The sure thing principle and the law of total probability are basic laws in classic probability theory. A disjunction fallacy leads to the violation of these two classical probability laws. In this paper, a new quantum dynamic belief decision making model based on quantum dynamic modelling and Dempster-Shafer (D-S) evidence theory is proposed to address this issue and model the real human decision-making process. Some mathematical techniques are borrowed from quantum mathematics. Generally, belief and action are two parts in a decision making process. The uncertainty in belief part is represented by a superposition of certain states. The uncertainty in actions is represented as an extra uncertainty state. The interference effect is produced due to the entanglement between beliefs and actions. Basic probability assignment (BPA) of decisions is generated by quantum dynamic modelling. Then BPA of the extra uncertain state and an entanglement degree defined by an entropy function named Deng entropy are used to measure the interference effect. Compared the existing model, the number of free parameters is less in our model. Finally, a classical categorization decision-making experiment is illustrated to show the effectiveness of our model.
The sure thing principle and the law of total probability are basic laws in classic probability theory. A disjunction fallacy leads to the violation of these two classical probability laws. In this paper, a new quantum dynamic belief decision making model based on quantum dynamic modelling and Dempster-Shafer (D-S) evidence theory is proposed to address this issue and model the real human decision-making process. Some mathematical techniques are borrowed from quantum mathematics. Generally, belief and action are two parts in a decision making process. The uncertainty in belief part is represented by a superposition of certain states. The uncertainty in actions is represented as an extra uncertainty state. The interference effect is produced due to the entanglement between beliefs and actions. Basic probability assignment (BPA) of decisions is generated by quantum dynamic modelling. Then BPA of the extra uncertain state and an entanglement degree defined by an entropy function named Deng entropy are used to measure the interference effect. Compared the existing model, the number of free parameters is less in our model. Finally, a classical categorization decision-making experiment is illustrated to show the effectiveness of our model.
Categorization is necessary for many decision making tasks. However, the categorization process may interfere the decision making result and the law of total probability can be violated in some situations. To predict the interference effect of categorization, some model based on quantum probability has been proposed. In this paper, a new quantum dynamic belief (QDB) model is proposed. Considering the precise decision may not be made during the process, the concept of uncertainty is introduced in our model to simulate real human thinking process. Then the interference effect categorization can be predicted by handling the uncertain information. The proposed model is applied to a categorization decision-making experiment to explain the interference effect of categorization. Compared with other models, our model is relatively more succinct and the result shows the correctness and effectiveness of our model.
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.
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.
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
Dempster-Shafer evidence theory is an efficient mathematical tool to deal with uncertain information. In this theory, basic probability assignment (BPA) is the basic structure for the expression and inference of uncertainty. In this paper, quantum entanglement involved in Dempster-Shafer evidence theory is studied. A criterion is given to determine whether a BPA is in an entangled state or not. Based on that, the information volume involved in a BPA is discussed. The discussion shows that a non-quantum strategy (or observation) can not obtained all information contained in a BPA which is in an entangled state.
How can we use quantum theory to model other phenomena of interest to researchers in cognition and decision making? Quantum theory is not easy for researchers in cognition and decision making to accept. In fact, quantum mechanics was not easy for physicists to accept either, but it was forced on them by several paradoxical findings that could not be explained using classical physics. We have a similar problem in cognition and decision making – there are numerous paradoxical findings that just seem irrational according to classic probability theory. For example, under some conditions, people judge the probability of event A and B to be greater than the probability of event B , which is called the conjunction fallacy (Tversky & Kahneman, 1983). Also, under the same conditions, they judge the probability of A or B to be less than the probability of event A (Carlson & Yates, 1989), which is called the disjunction fallacy. In this chapter we examine how quantum probability theory explains these and other puzzling results from human probability judgment research. This chapter has two main parts. In the first part, we use a quantum model to derive qualitative predictions for conjunction errors, disjunction errors, and other closely related findings. The first section provides a general set of predictions that do not depend on specific assumptions about the features used to represent events, and the predictions are parameter free. In the second part, we examine the quantitative predictions of the quantum model for a Bayesian inference task, which we use to explain order effects on inference.