Keywords

1 Introduction

Electoral periods are favorable to opinion polls. We keep in mind that opinion polls are intrinsically complex (see e.g. Gallup [14]) and give an approximates picture of a possible social reality. They are traditionally of two types: popularity polls for various outstanding political personalities and voting intention polls when a list of candidates is known. We have two different informations and to construct a link between them is not an easy task. In particular, the determination of the voting intentions is a quasi intractable problem! Predictions of votes classically use of so-called “voting functions”. Voting functions have been developed for the prediction of presidential elections in the United States. They are based on correlations between economical parameters, popularity polls and other technical parameters. We refer to Abramowitz [1], Lewis-Beck [22], Campbell [10] and Lafay [20].

We do not detail here the mathematical difficulties associated with the question of voting when the number of candidates is greater than three [2, 6, 9]. They conduct to present-day researches like range voting, independently proposed by Balinski and Laraki [3, 4] and by Rivest and Smith [24, 25]. It is composed by two steps: grading and ranking. In the grading step, all the candidates are evaluated by all the electors. This first step is quite analogous to a popularity investigations and we will merge the two notions in this contribution. The second step of range voting is a majority ranking; it consists of a successive extraction of medians.

In this contribution, we adopt quantum modelling (see e.g. Bitbol et al. [5] for an introduction), in the spirit of authors like Khrennikov and Haven [16, 17], La Mura and Swiatczak [21] and Zorn and Smith [28] concerning voting processes. Moreover, Wang et al. [27] present a quantum model for question order effects found with Gallup polls. The fact of considering quantum modelling induces a specific vision of probabilities. We refer e.g. to the classical treatise on quantum mechanics of Cohen-Tannoudji et al. [8], to the so-called contextual objectivity proposed by Grangier [15], or to the elementary introduction proposed by Busemeyer and Trueblood [7] in the context of statistical inference.

This contribution is organized as follows: we recall in Sects. 2 and 3 two quantum models for the vote developed previously [11, 12] and a first tentative [13] to connect these two models (Sect. 4). In Sect. 5, we develop the main idea of this paper. We construct a link between opinion polls and voting. This idea is tested numerically in Sects. 6 and 7 for synthetic data and a “real life” election.

2 A Fundamental Elementary Model

In a first tentative [11], we have proposed to introduce an Hilbert space \(\, V_\varGamma \,\) formally generated by the candidates \( \, \gamma _j \in \varGamma . \, \) In this space, a candidate \( \, \gamma _j \,\) is represented by a unitary vector \( \, | \, \gamma _j \! > \,\) and this family of \(n\) vectors is supposed to be orthogonal. Then an elector \( \, \ell \,\) can be decomposed in the space \(\, V_\varGamma \,\) of candidates according to

$$\begin{aligned} \displaystyle \displaystyle | \, \ell \! > \,\,=\, \, \sum _{j=1}^n \, \theta _j \,\, | \, \gamma _j \! > \, . \end{aligned}$$
(1)

The vector \( \, | \, \ell \! > \, \in V_\varGamma \,\) is supposed to be a unitary vector to fix the ideas. According to Born’s rule, the probability for a given elector \(\, \ell \, \) to give his voice to the particular candidate \( \, \gamma _j \, \) is equal to \( \, | \, \theta _j \, | ^2 . \, \) The violence of the quantum measure is clearly visible with this example: the opinions of an elector \( \, \ell \,\) never coincidate with the program of any candidate. But with a voting system where an elector has to choice only one candidate among \(n\), his social opinion is reduced to the one of a particular candidate.

