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

March 11, 2021

March 10, 2021

March 9, 2021

March 8, 2021

March 7, 2021

March 6, 2021

March 5, 2021

March 4, 2021

QEF08 – Quantizing Propensity

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 F \left( x \right) = \gamma \frac{P'(x)}{P(x)} where P is the probability curve. The energy involved in moving from one position x_1 to another x_2 is again going to involve a logarithm of the final propensity divided by the initial propensity \Delta E = \gamma \log \frac{P(x_2)}{P(x_1)}.

In the case of a normal propensity curve the entropic force turns out to be linear. It’s given by the equation s F \left( x \right) = \frac{-\gamma \left(x - \mu \right)}{\sigma^2} 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 \gamma = \frac{\hbar \omega}{2} and the associated mass is \gamma = \frac{\hbar}{2 \omega \sigma^2} 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.

Buyer, seller, and joint (shaded) propensity curves

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.

Further reading:

Orrell D (2020) A Quantum Model of Supply and Demand. Physica A 539: 122928.

Orrell D, Houshmand M (2021) Quantum propensity in economics. Arxiv.org/abs/2103.10938.

For an online app demonstrating the quantum supply and demand algorithm, see here.

Previous: QEF07 – The Prisoner’s Dilemma

Next: QEF09 – Threshold Effects in Quantum Economics

Playlist: Quantum Economics and Finance

March 3, 2021

QEF09 – Threshold Effects in Quantum Economics

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 A which are creating a context which influences the decision B. 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 \Delta E = \frac{\hbar \omega}{2} \log 3 \approx \frac{\hbar \omega}{2} 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 x_1 and x_2 has the objective utility function f\left(x_1\right) = \frac{x_2}{x_1+x_2} and f\left(x_2\right) = \frac{x_1}{x_1+x_2}. The preference reversal criterion then holds if \frac{x_2}{x_1} > 3 (or we can use e 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 \frac{7.12}{2.87} \approx 2.5 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 x_0 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 p due to coercion is given by the formula \Delta E \approx \frac{\hbar a x_0}{2} \log \frac{1}{p} = \frac{\hbar \omega}{2} where \omega = a x_0 \log \frac{1}{p} 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.

Further reading:

Orrell D (2021) The Color of Money: Threshold Effects in Quantum Economics. Quantum Reports 3(2), 325-332.

Orrell D (2021) Quantum Financial Entanglement: The Case of Strategic Default. (Under review.)

Previous: QEF08 – Quantizing Propensity

Next: QEF10 – A Quantum Option Pricing Model

Playlist: Quantum Economics and Finance

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