Evolution of Electoral Preferences for a Regime of Three Political Parties

In this article, we use a discrete system to study the opinion dynamics regarding the electoral preferences of a nontendentious group of agents. To measure the level of preference, a continuous opinion space is used, in which the preference (opinion) can evolve from any political option, to any other; for a regime of three parties, a circle is the convenient space. To model a nonbiased society, new agents are considered. Besides their opinion, they have a new attribute: an individual iterative monoparametric map that imitates a process of internal reflection, allowing them to update their opinion in their own way.These iterativemaps introduce six fixed points on the opinion space; the points’ stability depends on the sign of the parameter. When the latter is positive, three attractors are identified with political options, while the repulsors are identified with the antioptions (preferences diametrically opposed to each political choice). In this new model, pairs of agents interact only if their respective opinions are alike; a positive number called confidence bound is introducedwith this purpose; if opinions are similar, they update their opinion considering each other’s opinion, while if they are not alike, each agent updates her opinion considering only her individual map. In addition, agents give a certain level of trust (weight) to other agent’s opinions; this results in a positive stochastic matrix of weights which models the social network. The model can be reduced to a pair of coupled nonlinear difference equations, making extracting analytical results possible: a theorem on the conditions governing the existence of consensus in this new artificial society. Some numerical simulations are provided, exemplifying the analytical results.


Introduction
The arising of opinion in society has become a very interesting research topic in the last decades.For example, Ding and Mo [1] have studied how noise, from different sources, like TV, newspaper, and so on, contributes to eliminate the disagreement in a social group; Chen et al. [2] have considered how the social similarity influences the opinion dynamics between individuals; D' Aniello et al. [3] have used a fuzzy consensus approach to investigate decision-making in a group; Lu et al. [4] have investigated the impact of the community structure on the convergence time to reach a consensus in a social group.A recent survey on some opinion dynamics models and their applications is found in [5].
In particular, the opinion dynamics about political options has been studied broadly, using models borrowed from statistical physics, such as the Ising and Potts ones [6,7], in which agents are forced to choose between a finite set of options, no matter how much they agree with them.The set of options, naturally, is discrete as the corresponding opinion space for these models too, allowing tendentious agents.An alternative would be the use of a continuous opinion space, in which agents can show their level of agreement regarding the options; this space must also allow opinions to evolve from one option to another, options being equidistant from each other; these requirements suggest that an adequate opinion space, for -options, is an  − 2-sphere.For example, a circle is an appropriate opinion space for three options, while a sphere is suitable for four options.These ideas were used by Medina et al. [8] to study the consensus formation on a circle for a regime of three political parties; they used an iterative Figure 1: The figure shows attractors described by Theorem 2 and their respective attraction basins for system (7).Attractors with 0 <  < 2/3 are the associated electoral preferences.Attractors with −2/3 <  < 0, also called antioptions, are completely opposite to existing options.Agents attracted to one antioption radically refuse the one option remaining half trusting to the other options.monoparametric map which introduced six fixed points on the opinion space.The stability of these points is governed by the sign of the map's parameter, the personal parameter; if it is positive, the three attractors are identified with the political options, while the remaining three repellers are called the (political) antioptions.An agent having a negative personal parameter rejects all the available political options, being attracted to the antioptions.Since there is an alternating sequence of attractors and repellers on the circle, options and antioptions are diametrically opposed; see Figure 1.Hence an agent whose opinion is attracted to one antioption rejects the diametrically opposed political option, while she (or he) is insecure about the remaining two options.So in contrast, with the discrete option models, these agents are not forced to pick an option.In this sense, the iterative map allows agents to update their opinion taking into account their postures with respect to the choices, emulating an internal reflection process in them.The above agents interact by pairs updating their opinion considering their internal reflection processes and their partners' opinions with some weight (a subjective trust given to each other's opinion); if their opinions are not alike, they update their opinions considering only their own internal reflection processes.A positive number, the confidence bound, is used as a threshold to separate between these two cases.
Although agents, in Medina et al. [8] model, have different personal parameters, the same iterative map was used for the opinion updating.However, in a more realistic society, each person forms her opinion differently from others; hence it has sense to consider a different updating rule for each agent.So, in this work, we propose to give each agent a new attribute, a different iterative map that allows them to update their opinion in their own manner.Then we investigate the conditions under which a consensus arises, in this new artificial society, under these new updating rules.Finally, some numerical experiments are performed to validate the analytical results.
This work is sectioned as follows: Section 2 presents a new opinion dynamics model on the circle.Section 3, Main Results, gives a theorem stating the conditions under which the consensus emerges in the social network.Section 4 shows that some simulations with two different opinion updating rules are given.A section of concluding remarks and two appendixes end this communication.

