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Studies in systems, decision and control

How to Estimate Unknown Unknowns: From Cosmic Light to Election Polls

Talha Azfar, Vignesh Ponraj, Владик Крейнович, Nguyễn Hoàng Phương

Published Jan 1, 2024• 0 citations
Publisher
DOI
10.1007/978-3-031-63929-6_1

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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.
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’.