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arXiv:1811.00045

Quantum-like modeling of the order effect in decision making: POVM viewpoint on the Wang-Busemeyer QQ-equality

Aleksandr Lebedev, Andrei Yu. Khrennikov

Published Oct 31, 2018• 0 citations
arXivHTMLPDF

Abstract

In recent years, quantum mechanics has been actively used in areas outside of physics, such as psychology, sociology, theory of decision-making, game theory, and others. In particular, quantum mechanics is used to explain the paradoxes arising in cognitive psychology and decision making. Wang and Busemeyer invented a quantum model and approach as well as non-parametric equality (so-called QQ-equality), explaining the questions order effect. The primary objective of this note is to test the possibility to expand the Wang-Busemeyer model by considering questions which are mathematically represented by positive operator valued measures. We found that, for such observables, the QQ-equality can be violated. But, we also showed that, in principle, it is possible to reduce expanded model to the original Wang-Busemeyer model by expanding the context of the questions. This version of preprint is aimed to point out to annoying miscalculation in version 1. This miscalculation might mislead a reader who is not experienced in operating with POVMs. Otherwise the main line of construction and reasoning presented in version 1 is right and it can be easily completed by the reader on the basis of version 1 and the correction remark in version 2.

DOI
10.48550/arxiv.1811.00045

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In recent years, quantum mechanics has been actively used in areas outside of physics, such as psychology, sociology, theory of decision-making, game theory, and others. In particular, quantum mechanics is used to explain the paradoxes arising in cognitive psychology and decision making. Wang and Busemeyer invented a quantum model and approach as well as non-parametric equality (so-called QQ-equality), explaining the questions order effect. The primary objective of this note is to test the possibility to expand the Wang-Busemeyer model by considering questions which are mathematically represented by positive operator valued measures. We found that, for such observables, the QQ-equality can be violated. But, we also showed that, in principle, it is possible to reduce expanded model to the original Wang-Busemeyer model by expanding the context of the questions. This version of preprint is aimed to point out to annoying miscalculation in version 1. This miscalculation might mislead a reader who is not experienced in operating with POVMs. Otherwise the main line of construction and reasoning presented in version 1 is right and it can be easily completed by the reader on the basis of version 1 and the correction remark in version 2.
We continue to analyze basic constraints on human's decision making from the viewpoint of quantum measurement theory (QMT). As has been found, the conventional QMT based on the projection postulate cannot account for combination of the question order effect (QOE) and the response replicability effect (RRE). This was an alarm signal for quantum-like modeling of decision making. Recently, it was shown that this objection to quantum-like modeling can be removed on the basis of the general QMT based on quantum instruments. In the present paper we analyse the problem of combination of QOE, RRE, and the famous QQ-equality (QQE). This equality was derived by Busemeyer and Wang and it was shown (in the joint paper with Solloway and Shiffrin) that statistical data from many social opinion polls satisfies it. Now, we construct quantum instruments satisfying QOE, RRE, and QQE. The general features of our approach are formalized with postulates which generalize {\it Wang-Busemeyer} postulates for quantum-like modeling of decision making. Moreover, we show that our model closely reproduces the statistics of the famous Clinton-Gore Poll data with a prior belief state independent of the question order. This model successfully removes the order effect from the data to determine the genuine distribution of the opinions in the Poll. The paper also provides a psychologist-friendly introduction to the theory of quantum instruments - the most general mathematical framework for quantum measurements. We hope that this theory will attract attention of psychologists and will stimulate further applications.
We continue to analyze matching of some basic constraints on human's decision making with quantum theory of measurement. As was found, quantum measurement theory based on the projection postulate does not match with combination of the question order effect (QOE) and the response replicability effect (RRE). This was the alarm signal for quantum-like modeling of decision making. Recently, it was shown that this objection to quantum-like modeling can be removed on the basis of the general measurement theory based on quantum instruments. In the present paper, the problem of combination QOE, RRE with the famous QQ-equality is analyzed. This equality was derived by Busemeyer and Wang and it was shown (in the joint paper with Solloway and Shiffrin) that statistical data from many social opinion polls satisfies it. Now, we construct quantum instruments satisfying QQE, RRE, and QQ-inequality. Moreover, we show that this model reproduces the statistics of the famous Clinton-Gore Poll data almost faithfully with a prior belief state independent of the question order. This model successfully removes the order effect from the data to determine the genuine distribution of the opinion to the Poll. Independently of modeling of the concrete psychological effects, the paper provides psychologist-friendly presentation of the theory of quantum instruments - the most general mathematical framework for quantum measurements. We hope that this theory will attract attention of psychologists and generate further psychological applications.
