These are the notes to the first of a series of presentations on quantum economics and finance. For the video version see here.

Quantum economics and finance uses quantum mathematics to model phenomena including cognition financial transactions and the dynamics of money and credit. In these talks we’ll be talking about topics including: why quantum in the first place; quantum probability and logic; basics of quantum computing; quantum cognition ; quantum walk; quantum game theory; quantum supply and demand; threshold effects; option pricing; and the money bomb.

So why quantum in the first place? The quantum revolution in physics was born when physicists found that at the subatomic level energy was always exchanged in terms of discrete parcels which they called quanta, from the Latin for “how much”. In economics the equivalent is financial transactions, like when you buy an ice cream in Italy and you say “quanto costa” which makes the quantum connection a little clearer. So money behaves in some ways like an object but not a classical one. It shows signature properties of quantum systems such as discreteness, indeterminacy, entanglement, duality, interference and so on.

Perhaps the most obvious such property is the way that money jumps. In physics Erwin Schrodinger said “If we have to go on with these damned quantum jumps then I’m sorry that I ever got involved” but with financial transactions of course the same thing happens all the time. For example when you tap your card at a store, the money doesn’t flow out continuously, it just jumps.

In physics the position of a particle is fundamentally indeterminate and is in a sense constructed by the measurement procedure. It’s the same thing in markets. If you put your house up for sale you will have a fuzzy idea of the price but the actual monetary value is only determined at the moment of the sale.

And this is money’s job: it’s a way to collapse value down to number. Theorists often talk about money but one thing that isn’t often emphasized is the most basic feature which is is its connection with numbers. If you look at a US dollar bill for example you see it has a numerical 1 and a word “one” in each corner, so it’s got quite a few ones, and then it’s got “one dollar” down at the bottom, and a big “one” in the middle, and there’s a lot more ones on the other side, so they’re really emphasizing the connection with one, and that is money’s most basic property – that it combines the properties of a real owned thing with a virtual number.

These dual real/virtual properties are reflected in the two main historical theories of money which are bullionism – money is gold and nothing else as JP Morgan said – and chartalism, which is the idea that credit alone is money as Alfred Mitchell-Innes said the next year. But then Bitcoin comes along and on the one hand it seems to be completely virtual, but on the other hand it’s also real as you’ll notice if you happen to lose the hard drive in which your bitcoins are located.

The duality of money is therefore similar to the duality of light. Wave-particle complementarity has been reflected in theories of light that go back millennia – Aristotle thought light was a wave, and Newton thought it was particles, and this bounced back and forth until finally the quantum theory came along and showed that it has properties of both at the same time. It’s the same with money.

In economics we’re used to treating preferences as something like fixed and known objects, adjusted for some cognitive biases, but often our preferences are made up in response to questions which act like a kind of a measurement event. So thoughts and ideas behave in some ways like objects but they’re not classical objects. In physics, Bohr’s theory of wave particle complementarity was actually inspired by the observation from psychologists that we can hold opposite ideas in the mind at the same time in superposition, and in fact it’s these interference terms which play a very important role in quantum cognition as we’ll see.

In physics, particles can mysteriously become entangled so they act as a single system. In the financial system there is a much more direct form of entanglement where financial assets and virtual liabilities have these quantum characteristics of entanglement.

In economics there’s this idea of rational economic man, who is like a kind of robot. The picture which is emerging from quantum social science is a quantum economic person who’s entangled, indeterminate, dynamic, paradoxical and alive. As the philosopher Slavoy Zizek said, a fact rarely noticed is that quantum physics appears to defy our common sense view of material reality, but it seems to apply somewhat better to human reality where the human spirit encounters itself outside itself.

Further reading:

Orrell D and Chlupatý R (2016) The Evolution of Money. New York: Columbia University Press.

What does it mean when we say we want to quantize economics? Well, it doesn’t mean that we’re emulating (or somehow abusing) quantum physics. Quantum mathematics is about information and probabilities and observables and how they relate to each other, as Scott Aaronson said, and money is a form of information that does not behave like a classical object.