3 A Quantum Model for Range Voting

Our second model [12] is adapted to the grading step of range voting [3, 24]. We consider a grid \(\mathrm G\) of \(m\) types of opinions as one of the two following ones. We have \( \, m=5 \,\) for the first grid (2) and \( \, m=3 \,\) for the second one (3):

$$\begin{aligned}&++ \,\, \,\succ \, \,\, + \,\,\, \succ \,\,\, 0 \,\,\, \succ \,\,\, - \,\,\, \succ \,\,\, {-}{-}\end{aligned}$$
(2)
$$\begin{aligned}&+ \,\,\, \succ \,\,\, 0 \,\,\, \succ \,\,\, - \,. \end{aligned}$$
(3)

These ordered grids are typically used for popularity polls. We assume also that a ranking grid like (2) or (3) is a basic tool to represent a social state of the opinion. We introduce a specific grading space \( \, W_\mathrm{G} \,\) of political appreciations associated with a grading family \(\mathrm G\). The space \( \, W_\mathrm{G} \,\) is formally generated by the \(m\) orthogonal vectors \( \, | \, \zeta _i \! > \,\) relative to the opinions. Then we suppose that the candidates \( \, \gamma _j \,\) are now decomposed by each elector on the basis \( \, | \, \zeta _i \! > \,\) for \( \, 1 \le i \le m\):

$$\begin{aligned} \displaystyle \displaystyle | \, \gamma _j \! > \,=\, \sum _{i=1}^m \, \alpha _j^i \, \, | \, \zeta _i \! > \, , \quad \gamma _j \in \varGamma \, , \quad 1 \le j \le n \, . \end{aligned}$$
(4)

Moreover the vector \( \, | \, \gamma _j \! > \,\) in (4) is supposed to be by a unitary:

$$\begin{aligned} \displaystyle \displaystyle \sum _{i=1}^m\, | \, \alpha _j^i \, | ^2 \,\, = \,\, 1 \, , \quad \gamma _j \in \varGamma \, , \quad 1 \le j \le n \, . \end{aligned}$$
(5)

With this notation, the probability for a given elector to appreciate a candidate \( \, \gamma _j \,\) with an opinion \( \, \zeta _i \,\) is simply a consequence of the Born rule. The mean statistical expectation of a given opinion \(\, \zeta _i \,\) for a candidate \( \, \gamma _j \,\) is equal to \( \, | \, \alpha _j^i \, | ^2 \, \) on one hand and is given by the popularity polls \( \, S_{j \, i} \, \) on the other hand. Consequently,

$$\begin{aligned} \displaystyle | \, \alpha _j^i \, | ^2 \,=\, S_{j \, i} \,, \quad \gamma _j \in \varGamma \,, \,\, \zeta _i \in \mathrm{G} \,, \quad 1 \le j \le n \,, \quad 1 \le i \le m . \end{aligned}$$

4 A First Link Between the Two Previous Models

In [13], we have proposed a first link between the two previous models. We simplify the approach (1) and suppose that there exists some equivalent candidate \( \, | \, \xi \! > \, \in V_\varGamma \,\) such that the voting intention for each particular candidate \( \, \gamma _j \in \varGamma \,\) is equal to \(\, | <\xi \,,\, \gamma _j > |^2 \). We interpret the relation (4) in the following way: for each candidate \( \, \gamma _j \in \varGamma , \,\) there exists a political decomposition \( \, A \, | \, \gamma _j \! > \, \in W_\mathrm{G} \,\) in terms of the grid G. By linearity, we construct in this way a linear operator \(\, A :\, V_\varGamma \longrightarrow W_\mathrm{G} \,\) between two different Hilbert spaces. Preliminary results have been presented, in the context of the 2012 French presidential election.

5 From Opinion Polls to the Prediction of the Vote

In the space \( \, W_\mathrm{G} \,\) of political appreciations described in Sect. 3 of this contribution, the opinion polls allow through the relation (4) to determine some knowledge about each candidate \(\, \gamma _j \in \varGamma \,\) in the space \( \, W_\mathrm{G}\). We suppose that each candidate is represented by a unitary vector and the relation (5) still holds. The question is now to evaluate the probability for an arbitrary elector to vote for the various candidates.

