a theory of motion, than a theory of stability: each element sought its own level, following the same order with Earth on the bottom and Ether on top, and in fact would do so instantaneously in a vacuum. Aristotle deduced from this that a vacuum could not exist: nature abhors a vacuum.

The Greek model lasted for well over a thousand years – it was adopted by the Church, and was not finally overturned until the Renaissance. How did it manage to last for such a long time? And what lessons are there for economics?

One reason for its durability was that it could make accurate predictions of important events such as eclipses. Another reason, though, is related to aesthetics and the fact that, as Aristotle put it, man is a political animal. There was a strong parallel between the perceived order of the cosmos and the order of society. Greek society was structured as a well-ordered hierarchy, with slaves at the base, followed in ascending order by ex-slaves, foreigners, artisans, and finally the land-owning, non-working upper class. These men alone could be citizens, and oversaw everything from above, like the stars in the firmament (women did not take part in political life and took their social class from their male partner). A model of the universe which suggested that everything has its natural place in a beautiful, geometrically-governed cosmic scheme therefore supported the status quo. For this reason it would certainly have appealed to the male leisure class that ruled ancient Athens, and later to the Catholic Church.

The first cracks in the model appeared in 1543, when Copernicus proposed that the Earth might go around the Sun, rather than vice versa. European astronomers began to observe comets which passed between the planets, so if Aristotle’s crystalline spheres had actually existed, they would have broken through them. Finally, in the late seventeenth century, Isaac Newton derived his three laws of motion and the law of gravity. The static circles of classical geometry were replaced with dynamical equations, which had a different but equally powerful aesthetic appeal. Sometimes it takes a model to defeat a model.

So what does this have to do with economics? Well, there is an obvious parallel between the Greek circle model, and our modern model of the economy, because it too pictures the world as rational, stable, ordered, and efficient, and therefore favours the elite. Indeed mainstream economics, in its obsession with rationality and efficiency, sometimes sounds like the PR wing of the financial sector. The model can’t make predictions of things like crises, or indeed the economy in general, but it goes a step further by predicting that it can’t predict, as per efficient market theory. The main difference is that while Aristotle thought that a vacuum could not exist, because otherwise things would find their equilibrium immediately, efficient market theory assumes that prices do reach equilibrium instantaneously – so a vacuum does exist, and it is the market.

Both the Greek circle model and the economics model also show the strength of a mathematical model. Even something like the financial crisis only made a dent. As Paul Krugman wrote in 2018, “Neither the financial crisis nor the Great Recession that followed required a rethinking of basic ideas.” So like the Greek model, the orthodox model of the economy is incredibly resilient.

I think if there was a specific point where this narrative became compelling in economics, it was when economists changed their theory of value, while leaving the rest of the theory mostly the same. The switch was announced 150 years ago by William Stanley Jevons, in the second paragraph of his 1871 Theory of Political Economy, where he wrote that “Repeated reflection and inquiry have led me to the somewhat novel opinion that value depends entirely upon utility.” Here utility was a kind of energy-like quality which roughly equated to happiness. Classical economists such as Smith had followed a labour theory of value which acknowledged the role of power. Utility created a new narrative which flipped this on its head. Economics was now about pleasure and good times.

Of course utility couldn’t be measured directly. Another approach though was to simply assume that utility is reflected by price. Or as Jevons put it: “just as we measure gravity by its effects in the motion of a pendulum, so we may estimate the equality or inequality of feelings by the decisions of the human mind. The will is our pendulum, and its oscillations are minutely registered in the price lists of the markets.”

A side-effect of this emphasis on subjective utility, ironically, was that by reducing value to a number, subjective things like emotion or social power or dignity or ethics were usurped by the theory and thus stripped of all weight. As Tomáš Sedláček wrote for example in his 2011 book The Economics of Good and Evil, “The issue of good and evil was dominant in classical debates, yet today it is almost heretical to even talk about it.” Another thing which also of course didn’t fit into this rational utility approach – it doesn’t compute – was money, with its economyth-defying properties of entanglement, instability, and irrationality.

But still, why is it that more than a decade after the crisis economics still remains rooted in the past? Why is the orthodox narrative so resistant to change?

One reason again is that as with Aristotelian physics, the economyths form part of a connected structure. The story they tell is consistent and self-reinforcing – markets are efficient, stable, rational – and you can’t change one part without changing them all.

Instead, what happens is that economists attempt to fold in new ideas from other areas such as complexity or behavioural psychology, without changing the structure too much. Behavioural economics for example has in my view won acceptance exactly because it can be incorporated in this way, and viewed as an epicycle that can be wheeled out for particular situations. Economists also try to add in so-called “frictions” to their equilibrium models, but assume the equilibrium exists in the first place. Paul Krugman again: “We start with rational behaviour and market equilibrium as a baseline, and try to get economic dysfunction by tweaking that baseline at the edges.” A few more epicycles and it will be perfect, the thinking goes.

The story also offers a powerful restoration narrative, because it says that crises and upsets are caused by external events, and economic forces bring the economy back to this imagined equilibrium. It therefore taps into the human desire for order which is important in politics but also in fiction. And the idea that markets are rational and efficient also justifies the powerful position of the financial sector and the wealthy, while at the same time distracting from the workings of power and the role of money.

To change the narrative it isn’t enough to modify details of the model, instead we need to go back as the neoclassicals did to the fundamental idea of value and its relation with price – in other words, the question of how much something is worth, its numerical cost. There are a number of ways of going about this, but I would argue that a good place to start is with the math.