Now when we try to apply quantum ideas to areas outside of physics we encounter a lot of obstacles. One is that we’ve constantly been told by famous physicists and mathematicians that quantum mechanics is fundamentally incomprehensible, so we’ll never really understand it. We have these weird phenomena like Schrodinger’s cat which can be alive or dead at the same time. We’re often told that concepts such as superposition and entanglement only apply to the tiniest quantum particles and you’ll never encounter them in your everyday life. And then finally there’s this impenetrable mathematics. But let’s just forget physics for the time being, and think of something much simpler like a coin toss.

If we wanted to model the state of a coin toss where we don’t know the outcome, there are two possible outcomes heads or tails, so we would want a two-dimensional space. And we could represent the state of the coin, if it was a balanced coin, with a diagonal line (see screenshot of video below).

That balance between the two states is really the idea of quantum probability. We can represent heads as an up arrow in Dirac notation, which is used in quantum mechanics, or as a vector (1,0)^{T}. Tails can be a down arrow or a vector (0,1)^{T}. Our superposed state is a mix of these two, a balanced combination of up and down.

In order to get the probabilities we’ll just take the projections. The projection on the horizontal axis, when squared, is going to give the probability of heads, and the probability of tails is going to be the square of the projection onto the vertical axis. The reason we are squaring them is because we want the projections to always add to one, and by the Pythagorean theorem that will be the case.

So the basic difference between classical and quantum probability is that classical probability uses what’s called the 1-norm, so the options are heads or tails, while quantum uses this 2-norm which involves the square. Making this change from a 1-norm to a 2-norm leads to all these different quantum concepts such as superposition, negative probability, interference, and entanglement.

So what do we mean by negative probability. Well if we just flip our state over, mirroring it around the vertical axis, then now we’ve got a negative probability for heads. The size of the probability when we take the norm, because we’re squaring it, is obviously positive again so nothing has really changed, except that when we add probabilities together we can get a plus and a minus canceling out in what are called interference effects.

We can also imagine rotating the state by acting on it by a matrix such as the so-called Hadamard transformation. It has the effect here of rotating clockwise by 45 degrees. Our state is now perfectly aligned with the heads axis, so this is like a case where a coin is in the heads up state and it’s going to stay there. Our projections which were originally on the axes have also rotated around so each of these projections now have a tails component. One is positive and one is negative, so when they add together they cancel out. That’s an example of interference, so there’s now no tails component.

We can also think of more complicated systems. Imagine for example that we had two coins, so now there are four different things that we need to keep track of: there’s heads and tails for one coin, and heads and tells for the other coin. That’s four dimensions which is obviously not easy to draw. But quantum coins can be entangled so for example you might have the only possibilities as heads-heads or tails-tails. So what this means is that just by making this switch from a one norm to a two norm we’re allowing all of these quantum phenomena such as superposition, interference, entanglement and so on.

Further reading:

Aaronson S (2013) Quantum Computing Since Democritus. Cambridge: Cambridge University Press.

The approach which we’ll be following is based more on quantum computing than on quantum mechanics per se, so let’s have a look at how these quantum computers work.

Quantum computers perform their operations using qubits which are in a superposed state, so one big difference between them and classical computers is that they’re inherently probabilistic – you have to run a simulation many times and then sample the output .

The basic mathematical tool we’ll be using is the Hilbert space. This can be viewed as a generalization of Euclidean space, with the difference that there are complex coefficients. Quantum probability involves negative probabilities (in the sense of projections) so we need complex numbers for example when we take the square root of a probability.

The dual state is the complex conjugate of the transpose of , also called the Hermitian conjugate, which is written as .

The inner product between two elements and is denoted , and is analogous to the dot product in a normal vector space, with the difference that the result can again be complex. The norm of is given by which is a non-negative real number.

The outer product is denoted , and is like multiplying a column vector by a row vector.

A unitary matrix is one whose inverse is its conjugate transpose. Unitary matrices preserve probabilities and play a big role in quantum computing, where logic gates are represented by unitary matrices that act on qubits. An example of a gate is the NOT gate which has the effect of flipping a qubit, so if the input is then the output will be . There’s the Hadamard gate which we’ve already seen. It takes a an input of say and puts it into a superposed state. There’s a rotation gate which simply rotates the qubit by an angle , so if the input is then the output will be a superposed state which has the probability of being measured in and of being measured in . The symbol for measurement is the gauge symbol.