We denote by \( \, \varPi _j \equiv | \, \gamma _j \! > < \! \gamma _j \, | \,\) the orthogonal projector onto the direction of the state \(\, | \, \gamma _j \! > \). Then we introduce a density matrix \( \, \rho \,\) associated to a statistical representation of the voting population:

$$\begin{aligned} \displaystyle \displaystyle \rho \,=\, \sum _{j=1}^n \alpha _j \, \varPi _j \, \equiv \, \sum _{j=1}^n \alpha _j \, | \, \gamma _j \! > < \! \gamma _j \, | \, . \end{aligned}$$
(6)

It is classical that \( \, \mathrm{tr} \rho = \sum _{j=1}^n \alpha _j \,\) and if \( \, \alpha _j \ge 0 \,\) for each index \(j\), the auto-adjoint operator \( \, \rho \,\) is non-negative:

$$\begin{aligned} \displaystyle \displaystyle < \! \rho \, \zeta \,,\, \zeta \! > \, \, \ge \, \, 0 \,, \qquad \forall \zeta \in W_\mathrm{G} \, . \end{aligned}$$

It is then natural to search the coefficients \( \, \alpha _j \,\) such that

$$\begin{aligned} \displaystyle \left\{ \begin{array}{l} \displaystyle \alpha _j \,\ge \, 0 \,, \qquad 1 \le j \le n \\ \displaystyle \sum _{j=1}^n \alpha _j \, = \, 1 \, . \end{array} \right. \end{aligned}$$
(7)

In these conditions, the Esperance of election \( \, < \gamma _j > \,\) of the candidate \( \, \gamma _j \,\) is given through the relation

$$\begin{aligned} \displaystyle < \gamma _j > \,=\, \mathrm{tr} \big ( \rho \, \varPi _j \big ) \,, \quad 1 \le j \le n \, . \end{aligned}$$
(8)

We have the following calculus:

$$\begin{aligned} \displaystyle \mathrm{tr} \big ( \rho \, \varPi _j \big )&= \sum _{k=1}^n \, < \zeta _k \,, \, \rho \, \varPi _j \, \zeta _k > \,=\, \sum _{k=1}^n \sum _{\ell =1}^n \, < \zeta _k \,, \, \alpha _\ell \, \gamma _\ell > \, < \gamma _\ell \,, \, \gamma _j > \, < \gamma _j \,, \, \zeta _k > \\ \displaystyle&= \sum _{\ell =1}^n \, \alpha _\ell \, < \gamma _\ell \,, \, \gamma _j > \, \sum _{k=1}^n \, < \zeta _k \,, \, \gamma _\ell > \,< \gamma _j \,, \, \zeta _k > \\ \displaystyle&= \sum _{\ell =1}^n \, \alpha _\ell \, < \gamma _\ell \,, \, \gamma _j > \, \sum _{k=1}^n \, < \gamma _j \,, \, \zeta _k > \, < \zeta _k \,, \, \gamma _\ell >\\ \displaystyle \qquad&= \sum _{\ell =1}^n \, \alpha _\ell \, < \gamma _\ell \,, \, \gamma _j > \, < \gamma _j \,, \, \gamma _\ell > \,= \, \sum _{\ell =1}^n \, \alpha _\ell \, \mid < \gamma _\ell \,, \, \gamma _j > \mid ^2 \, . \end{aligned}$$

We introduce the matrix \(A\) composed by the squares of the scalar products of the vectors of candidates:

$$\begin{aligned} \displaystyle A_{j \, \ell } \,=\, \, \mid < \gamma _j \,, \, \gamma _\ell > \mid ^2 \, , \quad 1 \le j,\, \ell \le n \, . \end{aligned}$$
(9)

Then the previous calculus establishes that

$$\begin{aligned} \displaystyle < \gamma _j > \,=\, \sum _{\ell =1}^n \, A_{j \, \ell } \, \, \alpha _\ell \, . \end{aligned}$$
(10)

It is interesting to imagine a link between the Esperance of election \( \, < \gamma _j > \,\) of the candidate \( \, \gamma _j \,\) and the coefficient \( \, \alpha _j \,\) of the density matrix introduced in (6). In general they differ. In the following, we focus our attention to the particular case where these two quantities are proportional, id est

$$\begin{aligned} \displaystyle \exists \, \lambda \in \mathbb {C}\,, \, \quad \forall \, j = 1, \, 2, \, \dots \, , \, n \,, \, \quad < \gamma _j > \,=\, \lambda \, \alpha _j \, . \end{aligned}$$
(11)