Opinion Dynamics Modeled on a Circle
Through the work, the opinion space  is a circle, and it could be identified with the segment [0, 2] modulus 2 in radians or, equivalently, as the segment [0, 360] modulo 360 in degrees.On it, the opinion updating rules establish three attractors identified with the political options and three antioptions.Options and antioptions are distributed in an alternating sequence; see Figure 1.The antioption opposed to an option corresponds to a total rejection of the latter.The personal parameter space is denoted by ; it is identified with the segment [−, ] in radians or [−180, 180] in degrees.A set with an even number  of interacting agents is considered.Such interactions take place on discrete times   , for each  ∈ Z + ∪ 0, so (  ) ∞ =0 is a nonbounded sequence of positive real numbers such that  0 = 0.Each agent is characterized by three attributes: an opinion    ∈ ; a constant personal parameter   ∈  that allows acceptance or rejection of the presented options (the attracting nature of options and antioptions depends on the sign of   ; see below); and an opinion updating function Ξ  :  ×  → R. Like in [8,9], the social network describing the agent population is governed by the stochastic absolute weights matrix: where its elements are positive real numbers fulfilling 0 <   ≤ 1 and 0 ≤   < 1, ∀,  ∈ {1, 2, . . ., }, in such a way that the sum of each row is equal to one: From a social point of view, a possible interpretation to   could be the credibility given by agent  to the other agents' opinions in the social network, including his own opinion.The first inequality above 0 <   ≤ 1 establishes that each agent must give a nonzero credibility to its own opinion.
Since the matrix rows must add to one, the second inequality 0 ≤   < 1 establishes that the credibility given to other agent's opinion must be less than one.Since the interactions between the agents are pairwise, it is necessary to reevaluate the mutual trust between the agents.Hence we use relative weights, The relative weights fulfill these properties: On the same manner as well-known models of Deffuant et al. [10,11] and Hegsellman-Krause [12], we consider a confidence bound  > 0 that determines a criterion to set a limit to agents interaction; i.e., the opinions belonging to agents  and  are affine if and only if |   −    | < .Now we formulate the opinion updating rules on the circle.It is important to distinguish six special points on the opinion space: three options and three antioptions.To do that, we consider iterative monoparametric maps that introduce these six fixed points.As a construction element of the dynamic rules, we use the following functions: where where   : [−1, 1] → (0, 1] is a  1 class differential function such that   (0) = 1, ∀ ∈ {1, 2, . . ., }; the sine function, sin(3), introduces the fixed points,  = /3 for  ∈ Z; in the iterative map  +1 = Ξ(  ,   ): Instead of the sine function in (5), it would be possible to use the cosine function with the same purpose.However, this would only move the fixed points on the circle.Medina et al. [8] consider the same updating opinion rule for all agents with   () = 1.Now, we propose to use a different function   () for each agent, with the purpose to give them a different way to update their opinions; thus the functions Ξ  (  ,   ) model internal reflection processes for opinion updating in agents; they also provide the knowledge about the options and permit them to keep an attitude regarding these options; see Appendix A. In this sense, the new model introduces a more heterogeneous and less tendentious agent population.
On the recursive model, as initial conditions, a personal parameter and a random opinion are assigned to each agent; aside the stochastic matrix  is generated too.On the  temporal step, from the agents population, /2couples are randomly chosen to interact according to these rules: (i) When a pair of agents have similar opinions, |  −  | < , they update their opinions considering their respective internal reflection processes and considering each other's opinion, with some relative weights: (ii) But if they do not have similar opinions, |  −   | > , every agent in the pair uses only her iterative map to update her opinion, without considering other's opinion, consequently, using   =   = 1 and   =   = 0 in (7).

