The Future of Everything

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

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