Multiple qubits are denoted as a tensor product , so two qubits gives you a column vector with four elements.

A state in a Hilbert space is entangled if it does not factor as a tensor product of the form where and . For example , which has an equal probability of being observed in the state or , is not entangled because it can be written as a tensor product; however the state , which has an equal probability of being observed in the state or , cannot be similarly decomposed, so is entangled.

Gates can act on multiple qubits, an example is the C-NOT gate which here flips the state of the lower qubit depending on the state of the upper qubit: , , , .

The Toffoli gate on the left is similar but acts on three qubits: it flips the state of the first qubit depending on the state of the two control qubits. The figure below shows a quantum circuit with three qubits. The first gate is a Toffoli gate, the middle gate is a C-NOT gate which acts on two qubits, and the last gate on the right is a NOT gate which flips the bottom qubit.

If we analyse the effect of this circuit, we find that , , and so on. The circuit therefore has the effect of incrementing a binary counter.

Another circuit that we’re going to be using quite a lot is the two-qubit figure one shown in the video screenshot below. We have a unitary matrix acting on the first qubit which might for example rotate that qubit around in a particular direction, and the same thing for the lower qubit. Then the top qubit is going to be acting as a control on the second one. As we’ll see this circuit can be used in quantum cognition, where the top qubit can represent a context for a decision, or also in quantum finance where we’re going to be using this to model the debt relationship.

Now that we’ve got most of the mathematical and computational background out of the way, let’s get on to some applications in quantum cognition. Quantum methods were first adopted here because there are problems in behavioral economics which are not easily handled using classical logic, and it’s it’s easier to address them using quantum methods because you can use features such as interference and entanglement.

In quantum cognition we’re going to be modeling mental states using what amount of qubits. So imagine if we started off with an initialized qubit , and then we’re going to act on it by a gate which puts it into a certain state, and then we measure the result which will be a or . This could represent for example different outcomes or decisions.

One of the first applications of quantum cognition was to the order effect. This refers to the phenomenon seen with surveys where the response to questions depends very much on the order in which the questions are asked. One example was a survey done back in the 90s of whether Clinton and Gore were trustworthy and it turned out that the answer was sensitive to the order of the questions. This order effect can be modeled using the quantum formalism as a sequence of projections.

Here the main horizontal and vertical axes correspond to the frame for addressing the Clinton question, and then the the dashed lines represent the Gore axes which are at an angle to that. The initial state shown by the grey line is at angle of about 40 degrees so it’s roughly equally balanced in the Clinton axis for saying Yes or No. So let’s say that the response to the question is Yes, Clinton is trustworthy. Then that is then used as the starting point for the next question about Gore (dotted line) which gives one end point. But if the order of the question is reversed then you project first on to the Gore axis, and then onto the Clinton axis, and you get a different result. The reason is that there’s a kind of interference caused by the shift in the mental frame.

If we go through the exercise of calculating all the different probabilities for the case where the order is Clinton and then Gore we find the table of probabilities looks like this:

We can get exactly the same result if we use the circuit below, which we have already seen. Here we’ve got a rotation gate which in this case rotates by which is preparing our state, and then we’ve got a second qubit which is similarly rotated by the angle which represents this relative shift in mental frame.

More generally this same circuit can be used to simulate any decision B which is influenced by a context A. Some examples include preference reversal where we change our mind depending on the context, the endowment effect where we value something more if we own it than if we don’t own it, and the disjunction effect.

The disjunction effect goes back to a 1992 experiment from Tversky and Shafir. They asked students to imagine that they have a tough exam coming up, and they have an opportunity to buy a vacation to Hawaii at a very good price. Would they take the offer?

In one version of the test they were told the result of the exam. If the result was a pass then 54 percent chose to buy, if the result was fail 57 percent chose to buy. So in each case more than half. But then there was another version in which they were told they will not know the result, and in this case only 32 percent chose to buy. This is odd because the outcomes can only be pass or fail, so you’d expect it to be close to the average or about 55 and a half percent, but no only 32 percent chose to buy. So this is an example of some kind of mental interference effect, where the reasons effectively cancel out.

