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NeuroQuantology

Quantum Decision-making in Newcomb’s Problem: Effect of Reward Size

Nobuye Ishibashi-Ohmura, Taiki Takahashi

Published Jun 22, 2015• 0 citations
Publisher

Abstract

This study experimentally examined people’s decision making in the Newcomb’s problem. We observed that Savage’s sure-thing principle and Kolmogorovian law of probability was violated. Also, the degree of the violation increased as the reward size increased. By adopting quantum decision theory, we further quantified interference effect as a quantum phase factor. The quantum phase also depended on the reward size; i.e., it increased as the reward size in the unknown box in the Newcomb’s problem increased. Future directions in the application of the present theory to studies in quantum decision theory and neuroeconomics are discussed.

DOI
10.14704/nq.2015.13.3.857

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This study experimentally examined people’s decision making in the Newcomb’s problem. We observed that Savage’s sure-thing principle and Kolmogorovian law of probability was violated. Also, the degree of the violation increased as the reward size increased. By adopting quantum decision theory, we further quantified interference effect as a quantum phase factor. The quantum phase also depended on the reward size; i.e., it increased as the reward size in the unknown box in the Newcomb’s problem increased. Future directions in the application of the present theory to studies in quantum decision theory and neuroeconomics are discussed.
Recent work in cognitive psychology has revealed that quantum probability theory provides another method of computing probabilities without falling into the restrictions that classical probability has in regard to modeling cognitive systems and decision-making. This enables the explanation of paradoxical scenarios that are difficult, or even impossible, to explain through classical probability theory. In this work, we perform an overview of the most important quantum models in the literature that are used to make predictions under scenarios where the Sure Thing Principle is being violated (the Quantum-Like Approach, the Quantum Dynamical Model, the Quantum Prospect Theory and Quantum-Like Bayesian Networks). We evaluated these models in terms of three metrics: interference effects, parameter tuning and scalability. The first examines if the analyzed model makes use of any type of quantum interference to explain human decision-making. The second is concerned with the assignment of values to a large number of quantum parameters. The last one consists of analyzing the ability of the models to be extended and generalized to more complex scenarios. We also studied the growth of the quantum parameters when the complexity and the levels of uncertainty of the decision scenario increase. Finally, we compared these quantum models with traditional classical models from the literature. We conclude with a discussion of the manner in which the models addressed in this paper can only deal with very small decision problems and why they do not scale well to larger, more complex decision scenarios.
Two experimental tasks in psychology, the two-stage gambling game and the Prisoner's Dilemma game, show that people violate the sure thing principle of decision theory. These paradoxical findings have resisted explanation by classical decision theory for over a decade. A quantum probability model, based on a Hilbert space representation and Schrödinger's equation, provides a simple and elegant explanation for this behaviour. The quantum model is compared with an equivalent Markov model and it is shown that the latter is unable to account for violations of the sure thing principle. Accordingly, it is argued that quantum probability provides a better framework for modelling human decision-making.
One of the most complex systems is the human brain whose formalized functioning is characterized by decision theory. We present a "Quantum Decision Theory" of decision-making, based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of superposition of composite prospects, including many incorporated intentions, which allows us to explain a variety of interesting fallacies and anomalies that have been reported to particularize the decision-making of real human beings. The theory describes entangled decision-making, non-commutativity of subsequent decisions, and intention interference of composite prospects. We demonstrate how the violation of the Savage's sure-thing principle (disjunction effect) can be explained as a result of the interference of intentions, when making decisions under uncertainty. The conjunction fallacy is also explained by the presence of the interference terms. We demonstrate that all known anomalies and paradoxes, documented in the context of classical decision theory, are reducible to just a few mathematical archetypes, all of which allow the finding of straightforward explanations in the frame of the developed quantum approach.
The application of the mathematical formalism of quantum mechanics to model behavioral patterns in social science and economics is a novel and constantly emerging field. The aim of the so called 'quantum like' models is to model the decision making processes in a macroscopic setting, capturing the particular 'context' in which the decisions are taken. Several subsequent empirical findings proved that when making a decision people tend to violate the axioms of expected utility theory and Savage's Sure Thing principle, thus violating the law of total probability. A quantum probability formula was devised to describe more accurately the decision making processes. A next step in the development of QL-modeling in decision making was the application of Schrödinger equation to describe the evolution of people's mental states. A shortcoming of Schrödinger equation is its inability to capture dynamics of an open system; the brain of the decision maker can be regarded as such, actively interacting with the external environment. Recently the master equation, by which quantum physics describes the process of decoherence as the result of interaction of the mental state with the environmental 'bath', was introduced for modeling the human decision making. The external environment and memory can be referred to as a complex 'context' influencing the final decision outcomes. The master equation can be considered as a pioneering and promising apparatus for modeling the dynamics of decision making in different contexts.
FINDING: Human decision-making under uncertainty violates classical probability axioms and is better modeled by quantum probability theory — specifically non-commutative Hilbert space formalism with interference terms. MATH: - Classical (Kolmogorov) probability: P(A∪B) = P(A) + P(B) − P(A∩B); total probability law holds - Quantum probability: P(A) = |⟨ψ|A|ψ⟩|²; events as projection operators; non-commutativity [A,B] ≠ 0 - Interference term: P(A∪B) = P(A) + P(B) + 2√(P(A)P(B))·cos(θ) — the cos(θ) term has no classical analogue - Order effects: P(A then B) ≠ P(B then A) when projectors don't commute - Disjunction effect (Tversky-Shafir): P(A) > P(A∧B) + P(A∧¬B) — violates sure-thing principle - Social agent model (arXiv 1202.4918): quantum probabilities with entanglement-like correlations between agents; decision probability P_i = Tr(ρ·Π_i) where ρ is a joint density matrix CONNECTION: - The interference term cos(θ) introduces phase angles — the same mathematical structure underlying c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
The expected utility hypothesis and Savage's Sure-Thing Principle are violated in real life decisions, as shown by the Allais and Ellsberg paradoxes. The popular explanation in terms of ambiguity aversion is not completely accepted. As a consequence, uncertainty is still problematical in economics. To overcome these difficulties a distinction between risk and ambiguity has been introduced which depends on the existence of a Kolmogorovian probabilistic structure modeling these uncertainties. On the other hand, evidence of everyday life suggests that context plays a fundamental role in human decisions under uncertainty. Moreover, it is well known from physics that any probabilistic structure modeling contextual interactions between entities structurally needs a non-Kolmogorovian framework admitting a quantum-like representation. For this reason, we have recently introduced a notion of contextual risk to mathematically capture situations in which ambiguity occurs. We prove in this paper that the contextual risk approach can be applied to the Ellsberg paradox, and elaborate a sphere model within our hidden measurement formalism which reveals that it is the overall conceptual landscape that is responsible of the disagreement between actual human decisions and the predictions of expected utility theory, which generates the paradox. This result points to the presence of a quantum conceptual layer in human thought which is superposed to the usually assumed classical logical layer, and conceptually supports the thesis of several authors suggesting the presence of quantum structure in economics and decision theory.