Mathematical representation of bias and nudges centered on intangible goods using quantum information theory

arXiv:2411.08046 · physics.soc-ph, quant-ph · Submitted 2024-11-01 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Mathematical representation of bias and nudges centered on intangible goods using quantum information theory".

Kai: A model is proposed that mathematically expresses bias and nudges in relation to intangible goods by utilizing quantum information theory,

Mira: First, who's behind it and why it matters.

Paper summary: Kai: To summarize this paper, "Mathematical representation of bias and nudges centered on intangible goods using quantum information theory," the central thesis is that the relationship between bias and nudges can be mathematically expressed using quantum information theory, particularly when designing individualized nudges.

Mira: The paper claims to propose a model that takes into account the environment for intangible goods, defining a model of bias and nudges based on the value function of customer satisfaction which is subject to uncertainty because of the subjectivity in customer evaluations.

Lev: So, it's framing subjective experience as a mathematical system where uncertainty is inherent from the start. That's a big conceptual leap for modeling anything physical.

Kai: They then define an index of nudges from the mathematical properties of that value function derived from this economic model, suggesting that welfare gets impaired by bias through the structure of the gross social surplus derived as a social welfare function.

Mira: Furthermore, they argue that this mathematical structure of the gross social surplus can be made larger than it is in standard economics due to the inclusion of that emotional satisfaction term.

Lev: That claim about making the surplus larger is interesting, but I need to know if that's a theoretical upper bound or something we could actually observe experimentally.

Kai: The paper also notes that by defining decision utility and experienced utility through specific mathematical expressions, they establish an internality which is then mathematically expressed as the difference between those two utilities.

Mira: They define decision utility using equation two from standard economics, while experienced utility in behavioral economics includes a complex term with the cosine function to account for subjective perceptions of the environment and product quality.

Lev: That cosine term seems like it's where we need to worry about mapping classical physics onto quantum states; how do those phase differences translate into tangible system states?

Kai: When correlations exist between perceptions and their respective environments after provision or receipt, they further express the weight for emotional satisfaction, p(↑), by considering entangled states between the perception of intangible goods and the perception of the environment rules.

Mira: This entanglement is mathematically set up using specific POVMs, which results in an expression for pp(↑) that incorporates terms related to quantum probability distributions.

Lev: Entangled states are fragile things; any interaction with the environment could easily break that entanglement before we even get to calculating those probability distributions.

Kai: So, they're using this entire framework to connect behavioral concepts directly into the mathematical structure derived from quantum information theory for nudges.

Conclusion: Kai: Looking at this paper, "Mathematical representation of bias and nudges centered on intangible goods using quantum information theory," the main takeaway is that we can mathematically design customized customer experiences by enabling the prediction of how much behavioral improvement happens due to nudges.

Mira: The authors suggest that the relationship between bias and nudges linked to intangible goods can be represented using quantum information theory, which allows for the mathematical design of these experiences.

Lev: From a practical standpoint, I see it suggesting that heuristics could be used to derive solutions that are roughly correct, resulting in high levels of happiness or satisfaction.

Kai: They suggest that the environment rules themselves can be formulated using quantum information theory, which means we could potentially achieve measurement and control of the provider’s and the recipient’s perception using quantum computers.

Mira: This points toward a future where heuristics might be used to derive solutions that are roughly correct, leading to high levels of happiness or satisfaction in these intangible goods contexts.

Lev: If we look at the limitations they state, they flag that nudges are context-dependent and that bottlenecks vary across individuals due to their heterogeneity, which means the model's external validity needs more experimental validation.

Kai: So the paper provides a mathematical structure for this interaction, even if it requires further empirical testing to confirm how well those quantum concepts actually map onto real customer behavior.

Misao Fukuda

physics.soc-ph, quant-ph

Submitted: 2024-11-01

Updated: 2026-10-04

Comments: 20 pages, 5 figures; Added definitions of terms and variables, theoretical background, empirical methods for variables and discussion

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 65/100

The gist: A model is proposed that mathematically expresses bias and nudges in relation to intangible goods by utilizing quantum information theory, suggesting this approach can enhance the mathematical design

Key concepts

Hilbert Space: This is a mathematical space used to represent the possible states or values of customer satisfaction and dissatisfaction. It provides the foundational structure where all relevant aspects of the intangible good's value can be mathematically described, allowing for complex relationships between different perceptions.
Density Matrices: These matrices are used to model how the perception of an intangible good interacts with its environment (the rules). They help quantify correlations between what a customer perceives and the surrounding context, which is crucial for understanding how nudges affect behavior.
Entangled States: When perceptions of the intangible good and its environment are correlated after provision or receipt, they can be modeled as entangled states. This quantum concept captures deep interactions between the product's quality perception and the rules governing its use, providing a more nuanced view of behavioral change.
Gross Social Surplus (TS): This measures the total welfare generated by both customers and employees. The model shows that including emotional satisfaction terms, which represent bias, can result in a gross social surplus larger than what is predicted by standard economic models.

Terminology

Entanglement — 4× in this paper · explained in 3 episodes
Hilbert Space — 4× in this paper · explained in 2 episodes
Generalization — 2× in this paper · explained in 13 episodes
Trace — 2× in this paper · explained in 2 episodes

Summary

A model is proposed that mathematically expresses bias and nudges in relation to intangible goods by utilizing quantum information theory, suggesting this approach can enhance the mathematical design of customized customer experiences.

