One of the most puzzling findings in decision research field is the disjunction effect. Several studies demonstrate that the disjunction effect exists in a two-person prisoner’s dilemma game. This effect violates the sure-thing principle and cannot be explained by classical decision-making models. During the recent decade, quantum decision making models have been established on the basis of the mathematical structure and methodologies of quantum mechanics. Owing to their special theoretical structures, quantum decision-making models are well suited for explaining the disjunction effect, although these models continue to encounter difficulties. This study aims to overcome the difficulties in existing quantum decision-making models by developing a modified model. To achieve this goal, the deficiencies of the previous models were analyzed. We concluded three deficiencies: 1) Although previous quantum decision-making models can account for the disjunction effect, they can also obtain findings that defy the experimental results and common sense. 2) They cannot explain the disjunction effect with large values in certain experiments (e.g., the experiment of Shafir & Tversky, 1992 ). 3) They cannot properly illustrate the relationship between the utility of decision maker’s pay off and the scale of the disjunction effect. The reasons for these difficulties were investigated. An important reason is that previous quantum decision-making models ignore that the decision maker may consider another’s pay off based on different decision conditions. Another reason is the over-simplicity of the utility function. With the above analyses as basis, we adopted the equate-to-differentiate method to rebuild the quantum decision-making model. In this new model, whether the decision maker considers another’s pay off or not based on a specific decision condition is determined with the equate-to-differentiate method. In addition, the utility function is redefined by value function and hyperbolic tangent transformation. Results revealed that the new quantum decision-making model overcomes the difficulties in previous quantum decision-making models. The proposed model is an integration of heuristic and computational or mathematical models. This ideal model integration deserves much attention and has good theoretical significance and application prospects.
One of the most puzzling findings in decision research field is the disjunction effect. Several studies demonstrate that the disjunction effect exists in a two-person prisoner’s dilemma game. This effect violates the sure-thing principle and cannot be explained by classical decision-making models. During the recent decade, quantum decision making models have been established on the basis of the mathematical structure and methodologies of quantum mechanics. Owing to their special theoretical structures, quantum decision-making models are well suited for explaining the disjunction effect, although these models continue to encounter difficulties. This study aims to overcome the difficulties in existing quantum decision-making models by developing a modified model. To achieve this goal, the deficiencies of the previous models were analyzed. We concluded three deficiencies: 1) Although previous quantum decision-making models can account for the disjunction effect, they can also obtain findings that defy the experimental results and common sense. 2) They cannot explain the disjunction effect with large values in certain experiments (e.g., the experiment of Shafir & Tversky, 1992 ). 3) They cannot properly illustrate the relationship between the utility of decision maker’s pay off and the scale of the disjunction effect. The reasons for these difficulties were investigated. An important reason is that previous quantum decision-making models ignore that the decision maker may consider another’s pay off based on different decision conditions. Another reason is the over-simplicity of the utility function. With the above analyses as basis, we adopted the equate-to-differentiate method to rebuild the quantum decision-making model. In this new model, whether the decision maker considers another’s pay off or not based on a specific decision condition is determined with the equate-to-differentiate method. In addition, the utility function is redefined by value function and hyperbolic tangent transformation. Results revealed that the new quantum decision-making model overcomes the difficulties in previous quantum decision-making models. The proposed model is an integration of heuristic and computational or mathematical models. This ideal model integration deserves much attention and has good theoretical significance and application prospects.
The quantum model has been considered to be advantageous over the Markov model in explaining irrational behaviors (e.g., the disjunction effect) during decision making. Here, we reviewed and re-examined the ability of the quantum belief–action entanglement (BAE) model and the Markov belief–action (BA) model in explaining the disjunction effect considering a more realistic setting. The results indicate that neither of the two models can truly represent the underlying cognitive mechanism. Thus, we proposed a more realistic Markov model to explain the disjunction effect in the prisoner’s dilemma game. In this model, the probability transition pattern of a decision maker (DM) is dependent on the information about the opponent’s action, Also, the relationship between the cognitive components in the evolution dynamics is moderated by the DM’s degree of subjective uncertainty (DSN). The results show that the disjunction effect can be well predicted by a more realistic Markov model. Model comparison suggests the superiority of the proposed Markov model over the quantum BAE model in terms of absolute model performance, relative model performance, and model flexibility. Therefore, we suggest that the key to successfully explaining the disjunction effect is to consider the underlying cognitive mechanism properly.
During the recent decade, quantum decision-making models were established based on the mathematical structure and methodologies of quantum mechanics. Owing to its unique theoretical structure, the quantum decision-making models can be applied to explain problems that violate the classical decision models, especially for judgments under uncertainty and decisions under conflicts. Quantum decision-making models have been used to explain phenomena such as disjunction effect, conjunction fallacy and interference of categorization on decision making which are difficult to account for with classical decision models. A model called quantum question equality has been tested for its accurate prediction of order effects. Being a new research field contributing to analysis of decision making, quantum decision-making models worth further investigations both in theoretical and applicative levels.
In classical game theory, the rational behavior for the player is to make the decision which is approaching Nash equilibrium. The Prisoner's Dilemma, which is a canonical game, is often used to present the rationality. In real experiment in cognitive psychology which were performed by Tversky and Shafir, the statistical data show the existence of the irrational behaviors in reality. The phenomenon is called disjunction effect. To explain why it probably happens, we review the Asano-Khrennikov-Ohya model which is the mathematical modeling of the process of decision making in the game of Prisoner's Dilemma. It applies only the mathematical apparatus of quantum mechanics to the decision making process rather than the quantum physical model. In this paper, we present several numerical simulations for the Asano-Khrennikov-Ohya model together with the graphs of the von Neumann entropy for the solutions. By analyzing the simulation results, we explicitly and numerically present the existence of the irrational behavior for the player which is generated by the Asano-Khrennikov-Ohya 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.
The disjunction effect in human decision making is often taken to show that the classical law of total probability is violated, motivating quantum-like models. We re-examine this claim for the Prisoner's Dilemma disjunction effect. Under the mental-event reading of the opponent-choice events, the conventional classical decision-process model implicitly builds in a certainty-only premise: its standard partition assumptions leave no room for ambiguity, forcing every participant to be certain that the opponent will defect or will cooperate. We relax this by introducing a new classical model in which each participant carries a continuous expectation parameter representing the anticipated likelihood of opponent defection, and the participant pool is partitioned by expectation level; the resulting ambiguity set is precisely the union of the interior expectation bins. In contrast, under the quantum-like event semantics, ambiguous pure states are generic (dense and of full unitarily invariant measure on the unit sphere), so "certainty states" are mathematically exceptional. We prove that an instance of our classical model can realize any empirically observed triple of defection rates across the three information conditions, including strong disjunction-effect patterns, while strictly obeying the classical law of total probability. We further prove that for any such triple produced by a standard quantum-like model of the same experiment, there exists a classical instance reproducing it exactly. In this sense, classical and quantum-like approaches have the same observable-rate expressiveness; their substantive difference lies in how ambiguity is represented and in their respective event semantics, not in a breakdown of classical probability.
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.