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.

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