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Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Quantum generalized observables framework for psychological data: a case of preference reversals in US elections

Polina Khrennikova, Emmanuel E. Haven

Published Oct 2, 2017• 5 citations
Publisher

Abstract

Politics is regarded as a vital area of public choice theory, and it is strongly relying on the assumptions of voters’ rationality and as such, stability of preferences. However, recent opinion polls and real election outcomes in the USA have shown that voters often engage in ‘ticket splitting’, by exhibiting contrasting party support in Congressional and Presidential elections (cf. Khrennikova 2014Phys. ScriptaT163, 014010 ( doi:10.1088/0031-8949/2014/T163/014010 ); Khrennikova & Haven 2016Phil. Trans. R. Soc. A374, 20150106 ( doi:10.1098/rsta.2015.0106 ); Smithet al.1999Am. J. Polit. Sci.43, 737–764 ( doi:10.2307/2991833 )). Such types of preference reversals cannot be mathematically captured via the formula of total probability, thus showing that voters’ decision making is at variance with the classical probabilistic information processing framework. In recent work, we have shown that quantum probability describes well the violation of Bayesian rationality in statistical data of voting in US elections, through the so-calledinterference effectsof probability amplitudes. This paper is proposing a novel generalized observables framework of voting behaviour, by using the statistical data collected and analysed in previous studies by Khrennikova (Khrennikova 2015Lect. Notes Comput. Sci.8951, 196–209) and Khrennikova & Haven (Khrennikova & Haven 2016Phil. Trans. R. Soc. A374, 20150106 ( doi:10.1098/rsta.2015.0106 )). This framework aims to overcome the main problems associated with the quantum probabilistic representation of psychological data, namely thenon-double stochasticityof transition probability matrices. We develop a simplified construction of generalized positive operator valued measures by formulating special non-orthonormal bases with respect to these operators. This article is part of the themed issue ‘Second quantum revolution: foundational questions’.