One way to model this using the quantum method is to do something similar to the dual slit experiment in physics (see video image below), where light from a source gets split into two channels and forms an interference pattern when it recombines. In this case we take the test A which we can either pass (A+) or fail (A-). Then we have the decision to buy a vacation and again it can either split to B+ or B-. The setup is therefore similar to the order effect, where A plays the role of a context (the first question) and B represents the final decision, and again it can be represented using the same simple two-qubit circuit above.

In general, if we suppose that A represents a subjective context, and B represents an objective term such as a numerical payoff, then if we rate the overall attractiveness on a scale 0 to 1, and assume a uniform prior for the various probability terms, it is easily seen that the interference between the subjective and objective factors has an expected value equal to a quarter. Yukalov and Sornette (2015) call this result the quarter law. In the case of the disjunction effect, the interference is negative – if the person knows the outcome, more than half buy the vacation (average 55 percent), but if they are uncertain, this reduces to 32% (a reduction of 23% or about a quarter).

Another application of quantum cognition is to the question of debt. The state of a debt depends on whether or not the debtor is going to default. If the debtor is going to default the debt is worth nothing, if the debtor is not going to default for sure then that debt is worth its face value. An early example of a debt-based form of money was the tallies which were used in England in the middle ages. Suppose that the sovereign wanted to collect a tax debt. A tally stick would then be marked with the the value of the debt, and split down the middle. The sovereign would keep the longer version of the stick which was called the stock and they would hand the debtor the shorter piece of wood which was the foil. When the debt was repaid in the form of produce or whatever then the two sides of the stick were matched and destroyed to extinguish the debt.

We can model this using a version of the same two-qubit circuit. A NOT gate flips the first qubit to symbolize the creation of a debt. For the the debtor we can use the Hadamard gate for simplicity, which puts the qubit into a superposed state . The C-NOT gate has the debtor acting as a control. The outcomes are which is entangled. The debt is either in the state or which means one of these people is going to have the money, the debtor or the creditor, with a 50-50 chance.

Because the tallies represented a claim on a debt, that meant that they had monetary value and could circulate as money objects. So what we think of as a cognitive phenomenon – the decision on whether to default or not – is ultimately what creates the value for money. The sovereign’s job is to convince the debtor that they must not default on the debt, and as we’ll discuss later that involves a certain kind of a work or kind of energy which is really what forms the basis for money.

Further reading:

Busemeyer J and Bruza P (2012) Quantum Models of Cognition and Decision. Cambridge: Cambridge University Press.

Orrell D (2020) Quantum Economics and Finance: An Applied Mathematics Introduction. UK: Panda Ohana.

Wang Z, Solloway T, Shiffrin RS and Busemeyer JR (2014) Context effects produced by question orders reveal quantum nature of human judgments. Proceedings of the National Academy of Sciences 111(26): 9431–6.

Wendt A (2015) Quantum Mind and Social Science: Unifying Physical and Social Ontology. Cambridge: Cambridge University Press.

Yukalov VI and Sornette D (2015) Preference reversal in quantum decision theory. Frontiers in Psychology 6: 1–7.

For an online app which demonstrates the order effect, see here.

One algorithm which is used a lot in quantum computing and also in quantum cognition is the quantum walk, which is a quantum version of the classical random walk model.

The classical random walk was described by the statistician Karl Pearson in a 1905 paper using the example of a drunken man, who takes a step in one direction, another step in a different direction, and so on. The expected distance traveled is seen to grow with the square root of time but “the most probable place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point.”

The same idea had actually already been used by Louis Bachelier in his 1900 thesis on option pricing to argue that an investor’s expected profit or loss was zero because prices move randomly up or down, but the best forecast for an assets future price is its current price. The classical random walk can be modeled using a coin toss, so you toss the coin at each time step and move to the left if the coin is tails and move to the right if it’s heads up. If you repeat this many times, what you find is that many paths end up near the center, but the only way to get to one of the extremes is to get the same result at each coin toss (all heads or all tails) which is very unlikely. So the final probability distribution for position will be concentrated near the middle, and converges to a normal distribution.

