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

Polina Khrennikova, Emmanuel Haven

2017
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Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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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 US have shown that voters often engage in 'ticket splitting', by exhibiting contrasting party support in Congressional and Presidential elections, cf. [1], [2], [3]. Such types of preference reversals, cannot be mathematically captured via the formula of total
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... rmula 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 [4] and [2]. 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 POVMs (Positive Operator Valued Measures) by formulating special non-orthonormal bases in respect to these operators. 1 Decision theories under uncertainty and risk (expected utility theory under 2 risk by [5], subjective expected utility under uncertainty by, [6]) are applied 3 as key building blocks in modern economic and finance models, as well as 4 in public choice theory. In the recent decades, starting from the advent 5 of behavioural science and its penetration into the traditional domains of 6 economics and finance, expected utility theories encountered a wave of ex-7 perimental studies that challenged their axiomatic foundations. One of which 8 is the rational mode of information processing and decision formation, which 9 rests upon the canonical formulation of Kolmogorovian probability theory. 10 One of the core axioms of consequential reasoning (i.e. following classical 11 probability that implies the Bayesian mode of updating of new information) 12 namely the Savage's [6] Sure Thing Principle was shown to be not satisfied in 13 DM's (decision makers) preference formations. More specifically, the DM's 14 choice frequency for some option A without any certain information given 15 on a conditional event B, and its negation, B , was very often below the 16 conditional frequencies A|B and A|B , as well as below the total probabil-17 ity expressed via the disjunction of these conditional choice outcomes (given 18 by the formula of total probability, [7]). This type of decision making fal-19 lacy is also known as disjunction effect, which incorporates both sub-and 20 super-additivity of disjunctions in the formula of total probability. Findings 21 showed that disjunction effect was exhibited in a variety of decision making 22 contexts, involving both objective and subjective risk (uncertainty), related 23 to preferences for monetary payoffs [8], [9], [10], [11], [12], as well as voting 24 preferences, [1], [4], [2]. 25 Many of the applications related to the above discussed probabilistic fal-26 lacy, emerging under uncertainty and risk, were explained by generalized 27 models of quantum probability that relaxes the additivity and distributiv-28 ity axioms of classical probability theory, cf. [13], [14], [15], [8], [16], [17], 29 [18],[19],[20], [4], [21], and others. The usage of the generalized quantum 30 framework allowed to capture the observed probability sub-additivity effect 31 in the behaviour of decision makers via the so called interference term (emerg-32 ing due to interference of probability amplitudes, [5]). 33 We can witness that the collection of decision making situations, where 34 disjunction effect was detected is quite broad, and the emergence of non-35 classicality of human reasoning could be well presented through the geometric 36 2 properties of Hilbert space and decision-making projectors herein. However, 37 quantum probability models faced a major constraint, in terms of applicabil-38 ity of the 'genuine' quantum formalism. The so called transition probability 39 matrices for the conditional probabilities collected in decision making exper-40 iments exhibit stochasticity (due to the mutual exclusivity of choices in each 41 context), but violate double-stochasticity (a requirement for usage of Her-42 mitian operators), to represent random variables in a quantum framework 1 . 43 A body of literature approached the non-double stochasticity constraints via 44 the so-called generalized versions of Hermitian operators (POVMs), cf. [23] 45 and [24], where the latter work explores the possible origins of non-additivity 46 of transition probabilities for statistics in decision making and reasoning. 47 Unfortunately, there is no specified algorithm on construction of a POVM 61 observable, corresponding to an arbitrary collection of statistical data (in our 62 analysis and in related studies, we consider probabilities for two incompat-63 ible observables in combination with set of transition probabilities). More-64 over, up to our knowledge, no previous works focused on exploring, whether 65 for some arbitrary statistics, an operational POVM-representation with the 66 conventional Born rule would be possible at all. In this paper we suggest 67 a quantum-like (via a generalization of the constraints imposed on classical 68 POVM quantum measurement scheme) representation for statistical data on 69 voting preferences, with a non-doubly stochastic matrix of transition prob-70 abilities. To be able to describe precisely the decision making statistics, a 71 generalization of the conventional POVM structure is proposed. We relax 72 the condition of additivity up to the unit operator in eq. (3.1), i.e. for the 73 generalized projectors, Q k we can have k Q k = I. 74 To operate with such a generalization of quantum observables beyond the 75 standard POVM formalism, we proceed with a corresponding generalization 76 of the Born rule, in eq. 3.5, where the presence of the denominator differen-77 tiates it from the classical Born rule, given in eq. 3.2. 78 The devised paradigm has a straightforward geometric interpretation. In 79 the presented general framework, the eigenstates of our generalized observ-80 ables form non-orthogonal bases. Orthogonal bases correspond to the special 81 case of double stochasticity in our model and hence, a possibility of Hermi-82 tian operators representation. We also provide a numerical computation of 83 the devised generalized quantum operators, applied to the voting statistics 84 collected in previous studies, cf. section 6. 85 2 A short introduction to Politics and Voting 86 Theory 87 Politics is regarded as a vital area of social science and relies strongly on the 88 assumption of voters' rationality, implying a stability of preferences. People 89 would naturally follow the same principles, i.e. the axioms of rationality, 90 in their political decisions as well as in other situations, such as investment 91 decisions. The rational choice paradigm of modern decision theories was nat-92 urally conveyed into decision making domains other than economics, namely 93 political decisions, in particular voting theory. Political decision making is a 94 special sphere, where humans have to make decisions with far-reaching impli-95 cations for themselves and the society as a whole. The types of decisions can 96 involve ballot-casting in different types of elections from local to governmen-97 4 tal. Similarly, on the party level, the involved parties as political entities have 98 the responsibility to strategically plan their political actions, taking into con-99 sideration all possible consequences. Political decisions are by their nature 100 not as specific as the lottery choices in von Neumann-Morgenstern (vNM) 101 expected utility theories. It can be difficult to associate political decisions 102 with some concrete expected utilities derived from the monetary payoffs. At 103 the same time, some non-monetary benefits can always be attached to the 104 outcomes of different policies that are exercised at the governmental, as well 105 as at a local level. A large body of research in decision theory operates on the 106 assumption that the domain of the utility function can also be used to derive 107

doi:10.1098/rsta.2016.0391
pmid:28971943
fatcat:jvhvtidvbvf7ran5lvc3v3nswe