Main Results
Now, we present the conditions for consensus formation in the artificial society described by the before enunciated model.It is said that model ( 7) exhibits a strong consensus if the opinions of agents evolve to a stable fixed point.To simplify our notations and reduce model (7) to a system of only two nonlinear difference equations, we perform the following change of variables: The model becomes into where  +  = 1 and  +  = 1.
Analyzing the fixed points of the previous equations ( 9), we establish conditions for the existence and stability of the fixed points of the complete system (7), analogously as the model in [8].
Theorem 1.The point (, ) is a common fixed point of (7), if and only if  = /3 for some  ∈ Z.
By definition   () ̸ = 0, then sin(3) = 0 or  = /3 for some  ∈ Z. Conversely, a direct substitution shows that the point (, ) is a solution for (9), when  is an entire multiple of /3, like it is asked.
Note that these fixed points do not depend neither on the relative weights nor on personal parameters, so they are common to all pairs of (9) and they are fixed points of the full set of (7).Of course, (9) have other fixed points.However, when one considers the full system (7), fixed points from one pair of equations are not necessarily the same as another pair, so they may not be fixed points of the complete system.Some examples are presented in Appendix B.
Proof.For the linearized system (9), the Jacobian matrix is Under these considerations, the spectral radius () of  satisfies Since all eigenvalues of  are complex numbers with modulus less than 1, then (, ) is a stable fixed point; see [13,14].Theorem 2 gives the necessary and sufficient conditions for the existence of a consensus between a pair of agents; now Theorem 3 establishes the necessary and sufficient conditions for the existence of a consensus among all agents in the social web.
This theorem generalizes the one reported in [8] and it could be seen that the consensus in the social network does not depend on   functions, nor absolute weights matrix , nor number of agents .
A more realistic case is when agents have personal parameters in a subset containing both positive and negative values; then system (9), in general, will have distinct attractors for different couples of agents; see Appendix B. Intuitively, it is expected that these attractors be bounded in some subset of  × ; consequently the agents' opinions will evolve towards a subset  ⊊ ; hence, it is said that a weak consensus is present in the social network [8,9].

Some Examples
For the sake of clearance, some simulations are provided.Here we use a social web with 200 agents; it is described by a random matrix of absolute weights.Agents' starting opinions are uniform and randomly distributed on .Aside, a uniform time interval partition is considered.The parameters and variables will be expressed in degrees.Considering the same interacting dynamics between agents and two cases for updating rule opinion (5), The simulations use 100 time steps.
Example 4. In Figure 2, the personal parameters   are randomly and uniformly distributed on [0.5 ∘ , 38 ∘ ].Three different confidence bounds are considered: top rank  = 180 ∘ ;  = 120 ∘ in middle rank;  = 60 ∘ for bottom rank:  1 () to the left and  2 () to the right.When  = 180 ∘ all agents interact and one cluster appears.If  = 120 ∘ , agents interact with about two-thirds of total population at the most; initially, two clusters appear, but the bigger one absorbs the smaller one.For  = 60 ∘ , agents interact with one-third of population, and three clusters arise, each one with about one-third of total population.According to Theorem 3, strong consensus arise for all these cases.
Example 5.In Figure 3, the personal parameters   are randomly and uniformly distributed on [−10 ∘ , 10 ∘ ].Three different confidence bounds are considered too; top rank  = 180 ∘ ;  = 120 ∘ in middle rank;  = 60 ∘ for bottom rank:  1 () to the left and  2 () to the right.When  = 180 ∘ , all agents interact and one cluster appears, but with weak consensus.If  = 120 ∘ , the agents interact with about two-thirds of total population at the most, and two weak consensus clusters appear.For  = 60 ∘ , agents interact with one-third of population, but now strong and weak consensus states coexist.  is used on left column and  2 () on the right.In the top row, a single cluster appears.In the middle row, initially two clusters appear, which finally merge.In the lower row, three clusters, with about one-third of the population, appear.Strong consensus appears always, in all three cases, according to Theorem 3.