DOI
10.1098/rsta.2016.0391

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Politics is regarded as a vital area of public choice theory, and it is strongly relying on the assumptions of voters’ rationality and as such, stability of preferences. However, recent opinion polls and real election outcomes in the USA have shown that voters often engage in ‘ticket splitting’, by exhibiting contrasting party support in Congressional and Presidential elections (cf. Khrennikova 2014Phys. ScriptaT163, 014010 ( doi:10.1088/0031-8949/2014/T163/014010 ); Khrennikova & Haven 2016Phil. Trans. R. Soc. A374, 20150106 ( doi:10.1098/rsta.2015.0106 ); Smithet al.1999Am. J. Polit. Sci.43, 737–764 ( doi:10.2307/2991833 )). Such types of preference reversals cannot be mathematically captured via the formula of total probability, thus showing that voters’ decision making is at variance with the classical probabilistic information processing framework. In recent work, we have shown that quantum probability describes well the violation of Bayesian rationality in statistical data of voting in US elections, through the so-calledinterference effectsof probability amplitudes. This paper is proposing a novel generalized observables framework of voting behaviour, by using the statistical data collected and analysed in previous studies by Khrennikova (Khrennikova 2015Lect. Notes Comput. Sci.8951, 196–209) and Khrennikova & Haven (Khrennikova & Haven 2016Phil. Trans. R. Soc. A374, 20150106 ( doi:10.1098/rsta.2015.0106 )). This framework aims to overcome the main problems associated with the quantum probabilistic representation of psychological data, namely thenon-double stochasticityof transition probability matrices. We develop a simplified construction of generalized positive operator valued measures by formulating special non-orthonormal bases with respect to these operators. This article is part of the themed issue ‘Second quantum revolution: foundational questions’.
Politics is regarded as a vital area of social science and strongly relies on the assumptions of voters’ rationality and as such the stability of preferences (at least in decisions that are made simultaneously). The phenomenon of divided government that has dominated the US political arena for many election periods over the last 40 years is not consistent with the notion of stable preferences of voters. The authors claim that there is no well defined ranking of their preferences on their utility function of “political preferences.” Recently, this problem has been handled by using the novel approach based on the formalism of quantum mechanics. The theory of quantum decision-making is applicable to the modeling of irrational and biased behavior of voters as well as to the nonseparability of their preferences. Irrationality, biases, and nonseparability lead to a deeper uncertainty than classical probabilistic uncertainty. In particular, quantum probability describes well the violation of Bayesian rationality. The authors model this nonclassical uncertainty in voters’ preferences by using elements of the fundamental quantum information structures such as superposition and entanglement. The chapter is an introduction to the quantum modeling of the behavior of an electorate with unstable preferences (contextual preferences) and the authors use one of the most potent theories of adaptive dynamics, namely, the theory of open quantum systems. In this model a voter’s decision is created in the process of interaction with an information bath in which the mass media play the key role. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We search to devise a new paradigm borrowed from concepts and mathematical tools of quantum physics, to model the decision-making process of the US electorate. The statistical data of the election outcomes in the period between 2008 and 2014 is analysed, in order to explore in more depth the emergence of the so-called divided government. There is an increasing urge in the political literature which indicates that preference reversal (strictly speaking the violation of the transitivity axiom) is a consequence of the so-called non-separability phenomenon (i.e. a strong interrelation of choices). In the political science literature, non-separable behaviour is characterized by a conditioning of decisions on the outcomes of some issues of interest. An additional source of preference reversal is ascribed to the time dynamics of the voters' cognitive states, in the context of new upcoming political information. As we discuss in this paper, the primary source of political information can be attributed to the mass media. In order to shed more light on the phenomenon of preference reversal among the US electorate, we accommodate the obtained statistical data in a classical probabilistic (Kolmogorovian) scheme. Based on the obtained results, we attribute the strong ties between the voters non-separable decisions that cannot be explained by conditioning with the Bayes scheme, to the quantum phenomenon of entanglement. Second, we compute the degree of interference of voters' belief states with the aid of the quantum analogue of the formula of total probability. Lastly, a model, based on the quantum master equation, to incorporate the impact of the mass media bath is proposed.
The US political system in the recent years has been mainly formed by a Divided Government, which is regarded as a consequence of 'non-separability' of voters' preferences. The non- separability phenomenon emerges as a result of strong correlations that the voters establish between their preferences for the Congress and the White House contests. We investigate with help of the empirical data from the Smith et al (1999 J. Polit. Sci. 43 737–764) study what implications the upcoming information (encoded in the observables- questions) has on the non- separability emergence. We show that the informational context of the questions alters the preference frequencies that cannot be captured in a classical probabilistic framework. We attribute the changes to the incompatibility of observables C and P, which correspond to questions being asked. We embed our data in a quantum framework and model the voters' mental state evolution as it is impacted by the operators, to show the non-commutativity of the transition probabilities as a result of the question order effect.
In this contribution, we construct a connection between two quantum voting models presented previously. We propose to try to determine the result of a vote from associated given opinion polls. We introduce a density operator relative to the family of all candidates to a particular election. From an hypothesis of proportionality between a family of coefficients which characterize the density matrix and the probabilities of vote for all the candidates, we propose a numerical method for the entire determination of the density operator. This approach is a direct consequence of the Perron-Frobenius theorem for irreductible positive matrices. We applyour algorithm to synthetic data and to operational results issued from the French presidential election of April 2012.
Recently foundational issues of applicability of the formalism of quantum mechanics (QM) to cognitive psychology, decision making, and psychophysics attracted a lot of interest. In particular, in \cite{DKBB} the possibility to use of the projection postulate and representation of "mental observables" by Hermitian operators was discussed in very detail. The main conclusion of the recent discussions on the foundations of "quantum(-like) cognitive psychology" is that one has to be careful in determination of conditions of applicability of the projection postulate as a mathematical tool for description of measurements of observables represented by Hermitian operators. To represent some statistical experimental data (both physical and mental) in the quantum(-like) way, one has to use generalized quantum observables given by positive operator-valued measures (POVMs). This paper contains a brief review on POVMs which can be useful for newcomers to the field of quantum(-like) studies. Especially interesting for cognitive psychology is a variant of the formula of total probability (FTP) with the interference term derived for incompatible observables given by POVMs. We present an interpretation of the interference term from the psychological viewpoint. As was shown before, the appearance of such a term (perturbing classical FTP) plays the important role in cognitive psychology, e.g., recognition of ambiguous figures and the disjunction effect. The interference term for observables given by POVMs has much more complicated structure than the corresponding term for observables given by Hermitian operators. We elaborate cognitive interpretations of different components of the POVMs-interference term and apply our analysis to a quantum(-like) model of decision making.
Classical models of decision-making under uncertainty, grounded in expected utility theory and Bayesian probability, often fail to account for systematic cognitive anomalies such as preference reversals, order effects, ambiguity aversion, and context-dependent choices. Quantum cognition models offer a novel theoretical framework that applies the mathematical principles of quantum probability—such as superposition, interference, and non-commutativity—to cognitive processes without assuming physical quantum mechanisms in the brain. By representing cognitive states as probabilistic superpositions that collapse through decision “measurements,” these models provide parsimonious explanations for violations of classical rationality. This paper examines how quantum cognition frameworks capture contextual influence, belief updating, and interference effects in uncertain decision environments. Drawing on findings from cognitive psychology and behavioral economics, the study highlights the relevance of quantum cognition for applications in economic choice modeling, legal decision analysis, and artificial intelligence. The analysis suggests that quantum probability offers a robust alternative to classical approaches when modeling human decision-making in complex, ambiguous, and context-sensitive situations.