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IISE Annual Conference & Expo

Decision Engineering Homologies with Quantum Mechanics: EDRM Proximity and the Born Rule

Thomas J. Monroe, Mario G. Beruvides

Published Oct 1, 2024• 0 citations
Publisher

Abstract

Homologies between decision sciences, economics, and physics have been observed and applied for quite some time. Even von Neumann, who established the foundations of mathematics for quantum mechanics and game theory, noted what he considered obvious relationships between these fields of study. However, it is the physical world that is often used as a source of models for application in behavioral, micro, and macroeconomics rather than the other way around. This research seeks to advance prior work that identified an analogy between the Entropy Decision Risk Model's (EDRM) definition of proximity, which relates objective and subjective probability to explain how subjects perceive uncertainty, and the Born Rule of Quantum Mechanics. So far, there has been no derivation for the Born Rule, as it is based only upon an editor's remark in a footnote of Born's foundational 1926 paper, leaving an unanswered question. The goal of this research is to understand homologies and isomorphologies between quantum mechanics and decision engineering that will benefit both disciplines.

DOI
10.21872/2024iise_8211

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Homologies between decision sciences, economics, and physics have been observed and applied for quite some time. Even von Neumann, who established the foundations of mathematics for quantum mechanics and game theory, noted what he considered obvious relationships between these fields of study. However, it is the physical world that is often used as a source of models for application in behavioral, micro, and macroeconomics rather than the other way around. This research seeks to advance prior work that identified an analogy between the Entropy Decision Risk Model's (EDRM) definition of proximity, which relates objective and subjective probability to explain how subjects perceive uncertainty, and the Born Rule of Quantum Mechanics. So far, there has been no derivation for the Born Rule, as it is based only upon an editor's remark in a footnote of Born's foundational 1926 paper, leaving an unanswered question. The goal of this research is to understand homologies and isomorphologies between quantum mechanics and decision engineering that will benefit both disciplines.
Abstract In Everettian quantum mechanics, justifications for the Born rule appeal to self-locating uncertainty or decision theory. Such justifications have focused exclusively on a pure-state Everettian multiverse, represented by a wavefunction. Recent works in quantum foundations suggest that it is viable to consider a mixed-state Everettian multiverse, represented by a (mixed-state) density matrix. Here, we discuss the conceptual foundations for decoherence and branching in a mixed-state multiverse, and extend arguments for the Born rule to this setting. This extended framework provides a unification of ‘classical’ and ‘quantum’ probabilities, and additional theoretical benefits, for the Everettian picture.
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.
FINDING: Quantum cognition models use Born rule probabilities and non-commutative operators to describe human decision dynamics, with a derivation of the Born rule from measurement models. | MATH: Born rule: P(a)=∣⟨a∣ψ⟩∣2 P(a) = |\langle a|\psi\rangle|^2 P(a)=∣⟨a∣ψ⟩∣2; non-commutative operators [A,B]≠0[A,B] \neq 0[A,B]=0; Hilbert space structure; probability amplitudes. | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618), base-60, or crystallographic symmetries found in the provided sources. The Born rule derivation (arXiv:2408.06375v4) uses quantum measurement theory, not harmonic or lattice geometry. | DEPTH: 4 — The Born rule derivation is mathematically rigorous but does not reveal new geometric constants or symmetries. The quantum cognition application is interdisciplinary but mathematically standard (Hilbert space, complex amplitudes). No profound geometric harmony link. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We explain the measure problem (cf. origin of the Born probability rule) in no-collapse quantum mechanics. Everett defined maverick branches of the state vector as those on which the usual Born probability rule fails to hold — these branches exhibit highly improbable behaviors, including possibly the breakdown of decoherence or even the absence of an emergent semi-classical reality. Derivations of the Born rule which originate in decision theory or subjective probability (i.e. the reasoning of individual observers) do not resolve this problem, because they are circular: they assume, a priori, that the observer occupies a non-maverick branch. An ab initio probability measure is sometimes assumed to explain why we do not occupy a maverick branch. This measure is constrained by, e.g. Gleason’s theorem or envariance to be the usual Hilbert measure. However, this ab initio measure ultimately governs the allocation of a self or a consciousness to a particular branch of the wave function, and hence invokes primitives which lie beyond the Everett wave function and beyond what we usually think of as physics. The significance of this leap has been largely overlooked, but requires serious scrutiny.
A non-relativistic quantum mechanical theory is proposed that describes the universe as a continuum of worlds whose mutual interference gives rise to quantum phenomena. A logical framework is introduced to properly deal with propositions about objects in a multiplicity of worlds. In this logical framework, the continuum of worlds is treated in analogy to the continuum of time points; both “time” and “world” are considered as mutually independent modes of existence. The theory combines elements of Bohmian mechanics and of Everett’s many-worlds interpretation; it has a clear ontology and a set of precisely defined postulates from where the predictions of standard quantum mechanics can be derived. Probability as given by the Born rule emerges as a consequence of insufficient knowledge of observers about which world it is that they live in. The theory describes a continuum of worlds rather than a single world or a discrete set of worlds, so it is similar in spirit to many-worlds interpretations based on Everett’s approach, without being actually reducible to these. In particular, there is no splitting of worlds, which is a typical feature of Everett-type theories. Altogether, the theory explains (1) the subjective occurrence of probabilities, (2) their quantitative value as given by the Born rule, and (3) the apparently random “collapse of the wavefunction” caused by the measurement, while still being an objectively deterministic theory.
In physics, at the beginning of the twentieth century it was recognized that some experiments could not be explained by the conventional classical mechanics, but the same could be explained by the newly discovered quantum theory. It resulted in a new mechanics called quantum mechanics that revolutionized scientific and technological developments. Again, at the beginning of the twenty-first century, it is being recognized that some experiments related to the human decision-making processes could not be explained by the conventional classical decision theory but the same could be explained by the models based on quantum mechanics. It is now recognized that we need quantum mechanics in psychology as well as in economics and finance. In this paper we attempt to advance and explain the present understanding of applicability of quantum mechanics to the human decision-making processes. Using the postulates analogous to the postulates of quantum mechanics, we show the derivation of the quantum interference equation to illustrate the quantum approach. The explanation of disjunction effect experiments of Tversky and Shafir (Tversky A, Shafir E (1992) The disjunction effect in choice under uncertainty. Psych. Sci. 3(5):305–309) has been chosen to demonstrate the necessity of a quantum model. Further, to suggest the possibility of application of the quantum theory to the business-related decisions, some terms such as price operator, state of mind of the acquiring firm, etc., are introduced and discussed in context of the merger/acquisition of business firms. The possibility of the development in areas such as quantum finance, quantum management, application of quantum mechanics to the human dynamics related to healthcare management, etc., is also indicated.