Physics of risk and uncertainty in quantum decision making

The Quantum Decision Theory, developed recently by the authors, is applied to clarify the role of risk and uncertainty in decision making and in particular in relation to the phenomenon of dynamic inconsistency. By formulating this notion in precise mathematical terms, we distinguish three types of inconsistency: time inconsistency, planning paradox, and inconsistency occurring in some discounting effects. While time inconsistency is well accounted for in classical decision theory, the planning paradox is in contradiction with classical utility theory. It finds a natural explanation in the frame of the Quantum Decision Theory. Different types of discounting effects are analyzed and shown to enjoy a straightforward explanation within the suggested theory. We also introduce a general methodology based on self-similar approximation theory for deriving the evolution equations for the probabilities of future prospects. This provides a novel classification of possible discount factors, which include the previously known cases (exponential or hyperbolic discounting), but also predicts a novel class of discount factors that decay to a strictly positive constant for very large future time horizons. This class may be useful to deal with very long-term discounting situations associated with intergenerational public policy choices, encompassing issues such as global warming and nuclear waste disposal.

💡 Research Summary

The paper applies the recently developed Quantum Decision Theory (QDT) to elucidate how risk and uncertainty shape human choices, with a particular focus on dynamic inconsistency. By casting decision problems in the language of Hilbert spaces, QDT represents each prospect as a state vector and each choice as a measurement operator; the probability of selecting a prospect is given by the squared modulus of a complex amplitude. Crucially, the phase of this amplitude encodes “interference” between competing prospects, a feature absent from classical probability theory.
The authors formalize three distinct forms of dynamic inconsistency. First, time inconsistency occurs when a plan deemed optimal at one moment loses its optimality as time passes. Classical models explain this through non‑exponential discounting, but QDT attributes it to time‑dependent phase shifts and amplitude attenuation, which alter the interference pattern and thus the selection probabilities.
Second, the planning paradox describes the empirical observation that individuals often set long‑term goals yet later abandon them, contradicting the classical utility assumption of internal consistency. Within QDT, the paradox emerges naturally: the initial mental representation of a goal interferes destructively with the representation of immediate alternatives, producing a phase difference that suppresses the amplitude associated with the long‑term prospect. This quantum‑like suppression accounts for the systematic deviation without invoking ad‑hoc preference reversals.
Third, discounting inconsistency concerns the variety of discount functions used to model intertemporal choice. Traditional economics employs exponential or hyperbolic forms, both of which drive future utilities to zero as the horizon extends. To derive a more general class, the authors introduce a self‑similar approximation method. Starting from the evolution equation for the complex amplitudes, they construct a renormalization‑type hierarchy that respects the self‑similarity of the decision process across time scales. Solving this hierarchy yields a spectrum of discount factors: the familiar exponential and hyperbolic cases appear as special limits, while a novel family emerges that converges to a strictly positive constant for arbitrarily distant futures. This “baseline‑value” discounting implies that even infinitely remote outcomes retain a non‑vanishing weight, a feature especially relevant for intergenerational policy issues such as climate mitigation, nuclear waste management, and long‑term fiscal planning.
The paper also details a practical methodology for estimating the parameters of the QDT model. By fitting the self‑similar evolution equations to experimental or survey data, researchers can extract the amplitude phases and thereby quantify the degree of interference among prospects. This provides a systematic way to calibrate the quantum‑theoretic description of risk‑laden choices.
In the concluding discussion, the authors argue that the quantum‑theoretic perspective offers a unified explanation for phenomena that classical utility theory treats as anomalies. The interference‑based mechanism captures both the planning paradox and the nuanced shape of discount curves, while the self‑similar approximation supplies a mathematically rigorous route to generate realistic discount functions for very long horizons. They suggest that future work should focus on empirical validation across diverse decision contexts and on integrating QDT with existing behavioral models to enrich policy analysis. Overall, the study positions QDT as a promising framework for bridging the gap between the physics of complex systems and the psychology of human decision making, especially when risk, uncertainty, and long‑term consequences are at stake.