Because both \( \, < \gamma _j > \,\) and \(\, \alpha _j \,\) are positive, the coefficient \( \, \lambda \,\) must be a positive number. Moreover, due to (10), the relation (11) express that the non-null vector \( \, \alpha \in \mathrm{I}\! \mathrm{R}^n \,\) composed by the coefficients \( \, \alpha _j \,\) is an eigenvector of the matrix \(A\). Then, due to the hypothesis (7), we have \( \, \alpha _j \ge 0 \,\) and this eigenvector has non-negative components. If we suppose that the matrix \(A\) is irreductible (see e.g. in the book of Meyer [23] or Serre [26]), the Perron-Frobenius theorem states that there exists a unique eigenvalue (equal to the spectral radius of the matrix \(A\)) such that the corresponding eigenvector has all non-negative components. Moreover, all the components of this eigenvector are strictly positive. In other words, if the matrix \(A\) defined in (9) is irreductible and if the hypothesis of proportionality (11) is satisfied, the coefficients \( \, \alpha _j \,\) of the density matrix are, due to the second relation of (7), completely defined. In the following, we propose to determine the coefficients \( \, \alpha _j \,\) of the density matrix (6) and satisfying the conditions (7) as proportional to the positive eigenvector of the matrix \(A\) defined by (9).

The above model is not completely satisfactory for the following reason. The underlying order associated to the grading family \(\, G \,\) has not been taken into account. To fix the ideas, we suppose that each grade \( \, \nu _i \,\) is associated to a number \( \, \sigma _i \,\) such that

$$\begin{aligned} \displaystyle \sigma _1 \, > \, \sigma _2 \, > \, \cdots \, > \, \sigma _m \, . \end{aligned}$$
(12)

We introduce a “popularity operator” \( \, P_j \,\) associated to the \(j\)th candidate \( \, \gamma _j\):

$$\begin{aligned} \displaystyle P_j \, \equiv \, \sum _{i=1}^m \sigma _i \, \, | < \! \gamma _j \,, \, \zeta _i \! > |^2 \, | \, \zeta _i \! > < \! \zeta _i \, | \, . \end{aligned}$$
(13)

We can determine without difficulty the mean value of the operator \( \, P_j \,\) for the density configuration \( \, \rho \,\) defined in (6):

$$\begin{aligned} \displaystyle \mathrm{tr} \big ( \rho \, P_j \big )&= \sum _{k=1}^m \, < \zeta _k \,, \, \rho \, P_j \, \zeta _k > \\ \displaystyle&= \sum _{k=1}^m \sum _{\ell =1}^n \, < \zeta _k \,, \, \alpha _\ell \, \gamma _\ell > \, < \gamma _\ell \,, \, \sum _{i=1}^m \sigma _i \, \, | < \! \gamma _j \,, \, \zeta _i \! > |^2 \, | \, \zeta _i \! > < \! \zeta _i \, , \, \zeta _k >\\ \displaystyle&= \sum _{i=1}^m \sum _{\ell =1}^n \, \sigma _i \, \alpha _\ell \, < \! \zeta _i \, , \, \gamma _\ell \! > \, < \gamma _\ell \,, \, \zeta _i \! > \, | < \! \gamma _j \,, \, \zeta _i \! > |^2 \, \\ \displaystyle&= \sum _{i=1}^m \sum _{\ell =1}^n \, \sigma _i \, \alpha _\ell \, | < \! \gamma _\ell \,, \, \zeta _i \! > |^2 \, | < \! \gamma _j \,, \, \zeta _i \! > |^2 \,. \\ \end{aligned}$$

In other words, if we set

$$\begin{aligned} \displaystyle B_{j \, \ell } \, \equiv \, \sum _{i=1}^m \, \sigma _i \, | < \! \gamma _j \,, \, \zeta _i \! > |^2 \, \, | < \! \gamma _\ell \,, \, \zeta _i \! > |^2 \, \,, \end{aligned}$$
(14)

we have:

$$\begin{aligned} \displaystyle < P_j > \,\equiv \, \mathrm{tr} \big ( \rho \, P_j \big ) \,=\, \ \sum _{\ell =1}^n \, B_{j \, \ell } \, \, \alpha _\ell \, . \end{aligned}$$
(15)