The random walk was first used by Bachelier to price options, then it was used in nuclear physics, and then it fed back from there into finance again, and now it’s ubiquitous in finance and it forms the core of all the models used to calculate things like the price of options and financial derivatives in general.

There is a quantum circuit which does the quantum version of a random walk. It consists of a gate we have already seen which increments a counter (interpreted as a step to the right), another which decrements a counter (a step to the left), and in addition there is a Hadamard gate which puts the bottom control qubit into a superposition state. The empty control circle means the control is active when the qubit is in the state rather than .

The part of this lower qubit which is in the state is going to increment the counter, the part which is in is going to decrement it. The result, when repeated in a loop, is a wave function which is evolving over time. The only random part comes at the end when we collapse the wave function down during measurement. The quantum walk explores numerous different paths (see screenshot from video at end of post below) and when these paths meet somewhere they don’t have to add together, instead they can subtract from one another because of interference. The resulting distribution is quite interesting, and it’s very different from the the classical one as seen in the figure below.

The classical random walk (dashed line) converges as mentioned to the normal distribution, while the quantum walk (solid line) has these strange peaks on either side and is actually quite low in the middle because of the interference. This is interesting because in traditional finance theory we’re always thinking about everything being normally distributed, so prices stay close to their starting point. But while this might be objectively fairly accurate, when you think about the mental frame of investors it’s quite different. Most people buying options on a stock aren’t thinking about that stock staying at almost the same price as it is now, instead they’re going to be thinking about it either increasing by a certain amount per year or decreasing by a certain amount per year kind. They might be biased towards one picture or the other but they’re also going to be holding the opposite picture in their head at the same time.

Another difference between the classical and quantum walks is that the latter grows much more quickly, linearly in time rather than with the square-root of time. This feature of the quantum walk is one reason why it’s used a lot in quantum computing for things like search algorithms. Now we can make the the quantum walk act in a more classical fashion if we want by adding decoherence, as occurs when you get some random noise. Decoherence is the enemy of quantum computers because it makes the quantum computers behave in a more classical fashion. A big part of the challenge of quantum computing is to isolate the qubits from their environment so that they don’t get this decoherence. If we want, we can add some decoherence to a quantum walk model quite simply in the algorithm just by adding some random noise at each time step, and if enough is added then the quantum walk converges to the normal distribution. But is this really what we want or do we want to be able to exploit these quantum properties in some way?

In quantum cognition the quantum walk model has been used to model various cognitive effects including signal detection, how people assign ratings to stimuli, and general decision making. In terms of neural processes you can think of it as modeling a kind of a parallel cognitive architecture that involves both cooperative and competitive interactions, which results in interference effects. So as a model of how we think about the future it seems like an improvement over the classical random walk model, which is really more a model of something like a dust particle which doesn’t think about the future. Later on we’re going to be using the quantum walk to model subjective beliefs about future asset price changes as opposed to the actual observed changes which again are not the same thing.

Further reading:

Kempe J (2003) Quantum random walk – an introductory overview. Contemporary Physics 44: 307.

Kvam PD, Pleskac TJ, Yu S and Busemeyer JR (2015) Quantum interference in evidence accumulation. Proceedings of the National Academy of Sciences 112 (34): 10645-10650.

Game theory is very important in economics and it’s interesting to ask how games change when they’re played using quantum logic instead of classical logic.

Quantum game theory seems to have started back in 1999 when quantum computing was still in its infancy. There are two games that we’re going to talk about, one is the prisoner’s dilemma which will be the subject of the next segment, and the first one is the penny flip game. This is an extremely simple game where we have two players A and B. Player A starts by positioning a coin in the up state and player B can choose to flip the coin or not without seeing. Then player A can choose to flip the coin or not without B seeing. Player B then chooses to flip the coin or not and if the coin ends heads up then B wins, otherwise A wins.

We’ll denote the up/down states of the coin as usual , so up will be or (1,0)^{T} and down will be or (0,1)^{T}. The choices to flip or not flip a coin can then be represented by the NOT gate which is flip and the identity I which keeps things the same. A quantum circuit for this game would look like this where we’ve got the various different moves, and measure the final outcome to see whether it’s heads up or tails up.

