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 2014 Phys. ScriptaT163, 014010 (doi:10.1088/0031-8949/2014/T163/014010); Khrennikova & Haven 2016 Phil. Trans. R. Soc. A374, 20150106 (doi:10.1098/rsta.2015.0106); Smith et al. 1999 Am. 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-called interference effects of 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 2015 Lect. Notes Comput. Sci.8951, 196-209) and Khrennikova & Haven (Khrennikova & Haven 2016 Phil. 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 the non-double stochasticity of 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'.
Highlights
Decision theories under uncertainty and risk are applied as key building blocks in modern economic and finance models, as well as in public choice theory
We have shown that quantum probability describes well the violation of Bayesian rationality in statistical data of voting in US elections, through the so-called interference effects of probability amplitudes
One of which is the rational mode of information processing and decision formation, which rests upon the canonical formulation of Kolmogorovian probability theory
Summary
Decision theories under uncertainty and risk (expected utility theory under risk by von Neumann [1], subjective expected utility under uncertainty by Savage [2]) are applied as key building blocks in modern economic and finance models, as well as in public choice theory. A body of the literature approached the non-double stochasticity constraints via the so-called generalized versions of Hermitian operators (positive operator valued measures, POVMs; cf [22,23]), where the latter work explores the possible origins of non-additivity of transition probabilities for statistics in decision making and reasoning. If the matrix of transition probabilities would be doubly stochastic, psychological observables could be described by the conventional quantum probability framework, through the usage of Hermitian operators (cf a mathematical representation of such a measurement scheme in §4, equations (4.8)–(4.13)). We suggest a quantum-like (via a generalization of the constraints imposed on the classical POVM quantum measurement scheme) representation for statistical data on voting preferences, with a non-doubly stochastic matrix of transition probabilities. We provide a numerical computation of the devised generalized quantum operators, applied to the voting statistics collected in previous studies (cf. §6)
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