Conclusions
In this paper, a new model for opinion dynamics of political preferences about three parties options has been presented; it generalizes the model in Medina et al. [8] to include new individual updating rules: a set of monoparametric iterative maps, which emulate different internal reflection processes in agents.For some positive values of the personal parameter  (map's parameter), these rules introduce three equidistant attractors on a circular opinion space, which correspond to three choices, that can be interpreted with a different political party.For some negative values of the personal parameter, choices become repulsors, so agents with a negative personal parameter reject the political options.The value of the So, the key point of the present model is a new agent who is characterized by three attributes: opinion , personal parameter , and an iterative map to update her own opinion differently as her interlocutors.This kind of agent becomes a fundamental piece of a more heterogeneous artificial society.
The presented model, also, considers pairwise interactions among agents and uses the confidence bound  like Hegselmann & Krause and Deffuant & Weisbuch classical models to set a limit for dissimilar agents opinion case and random absolute weights to describe the subjective trust given to agents' opinions, defining an artificial social web.
The main interest of this work was to study consensus arising in this new artificial society; two classes of consensus were considered: a strong one which is the convergence of the opinions to a fixed point and a weak one which is the convergence of the opinions to a subset of the opinion space.Theorem 3 provides the necessary and sufficient conditions for the arising of a strong consensus in this artificial heterogeneous society.
Finally two sets of simulations were performed; in each set, a different iterative map and three different confidence bounds were used.In each simulation, were used either positive different personal parameters or mixed (positive and negative) personal parameters, random absolute weights, or random initial opinions, but the same iterative map for all agents.However, simulations show practically the same behaviors and convergence times as found in [8]; some of them show the emergence of strong consensus or the existence of weak consensus regions.
As perspectives of future research work, the model could be used to study the opinion dynamic of the electoral preferences regarding three political parties when a charismatic political actor (a very persuasive agent) is present.It is also possible to generalize the model to opinion spaces with four or more options, spheres of two or more dimensions.These latter models would be useful to study the situation when there are four or more political parties.

A. On the Role of the Personal Parameter
From a mathematical point of view, iterative maps present a vast range of behaviors; some of them can be visualized in the corresponding bifurcation diagrams; see Marotto [13].Some of these mathematical behaviors are used to model how agents update their own opinion: with certainty, if the opinion converges without oscillating; with doubt, if the opinion converges oscillating.The figure shows a parametric plot of families of curves for system (9) for Ξ () = −   1 (sin (3))sin (3),   = −0.66,and different relatives weights  = 0.1, 0.2, . . ., 0.9 (red lines), Ξ () =  −    2 (sin (3))sin (3),   = 0.66 and different relative weights  = 0.1, 0.2, . . ., 0.9 (blue lines).The intersections correspond to different fixed points of different pairs of equations.Notice that only those fixed points of the form (, ) are common to all equations in system (7); however, they are unstable.Other fixed points depend on parameters' values.

Figure 2 :
Figure 2: The confidence bounds are  = 180 ∘ (top row);  = 120 ∘ (middle row), and  = 60 (bottom row).  ∈ [0.5 ∘ , 38 ∘ ].Function  1 ()is used on left column and  2 () on the right.In the top row, a single cluster appears.In the middle row, initially two clusters appear, which finally merge.In the lower row, three clusters, with about one-third of the population, appear.Strong consensus appears always, in all three cases, according to Theorem 3.

Figure 4 :Figure 5 :
Figure 4: The figure shows a bifurcation diagram for the iterative map  +1 =   −  cos 4 (sin(3  )) sin(3  ); an alternating sequence of tree-like graphs goes upwards and downwards.Here are depicted the options for 0 <  < 2/3 and antioptions for −2/3 <  < 0. Stable fixed points bifurcate in a 2-cycle for  = 2/3; then a cascade of bifurcations appears, leading to chaos.The 2-cycle may model a dilemma, since agent's opinion jumps from one posture to another.Other -cycles may model other doubt processes in agents.