When you play the game with random moves you find that half the time the result is going to be up, so B wins, and half the time the result is going to be down, so A wins. Each player should therefore win 50 percent of the time. Suppose though that after playing a number of games, player B wins every time.

The situation is a bit like a trick performed by magician Darren Brown, where he takes somebody from the audience onto the stage, and that person holds a a coin behind their back and then holds both hands out in front, and Darren Brown has to guess which hand is holding the coin, and he does this several times in a row. What is going on?

In his case of course the answer is magic, but for our case it’s that player B is cheating by applying the Hadamard transformation (see video screenshot below). This puts the coin in a superposed state of up and down. Whether the coin is then flipped or not flipped by A has no effect on the superposed date. Player B then gets to apply the Hadamard transformation again at the end because they get the last move, and that has the effect of always putting the coin back in the up state. The answer is therefore always up and B wins.

A classical analogy of this would be that player B turns a normal coin by 90 degrees so it’s on its edge. If player A flips the coin or not it is still going to remain on its edge, and then player B turns it by 90 degrees again so it is face up and and B wins the game. Of course in the classical version a coin on its edge has a 50-50 chance of falling either way but the quantum coin can exist in a superposed state. As David Meyer, who invented this game back in 1999, pointed out quantum strategies can be more successful than classical ones and the reason quantum computers promise to have vastly stronger computational power than classical computers is because they can they can play these quantum tricks, and do moves which are simply not possible using classical computers. It is also exactly these moves that seem to play such a key role in human cognition.

Further reading:

Meyer DA (1999) Quantum Strategies. Physical Review Letters 82, 1052.

One game from game theory which has applications to economics is the prisoner’s dilemma. This game involves two imaginary members of a criminal gang who have been arrested for a crime and are being held separately. The prosecutor offers each prisoner a choice: they can testify that the other person committed the crime, or they can remain silent. Their penalties will then be as follows: if only one prisoner defects and betrays the other, that prisoner gets off and the other gets the full five years. If both prisoners remain silent, then they both get two years on a lesser charge. And if both prisoners betray the other, they both get four years.

The payoffs in the the table below represent the number of years saved from the maximum penalty for each case. The strategies are denoted C for cooperate or D for defect. The strategy where they both defect is a Nash equilibrium because if either of them changes their strategy unilaterally then they will be worse off.

Game theory was invented in large part by the mathematician John von Neumann along with the theory of expected utility. He was serving as an advisor to president Eisenhower on the use of the bomb and used results from game theory to argue in favor of the first strike doctrine. After the Soviets developed their own weapons this morphed into another kind of a strategy called mutually assured destruction or MAD.

Game theory does seem to give a rather bleak view of human nature and later experiments showed that in fact people cooperate much more often than rational choice would suggest. In fact in one typical experiment some 37 percent of people chose to cooperate in the game and this fell to 10 if they knew the other person’s strategy. So there’s something missing from the classical version.

In the quantum version each player’s strategy is now going to be encoded by a qubit which we can denote or . The joint strategy therefore exists in a two-dimensional Hilbert space spanned by these basis vectors. For example if the strategy for A is and that of B is then the joint strategy can be denoted . The strategies for the two players will be given by unitary operators and that only act on the player’s own qubit. Our circuit will then be the one shown in the diagram below.

The two qubits initialized in the state are going to be acted on by a unitary matrix whose role is to entangle the two qubits. Each of the qubits are then acted on by the individual strategies. Then we apply the inverse of which ensures that if you put a classical strategy in then you get a classical strategy out at the end. And finally we measure the output state and calculate the payoffs in the same way as in the classical version.

As an example we can choose the entanglement matrix to be a matrix with 1’s on the diagonal, and ‘s on the anti-diagonal, all divided by . After operating with we get which is now entangled. We then calculate the final state, which depends on the individual strategies, and the payoff, which is the penalty for each of the possible outcomes weighted by their probability.

The different possibilities are shown in the table in the video screenshot below. In the classical version (top four rows) the moves are only the identity or the NOT gate which flips from cooperate to defect. But in the quantum version we have extra quantum moves such as the Hadamard transformation. This leads to a new Nash equilibrium which is better than the old one.