This might seem counter-intuitive. After all, it is something of a cliché to say that economics is too mathematical. As one recent book put it, “today mainstream economics follows a path of great mathematical rigor that . . . does not make much room for other accounts of economic life.” However rigor isn’t useful if you are using the wrong kind of mathematics in the first place. And while Keynes wrote that “practical men … are usually the slaves of some defunct economist” we could also say that those economists are themselves slave to beliefs about number and value that are inherited in large part from a mathematical model.

Orthodox theory for example is based on the core idea that markets are stable and price is a measure of inherent value. It is best represented by the X-shaped figure of supply and demand. This plots supply as a line which increases with price, and demand as a line which decreases with price. The point where they match in an X represents the stable equilibrium where the market clears. This diagram appears in all introductory economics textbooks, but it is also there in the mathematical models used to simulate the economy.

An odd feature of the graph is that price appears on the vertical axis, and is assumed to be determined uniquely and passively by unknown forces of supply and demand which are in balance, so cancel out. While neoclassical economics is often described as Newtonian, it assumes equilibrium and has no real concept of mass or dynamics or force. The problem though is we never observe supply or demand independently, we only observe transactions. Like the crystalline spheres, these are imaginary constructs. Indeed the whole idea of representing a complex dynamic system as the stable intersection of two lines is very strange and is not done in other areas such as biology, where the only things that are at equilibrium are dead.

To make the diagram more scientific, a first step is to flip it around so that price is the independent variable on the horizontal axis. This seems a trivial change but is actually key to the whole story. Instead of utility, we can then plot the propensity curves for the buyer and seller, which represent the probability of transactions as a function of price. The fact that price is somewhat arbitrary and decisions are subject to effects such as context means these curves are inherently probabilistic. The X-shaped lines of supply and demand from the classical diagram are therefore replaced by probabilistic waves which only collapse down to a particular price during a transaction. This model therefore acknowledges that value is a soft and fuzzy concept, while price is a measure which is subject to intrinsic uncertainty.

The next step is to acknowledge that people are not separate atoms who only communicate by bouncing off one another, instead they are entangled beings who talk to each other, and share things like social norms. An example is the prisoner’s dilemma game, another staple from the textbooks: in classical theory everyone defects and rats out on the other person, but experimental results show people choose to collaborate between a third and a half of the time, which suggests a high degree of entanglement.

We also need to address the fact that people make decisions based on a mix of objective and subjective factors which are entangled in the mind and may interfere with one another. And above all we need to include the dynamics of money, which behaves more like a kind of information than a classical physical object.

The correct mathematical framework for this theory has already been developed by a group of radical thinkers over a century ago. Unfortunately this mathematics has until recently been reserved for the esoteric area of subatomic particles. I’m talking of course about the quantum formalism. For our purposes this refers, not to quantum physics, but to a kind of logic and probability which allows for effects such as interference, entanglement, and the idea of a measurement procedure. The word quantum is from the Latin for “how much” which applies naturally to the economy, where prices are measured through transactions. The point is not that there is a direct map between subatomic particles and humans, but that we can use similar mathematical tools to model each, which is a subtle but important distinction.

While quantum ideas have been around for a while, they are only now starting to reach critical mass in the social sciences. This new quantum narrative is under construction, in economics and finance but also in other areas such as psychology and political science. For example the Carnegie Foundation is funding a series of quantum bootcamps for social scientists at Ohio University starting this summer. There is a new anthology coming up this year from Oxford University Press on Quantizing International Relations. Danah Zohar has been bringing quantum principles into management theory for some time. The area which is seeing the most rapid adoption of quantum ideas is finance, because of quantum computing. As far as I know you can’t take a university course in quantum finance, but you can get a job in it right now in financial centers such as Paris, New York, Toronto, and so on. The development of classical computers in the post-war era changed the way we model and think about systems including the economy, and quantum computers – which have entanglement built in – are now doing the same thing.

What counts for the purposes of today’s discussion though is not the math, but the story told by the math. To summarise, the core narrative of mainstream economics is that people behave like classical atoms: hard, independent, stable. The economy can therefore be modelled as an equilibrium system. The main message of quantum economics is that people are entangled: with their own subjective feelings, with other people, with what they read in the news, and above all through the money system. The economy is a complex, dynamic, living system which can be modelled using a mix of techniques, such as ones from complexity science or systems dynamics, so long as they respect the indeterministic and entangling nature of both mind and money; and particularly the ability of the money system to scale up cognitive and financial entanglements to the societal level.

So as a one word description of the new narrative I would choose quantum or maybe entanglement. If you don’t want to do quantum mathematics, which is understandable, it doesn’t matter because what counts is the idea that the economy is best seen as a complex system which is entangled through a mix of financial and social effects. In practical terms this means that all the so-called “soft” ideas such as subjectivity, emotion, social dynamics, power, money, value, fairness and ethics that have been exiled from economics are now back in play. Debtors and creditors are entangled, shareholders and stakeholders are entangled, and we are all entangled with the climate system. Obviously the quantum approach doesn’t have all the answers, but its built-in emphasis on uncertainty can be liberating, and encourages a pluralistic response. Perhaps it is the model which teaches us to sometimes at least let go of models, because they can’t capture enough of the complex reality. And my hope is that the quantum approach and the idea of entanglement resonates with some of the other ideas and narratives discussed today.

Again, the idea that a new narrative should begin with our system of logic and probability may seem strange but history shows that mathematical models have great influence. And sometimes, as mentioned, it takes a model to beat a model. A new narrative which marks the next evolutionary step of capitalism is going to need a new mathematical framework, if only to better define its language, and help to do an audit on what ideas one may inherited, perhaps unconsciously, from the classical model. It is ironic that we live in an age characterised by volatility, uncertainty, complexity, and ambiguity but our economic theory assumes a deterministic state of placid equilibrium. It is therefore well overdue for an upgrade.

Further reading: quantum economics resources

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