Mathematical Framework for Intangible Goods

The study establishes a mathematical representation of intangible goods within a Hilbert space, where the value function of customer satisfaction is expressed using terms for emotional satisfaction and dissatisfaction. Specifically, Smerit represents customer satisfaction and Sdemerit represents customer dissatisfaction, defined by equations (1) through (5). Decision utility in standard economics is defined by equation (2), while experienced utility in behavioral economics incorporates a complex term involving the cosine function to account for subjective perceptions of the environment and product quality. The internality, denoted as Λ, is then mathematically expressed as the difference between decision utility and experienced utility (equation 6).

Incorporating Environment and Heterogeneity

The perception of intangible goods is modeled by considering its interaction with the environment (rules) through density matrices. The weight for emotional satisfaction, p(↑), which affects the repeat rate, is expressed using a trace operation over the tensor product of provider and recipient perceptions, as shown in equation (7). When correlations exist between perceptions and their respective environments after provision or receipt, p(↑) is further expressed by considering entangled states between the perception of intangible goods and the perception of the environment (rules). This entanglement is mathematically set up using specific POVMs, leading to an expression for pp(↑) that incorporates terms related to quantum probability distributions (equation 13).

The Mathematical Model of Nudges

The study connects behavioral concepts to the mathematical structure derived from quantum information theory. A customer repeating an action due to loss aversion is linked to averting losses resulting from lost operational rationality due to biases in preferences and probabilistic judgments. Within the standard economic framework, the trade-off for optimal nudges involves correcting biased individuals’ internalities and the psychological costs of nudges to all individuals (Jimenz-Gomez, 2018). For this study's model, a rapid increase in bias leads to perceived loss due to a rapid decrease in customer satisfaction, then to an increase in the repeat rate through loss aversion, and finally to greater effects of nudges (Figure 2).

Gross Social Surplus and Welfare

The gross social surplus (T S) is derived based on the first-order conditions for maximizing quasi-linear utility. This surplus is expressed as the sum of customer satisfaction and employee satisfaction, which includes an emotional satisfaction term: T S = CS + PS = pr(↑) log x pr(↑) + 1 + pp(↑) log x pp(↑) + 1. The presence of the emotional satisfaction term introduces bias and reduces welfare, and the resulting gross social surplus is stated to be larger than that in standard economics due to the presence of the emotional satisfaction term.

Design of Customized Customer Experiences

The design of individually customized customer experiences is mathematically equivalent to determining the coefficients of underlying factors in individual customers’ subjective perception (ρ) of receiving intangible goods. This design aims to maximize the value function for customer satisfaction by aligning these coefficients with the underlying factors and their coefficients for customers’ perception (P(↑)) of the process and, in some cases, in accordance with nudges (Figure 3). The findings indicate that Designing customized customer experiences is possible by mathematically expressing a default effect of nudges.

Conclusion and Implications

The new findings suggest that the relationship between bias and nudges linked to intangible goods can increase the feasibility of mathematical design of customer experiences by enabling the prediction of the rate of behavioral improvement due to nudges. The approach based on quantum information theory can potentially be used for an economic model in the theory of design of intangible goods and as a mathematical model for designing customer experiences. Furthermore, The environment (rules) can be formulated based on quantum information theory, thereby realizing measurement and control of the provider’s and the recipient’s perception using quantum computers. This suggests that heuristics can be used to derive solutions that are roughly correct, resulting in high levels of happiness/satisfaction. The ability to control biases through partial extraction of context (generalization of the environment or rules) is highlighted as a potential mechanism for control.

Summary

The relationship between bias and nudges linked to intangible goods can be mathematically represented using quantum information theory, allowing for the mathematical design of customized customer experiences by enabling prediction of behavioral improvement rates. The resulting gross social surplus model is capable of being larger than in standard economics due to the inclusion of emotional satisfaction terms. This framework suggests that environmental rules can be formulated using quantum information theory for measurement and control, potentially leading to high economic benefits in designing intangible goods experiences.

The gist

A mathematical model is proposed that expresses bias and nudges in relation to intangible goods by utilizing quantum information theory, suggesting this approach can enhance the mathematical design of customized customer experiences.

Improvements for AI systems

Here are specific improvements for AI systems based on the concepts presented in this paper:

The AI system can develop a sophisticated, mathematically grounded model for designing personalized nudges for intangible goods (e.g., digital services, subscription models, or abstract content).
The system can optimize customer experience design by mathematically quantifying and balancing the trade-off between correcting inherent customer biases (internalities) and minimizing psychological costs imposed on the user population through interventions.
The AI can predict the rate of behavioral improvement resulting from a nudge by mathematically modeling how bias, environment (context/rules), and customized experiences interact within a quantum information framework (specifically using entanglement in Hilbert space).
The system can move beyond traditional rational agent assumptions by utilizing quantum decision theory to model and understand irrational or biased decision-making processes in users interacting with intangible goods.
The AI can design individually customized nudges by mathematically determining the optimal coefficients for subjective perception models that maximize customer satisfaction, effectively tailoring the experience to each individual's unique state of bias and environmental context.
The system can quantify the Gross Social Surplus generated by a nudge, showing how it exceeds standard economic predictions due to the inclusion of emotional satisfaction terms derived from quantum information theory. This allows for better justification and design of these interventions.
The AI can formulate the environment or rules associated with intangible goods using quantum information theory, potentially leading to novel methods for measuring and controlling provider/recipient perceptions through advanced computational techniques (quantum computers).