So what does this entanglement mean? Well in quantum game theory it is usually thought of as representing some kind of social contract, so it can be used to model things like societal norms or altruism. Another interpretation is to identify player B with the person’s objective outlook, as they think about their own strategy, and player A with their subjective beliefs about what the other person’s going to do. So that context, represented by the top qubit, is going to act as a control. If we choose the entanglement matrix now to be the C-NOT gate, then it has no effect on the initialized input and we can omit it from the the beginning of the circuit. We are therefore left with the same two-qubit entanglement circuit that we’ve been using for projection sequences or decisions in quantum cognition where we have some subjective factors acting on the top qubit and some objective factors acting on the lower qubit.

It follows that we can apply the quarter law to this as before. The uncertainty about the other person’s strategy is going to take the base rate of cooperation, which is 10 percent when you know what the other person’s going to do, and it would add about a quarter to that, which would bring you to around 35 percent expected to cooperate, which is in good agreement with experiment.

Further reading:

Eisert J, Wilkens M and Lewenstein M (1999) Quantum Games and Quantum Strategies. Physical Review Letters 83, 3077 – 3080.

Perhaps the biggest difference between quantum economics and classical economics is that classical economics is based on the idea of a utility function, while quantum economics is based on the idea of propensity which is our probability of transacting. So instead of having a utility curve to model a consumer or supplier we’re going to use a propensity curve which describes the probability of buying or selling at a particular price. Now, let’s say that we start with a probability distribution, how would we get the quantum model from that? Well, a propensity curve describes information which is related to energy through the concept of entropic force, and in order to quantize the system the first step is to derive the entropic dynamics.

This concept of an entropic force was illustrated by the physicist Leo Szilard who imagined a thought experiment involving a simplified heat engine. A single particle is in a chamber at a particular temperature and we’re going to divide the chamber into two parts, denoted 0 and 1, so you can imagine this as a kind of minimal representation of a logical bit where the particle can be in the state 0 or it can be in the state 1. Now, let’s say that we know the particle is in state 1 so we have information about this system. In that case we could move the piston, with no force because it’s not going to encounter the particle, and then we could allow it to open up again and by doing that extract work from the system. The formula for the work done depends on the logarithm of the final volume over the initial volume which in this case is going to be logarithm of 2.

This thought experiment showed that having information means we can get a kind of a force out of it. Conversely a probability distribution can be viewed as the product of a corresponding entropic force. For example if we can say that there’s a likelihood that a particular particle is going to be located within a certain zone but not outside that zone you can imagine there’s a force which is acting on that particle to keep it in that area. And instead of a particle it could be an idea or in economics something like a price estimate. The equation for the entropic force is where is the probability curve. The energy involved in moving from one position to another is again going to involve a logarithm of the final propensity divided by the initial propensity .

In the case of a normal propensity curve the entropic force turns out to be linear. It’s given by the equation s which of course is the equation for a spring system, so you can imagine there’s a sort of spring force which is constraining the probability to stay within a certain range.

So let’s say that we have this entropic force – how can we then quantize the system to get a probabilistic wave function? Well the quantum version of a spring system is just the quantum harmonic oscillator. The ground state is a normal distribution with and the associated mass is which is quite nice because it allows us express mass in terms of the inverse variance.

When you get a buyer and a seller coming together, the propensity curve for the seller is going to be at a higher price and the propensity curve for the buyer is going to be shifted towards lower prices, and the active part of these curves is going to be the parts near the mid price. The probability of a transaction occurring is going to be the product of the individual propensity curves, and that turns out to be a scaled normal curve. The net associated entropic force is just the sum of the buyer and seller forces.

This is a very intuitive way of understanding transactions. The buyer has a certain force pulling down towards the lower price, and the seller is trying to pull it up to a higher price. The probability of transacting is going to scale depending on a number of factors including the spread or the distance between the buyer and the seller optimal prices – if there’s a big gap between them then the probability of a transaction occurring will be lower.