We use a positive parameter \( \, t \,\) and search the coefficients \(\, \alpha _j \,\) in such a way that the mean value of the candidate \(\gamma _j\) with some “upwinding” associated to its popularity is proportional to the above coefficients. In other words, due to (10) and (15), the mean value \( \, < \gamma _j > \, + \, t < P_j > \, \) takes the algebraic form

$$\begin{aligned} \displaystyle < \gamma _j > \, + \, t < P_j > \,=\, \sum _{\ell =1}^n \, \big ( A_{j \, \ell } + t \, B_{j \, \ell } \big ) \, \alpha _\ell \, . \end{aligned}$$
(16)

Under the condition that all the coefficients \( \, A_{j \, \ell } + t \, B_{j \, \ell } \,\) are positive, id est that the parameter \(t\) is small enough, we compute the coefficients \( \, \alpha _\ell \,\) with the help of the Perron-Frobenius theorem as presented previously.

6 A First Numerical Test Case

Our first model uses synthetic data. We suppose that we have three candidates (\(n=3\)) and two (\(m=2\)) levels of “political” appreciation. We suppose that

$$\begin{aligned} \displaystyle \left\{ \begin{array}{l} \displaystyle | \gamma _1 \! > \,= \, \cos \Big ( {\pi \over 6} \Big ) \, | \zeta _1 \! > \,+\, \sin \Big ( {\pi \over 6} \Big ) \, | \zeta _2 \! > \\ \displaystyle | \gamma _2 \! > \,= \, \cos \Big ( {\pi \over 4} \Big ) \, | \zeta _1 \! > \,+\, \sin \Big ( {\pi \over 4} \Big ) \, | \zeta _2 \! > \\ \displaystyle | \gamma _3 \! > \,= \, \cos \Big ( {\pi \over 3} \Big ) \, | \zeta _1 \! > \,+\, \sin \Big ( {\pi \over 3} \Big ) \, | \zeta _2 \! > \, . \end{array} \right. \end{aligned}$$
(17)

With the choice \( \, \sigma _1 = 1 \) and \( \, \sigma _2 = 0 \, \) in a way suggested at the relation (12), we can simulate numerically the process presented in the Sect. 5. The results are presented in Fig. 1. When the variable \(t\) is increasing, the first candidate has a better score, due to his best results in the grading evaluation (17).

Fig. 1.
figure 1

Result of the vote obtained by a quantum model from the opinion poll, with synthetic data proposed in (17).

Full size image

7 Test of the Method with Real Data

We have also used data coming from the “first tour” of the French presidential election of April 2012. Popularity data [18] and result of voting intentions [19] are displayed in Table 1. The names of the principal candidates to the French presidential election are proposed in alphabetic order with the following abbreviations: “Ba” for François Bayrou, “Ho” for François Hollande, “Jo” for Eva Joly, “LP” for Marine Le Pen, “Mé” for Jean-Luc Mélanchon and “Sa” for Nicolas Sarkozy. In Table 1, we have also reported the result of the election of 22 April 2012.

Table 1. Popularity, sounding polls and result, April 2012 [18, 19].
Full size table
Fig. 2.
figure 2

Result of the vote obtained by a quantum model from the opinion poll. Data issued from the April 2012 French presidential election.

Full size image

This test case corresponds to \(n=6\) and \(m=3\). The numerical data relative to the relation (12) are chosen such that \( \, \sigma _1 = 1 \), \( \, \sigma _2 = 0 \), and \( \, \sigma _3 = -1 \). Then the above Perron-Frobenius methodology is available up to \( \, t = 2.2 \). The numerical result are presented in Fig. 2. It reflects some big tendances of the real election. But the correlation between the popularity and the result is not always satisfied, as shown clearly by comparison between our simulation in Fig. 2 and the result of the election shown in the last column of Table 1.

8 Conclusion

In this contribution, we have used a given quantum model for range voting in the context of opinion polls. From these data, we have proposed a quantum methodology for predicting the vote. We introduce a density operator associated to the candidates. The mathematical key point is the determination of a positive eigenvector for a real matrix with non-negative coefficients. Our results are encouraging, even if the confrontation to real life data shows explicitly that other parameters have to be taken into account.