# The Future of Everything

## 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.

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

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Next: QEF10 – A Quantum Option Pricing Model

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