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Journal of Mathematical Finance

Quantum Mechanics Approach for Risk Aversion, Prudence, and Temperance

Miwaka Yamashita

Published Jan 1, 2024• 3 citations
Publisher

Abstract

The ideas from quantum mechanics (QM) have been used as one of problem-solving methods in the field of economics, especially in game theory and decision theory, starting about “coin flip” and “prisoner’s dilemma” and now days “decision paradoxes”. In this paper, the concept of QM is applied to prudence and temperance. Classically, risk aversion, prudence, and temperance are characterized by the risk attitude toward losses and its volatility (variance), skewness, and kurtosis as well as by utility theory, where derivatives of the utility are related to risk aversion, prudence, and temperance. Here those are treated as decision paradoxes and in the QM model, probabilities of alternatives are tentatively set as unknown and a person’s subjective probabilities toward the alternatives are set as parameterized. Investigating the utility difference before averaging can show the difference among risk aversion, prudence, and temperance. In that sense, a new QM interpretation of risk aversion, prudence, and temperance as opposed to the classical interpretation was founded in the first time.

DOI
10.4236/jmf.2024.141007

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The ideas from quantum mechanics (QM) have been used as one of problem-solving methods in the field of economics, especially in game theory and decision theory, starting about “coin flip” and “prisoner’s dilemma” and now days “decision paradoxes”. In this paper, the concept of QM is applied to prudence and temperance. Classically, risk aversion, prudence, and temperance are characterized by the risk attitude toward losses and its volatility (variance), skewness, and kurtosis as well as by utility theory, where derivatives of the utility are related to risk aversion, prudence, and temperance. Here those are treated as decision paradoxes and in the QM model, probabilities of alternatives are tentatively set as unknown and a person’s subjective probabilities toward the alternatives are set as parameterized. Investigating the utility difference before averaging can show the difference among risk aversion, prudence, and temperance. In that sense, a new QM interpretation of risk aversion, prudence, and temperance as opposed to the classical interpretation was founded in the first time.
Following the ideas of Bohr, Von Neumann, and Benioff, we formulate quantum decision theory (QDT) as the quantum-mechanical theory of measurement for probability operators. QDT captures the effect of superposition of composite prospects, including many incorporated intentions. It is based on the hypothesis that the thought processes of real human beings involved in the definition and analysis of alternative prospects and scenarios do not necessarily separate them according to the recipes of standard probability theory and of classical utility theory. Our QDT formalizes systematically a broader class of decision making processes in which prospects can interact, interfere and remain entangled. The mathematical QDT is developed so as to be applicable to real-life decision making processes. We demonstrate that all known anomalies and paradoxes documented in the context of classical utility theory are reducible to just a few mathematical archetypes, all of which finding straightforward explanations in the framework of QDT. Interference terms, which are essential for resolving the paradoxes, quantify the aversion of human beings to uncertainty and/or to perceived potential loss resulting from their decisions.
The core assumptions of modern economic utility theories such as rationality have been placed under scrutiny by recent studies in decision theory and behavioural economics and fi?nance. The main questions that were explored in these studies could be expressed as follows: do people always make rational choices? Do they make use of the normative rules of traditional logic and probability when dealing with risk and uncertainty? Are there any alternative methods to capture choices and reasoning when individuals are faced with uncertainty and risk? In this thesis I will elaborate on an interdisciplinary application of so-called QL (quantum-like) models to decision-making problems in ?finance and other socio-economic contexts. The core inquiry of this work is related to experimental tests of decision making with the help of investment related questions. Calculus and logic borrowed from quantum physics will be used to develop probabilistic models for the formation and evolution of decisions in ?finance context with the potential to generalize to other social decision- making settings. The main fi?ndings of this thesis can be summarized as follows: individuals exhibit preference reversals and context-dependent behaviour in different informational contexts. In particular, in condition of risk individuals exhibit uncertainty aversion and thus aversion to participating in other investment task, violating the postulates of classical probability theory. The core principle of rational (consequential) behaviour conceived by Savage (1954) [207], that is the "Savage Sure Thing" principle, was shown to be violated. In mathematical language, this violation is equivalent to the violation of the "Law of total probability". Implying contextuality of human judgement, different events cannot be jointly evaluated in a single probability space. Moreover, additional experimental fi?ndings showed that participants exhibited reference dependence in their risk attitudes depending on the preceding gain respective loss. These fi?ndings were statistically signi?cant and a quantum model for accommodating the above effects of contextual behaviour is introduced. A geometric depiction of the experimental fi?ndings was proposed. A so-called `state reconstruction' procedure from statistical data was performed via the inverse Born's rule (1926) [47], allowing the construction of a complete representation of agents' resolution from indeterminacy.
Classical models of decision-making under uncertainty, grounded in expected utility theory and Bayesian probability, often fail to account for systematic cognitive anomalies such as preference reversals, order effects, ambiguity aversion, and context-dependent choices. Quantum cognition models offer a novel theoretical framework that applies the mathematical principles of quantum probability—such as superposition, interference, and non-commutativity—to cognitive processes without assuming physical quantum mechanisms in the brain. By representing cognitive states as probabilistic superpositions that collapse through decision “measurements,” these models provide parsimonious explanations for violations of classical rationality. This paper examines how quantum cognition frameworks capture contextual influence, belief updating, and interference effects in uncertain decision environments. Drawing on findings from cognitive psychology and behavioral economics, the study highlights the relevance of quantum cognition for applications in economic choice modeling, legal decision analysis, and artificial intelligence. The analysis suggests that quantum probability offers a robust alternative to classical approaches when modeling human decision-making in complex, ambiguous, and context-sensitive situations.
In cognitive psychology, some experiments of games were reported [1, 2, 3, 4], and these demonstrated that real players did not use the "rational strategy" provided by classical game theory. To discuss probabilities of such "irrational choice", recently, we proposed a decision-making model which is based on the formalism of quantum mechanics [5, 6, 7, 8]. In this paper, we briefly explain the above model and calculate the probability of irrational choice in several prisoner's dilemma (PD) games.
The applications of techniques from statistical (and classical) mechanics to model interesting problems in economics and finance has produced valuable results. The principal movement which has steered this research direction is known under the name of `econophysics'. In this paper, we illustrate and advance some of the findings that have been obtained by applying the mathematical formalism of quantum mechanics to model human decision making under `uncertainty' in behavioral economics and finance. Starting from Ellsberg's seminal article, decision making situations have been experimentally verified where the application of Kolmogorovian probability in the formulation of expected utility is problematic. Those probability measures which by necessity must situate themselves in Hilbert space (such as `quantum probability') enable a faithful representation of experimental data. We thus provide an explanation for the effectiveness of the mathematical framework of quantum mechanics in the modeling of human decision making. We want to be explicit though that we are not claiming that decision making has microscopic quantum mechanical features.
Abstract We provide here a general mathematical framework to model attitudes towards ambiguity which uses the formalism of quantum theory as a “purely mathematical formalism, detached from any physical interpretation”. We show that the quantum-theoretic framework enables modelling of the Ellsberg paradox, but it also successfully applies to more concrete human decision-making tests involving financial, managerial and medical decisions. In particular, we elaborate a mathematical representation of various empirical studies which reveal that attitudes of managers towards uncertainty shift from ambiguity seeking to ambiguity aversion, and viceversa, thus exhibiting hope effects and fear effects. The present framework provides a promising direction towards the development of a unified theory of decisions in the presence of uncertainty.