The propensity diagram that we get is in some ways similar to the classical X-shaped supply and demand diagram, but in other ways it is quite different. The curves are now representing a probabilistic propensity so there’s no unique static equilibrium. There’s also no assumption that the market will clear and and so on. Simulations are obviously going to be stochastic because there’s only a probability of transactions occurring. Stochastic models are used very widely in areas such as systems biology where it’s important to take this kind of randomness into account. The video screenshot below shows results from a simple model of a supply chain where the amount of units sold in a particular week fluctuates up and down randomly because the system is inherently stochastic.

One difference between the quantum harmonic oscillator and a classical oscillator is that it has excited states with higher energies. The ground state is a normal curve but at very high energies we get a kind of jagged shape which is a bit reminiscent of the quantum walk. The higher states are not going to be used too much here but just by using the ground state and a couple of the next higher energy states we find that it’s possible to fit things like asset price fluctuations in stock markets very well.

Some of the strongest empirical evidence for quantum effects in the social sciences are shown by threshold effects. Consider our circuit where we have subjective factors which are creating a context which influences the decision . If we assume a uniform prior for the various probability terms then as we’ve seen we can assume that interference will add or subtract 25 percent according to quantum decision theory. The difference between a favorable context and an unfavorable one therefore leads to an expected factor of three difference in propensity. Applying our energy formula, we find the change in energy associated with this gap is which is the base energy of a quantum harmonic oscillator. This energy could correspond for example to the energy needed to convert a non-buyer into a potential buyer in a transaction. So this energy acts as a kind of threshold that needs to be overcome in order for a transaction to take place.

Many cognitive phenomena show a threshold effect. An example is preference reversal, where a switch from one context to another creates a large change in the propensity. Quantum decision theory normalises the objective terms to create a utility function which describes the objective factors. For example a decision between two possible options with associated costs and has the objective utility function and . The preference reversal criterion then holds if (or we can use as before for mathematical convenience).

A related phenomenon is the endowment effect, where people assign a higher value to an object that they own and are selling, than to one that they do not own and are buying. This can be viewed as another example of preference reversal, since the context has changed from selling to buying. The effect has been illustrated in a number of experiments, the best-known being one in which subjects were given a mug and then offered the chance to sell or exchange it. The experimenters found that people demanded more than twice as much in exchange for the mug (a median selling price of $7.12), as they were willing to spend to purchase the mug themselves (median buying price of $2.87). The price ratio is which is again close to e.

Another example is the ultimatum game. Here two subjects are offered an award of say ten dollars, but are given an ultimatum: one must decide how to split the money, and the other has to decide whether to accept the offer. If the offer is rejected, all the money is returned, so they both lose. Standard theory, based on rational utility maximizing behavior, would imply that any offer would be accepted, no matter how low, because it is better than nothing – however the game has been performed in many countries around the world, and the results consistently show that people reject an offer that is overly cheap, with about half of all responders rejecting offers below three dollars. Following the same procedure as above for this threshold gives a utility ratio of 2.33. Again, this could be viewed as a variant of preference reversal, since the context has changed from price setter to price taker.

Such experiments are usually carried out under controlled conditions, however a natural experiment for preference reversal was provided by the observed rate of strategic default during the US housing crisis. According to objective utility maximization, default makes sense if the costs associated with staying in a home exceed the costs associated with selling it – but according to a report from the Federal Reserve, the “median borrower walks away from his home when he is 62 percent underwater” which surprised many observers. Assuming a small downpayment, the cost ratio of finding a replacement at the new lower price, to the cost of staying in the home, is therefore about 2.63, which again is close to e. This threshold effect was important for lenders, because it would have cost an estimated $745 billion to restore all underwater borrowers.

We can also apply this threshold idea to the money objects which we discussed earlier. For a tally stick with a face value of the energy gap in changing from a default probability of 1 in the absence of coercion (so a 100 percent chance that the person is not going to pay the debt) to a smaller probability of default due to coercion is given by the formula where is a frequency parameter. In physics, the frequency of a photon is associated with color and in American idiom the color of someone’s money means proof that someone is going to pay you, so the quantum interpretation gives another angle on that.

In general these threshold effects occur when a minimum energy is needed in order to effect change or to close a deal. They are similar to the photoelectric effect in physics which occurs because a quantum of energy is required in order to dislodge an electron from an atom, with money objects playing the role of photons.