The order effect is a well-known phenomenon in cognitive psychology which violates specific axioms of the classical probability theory and refers to different probabilities of measurement outcomes (answers to questions) depending on their order. In this research, the order effect was analyzed using quantum probabilistic modeling of cognitive processes. The Quantum Question (QQ) model suggested by Wang and Busemeyer (2013) was tested on a sample of participants that had answered three yes/no questions about a painting, in different orders. For that purpose, three problems were considered. The first problem was to examine the presence of the order effect in participants’ answers and the results showed order effects for all three questions. The second problem was to test data fitness with a model prediction called QQ equality and the data proved to be a good fit to the model. The purpose of the third problem was to examine the dependency of the q value derived from the model on a third question. Given that the QQ equality is a prediction which includes only two yes/no questions, an additional prediction was derived from the model which includes a third question and according to which, the q value is independent of a third question. The results showed a limitation in the model in explaining empirical data. A proposal was made how to expand the theory, so it would include models with three or more questions.
Question order effects are commonly observed in self-report measures of judgment and attitude. This article develops a quantum question order model (the QQ model) to account for four types of question order effects observed in literature. First, the postulates of the QQ model are presented. Second, an a priori, parameter-free, and precise prediction, called the QQ equality, is derived from these mathematical principles, and six empirical data sets are used to test the prediction. Third, a new index is derived from the model to measure similarity between questions. Fourth, we show that in contrast to the QQ model, Bayesian and Markov models do not generally satisfy the QQ equality and thus cannot account for the reported empirical data that support this equality. Finally, we describe the conditions under which order effects are predicted to occur, and we review a broader range of findings that are encompassed by these very same quantum principles. We conclude that quantum probability theory, initially invented to explain order effects on measurements in physics, appears to be a powerful natural explanation for order effects of self-report measures in social and behavioral sciences, too.
Recently, quantum formalism started to be actively used outside of quantum physics: in psychology, decision-making, economics, finances, and social science. Human psychological behavior is characterized by a few basic effects; one of them is the question order effect (QOE). This effect was successfully modeled (Busemeyer–Wang) by representing questions A and B by Hermitian observables and mental-state transformations (back action of answering) by orthogonal projectors. However, then it was demonstrated that such representation cannot be combined with another psychological effect, known as the response replicability effect (RRE). Later, this no-go result was generalized to representation of questions and state transformations by quantum instruments of the atomic type. In light of these results, the possibility of using quantum formalism in psychology was questioned. In this paper, we show that, nevertheless, the combination of the QOE and RRE can be modeled within quantum formalism, in the framework of theory of non-atomic quantum instruments.
The hypothesis that human reasoning obeys the laws of quantum rather than classical probability has been used in recent years to explain a variety of seemingly "irrational" judgment and decision-making findings. This article provides independent evidence for this hypothesis based on an a priori prediction, called the quantum question (QQ) equality, concerning the effect of asking attitude questions successively in different orders. We empirically evaluated the predicted QQ equality using 70 national representative surveys and two laboratory experiments that manipulated question orders. Each national study contained 651-3,006 participants. The results provided strong support for the predicted QQ equality. These findings suggest that quantum probability theory, initially invented to explain noncommutativity of measurements in physics, provides a simple account for a surprising regularity regarding measurement order effects in social and behavioral science.