Condensed Matter Physics, 2017, Vol. 20, No 1, 13801: 1–11 DOI: 10.5488/CMP.20.13801 http://www.icmp.lviv.ua/journal Negative response to an excessive bias by a mixed population of voters ∗ V.S. Dotsenko1, C. Mejía-Monasterio2, G. Oshanin1 1 Laboratoire de Physique Théorique de la Matière Condensée, UPMC, CNRS UMR 7600, Sorbonne Universités, 4 Place Jussieu, 75252 Paris Cedex 05, France 2 Laboratory of Physical Properties, Technical University of Madrid, Av. Complutense s/n 28040 Madrid, Spain Received January 27, 2017, in final form February 28, 2017 We study an outcome of a vote in a population of voters exposed to an externally applied bias in favour of one of two potential candidates. The population consists of ordinary individuals, that are in majority and tend to align their opinion with the external bias, and some number of contrarians— individuals who are always hostile to the bias but are not in a conflict with ordinary voters. The voters interact among themselves, all with all, trying to find an opinion reached by the community as a whole. We demonstrate that for a sufficiently weak external bias, the opinion of ordinary individuals is always decisive and the outcome of the vote is in favour of the preferential candidate. On the contrary, for an excessively strong bias, the contrarians dominate in the population’s opinion, producing overall a negative response to the imposed bias. We also show that for sufficiently strong interactions within the community, either of two subgroups can abruptly change an opinion of the other group. Key words: non-linear and negative response, external bias, population of voters PACS: 89.65.Ef, 89.75.-k 1. Introduction One often encounters situations in which trying harder, pushing stronger or, in general, making any excessive effort appears to be counterproductive and leads to a smaller (or even a negative) effect, as com- pared to the outcome achieved with a more modest investment. This is so, in particular, for long-lasting human relationships and, on a larger scale, for a cohabitation of different countries. In condensedmatter physics, such an effect takes place in many instances. To name but a few, we mention electron transfer in semiconductors at low temperatures [1–4], hopping processes in disordered media [5, 6], transport of electrons inmixtures of atomic gases [7], somemodels of Brownianmotors [8–11], transport in kinetically constrained models of glass formers [12] and in lattice gases [13–21], which also exhibits other spectacu- lar anomalies beyond the linear-response regime [22–25]. In all these physical examples, exerting more force on an object dragged through a system yields a disproportionally strong response of the medium, so that the resulting velocity of the object appears to be less for strong forces than for the moderate ones. Similar phenomena also occur, due to other physical reasons, in far-from-equilibrium quantum spin chains [26], systems which exhibit a “freezing-by-heating” behavior (see, e.g., [27]) and different types of non-equilibrium systems [28]. In the present paper we discuss a toy model of opinion formation in a mixed society in which ap- plying an excessive external bias appears to be counterproductive. We consider here a big society which consists of many small communities, each comprising N individuals. These individuals have to vote for and eventually to elect either of two candidates — candidate 1 and candidate 2. The individuals within each community interact between themselves, all with all, trying to achieve a consensus — an opinion reached by the group as a whole. The society lives at some effective temperature T , which permits for fluctuations in opinions within a given community. ∗This paper is dedicated to Yu. Holovatch on the occasion of his 60th anniversary. © V.S. Dotsenko, C. Mejía-Monasterio, G. Oshanin, 2017 13801-1 https://doi.org/10.5488/CMP.20.13801 http://www.icmp.lviv.ua/journal V.S. Dotsenko, C. Mejía-Monasterio, G. Oshanin The voters are exposed to an external bias, e.g., due to mass media, prompting them to choose a preferential candidate. With respect to the external bias, each community is divided into two subgroups, of sizesN1 = (1−ρ)N andN2 = N−N1 = ρN with 0 É ρ É 1, respectively. The individuals that belong to the first subgroup— we call them “ordinary voters” — tend to adjust their opinion to the external bias. The voters that form the second subgroup— a sort of “contrarians” in amodel of Galam [29, 30]— are opposed to the external bias and tend to have an opinion different from the externally imposed one.1. The opinions of the ordinary voters and of the contrarians may have a different strength. Indeed, as it usually happens, the contrarians may be more persuasive, maymobilize themselves more readily and also may have more influence on the community than the ordinary voters. We are interested to know the eventual outcome of the elections in the whole society and, more specifically, to get a qualitative, conceptual understanding of how an intensity of the external bias affects this outcome. We proceed to show that upon exposing the system to a bias and gradually increasing its magnitude, a series of interesting phenomena may take place, including, e.g., an abrupt change of the opinion of each of the subgroups when the “ties” within a given community are sufficiently strong. We will also demonstrate that one first achieves an opinion aligned with the bias. Increasing the bias further up to some critical value, gives a maximal effect with a saturation of the bias-induced opinion, and then, upon exceeding this critical value, one observes a decrease of the latter and eventually, a formation of an opinion which is antagonistic to the bias. 2. The model To put our toy model into formal terms, we assign to each of the individuals a “spin” variable char- acterising the strength of his or her opinion in favour of one of the candidates. For ordinary individuals, this “spin” variable is defined as si (with index i labelling these individuals, i = 1, . . . , N1) and is rigidassuming only two values: si =+1 corresponding to the opinion in favor of the candidate 1, and si =−1 corresponding to the preference for the candidate 2. To describe next the strength of the opinion of each of the contrarians and to take into account their persuasive ability, we introduce a soft “spin” variable φ j ( j = 1, . . . , N2), and suppose that it can, in principle, attain any real positive or negative value, but itsdistribution P (φ j ) is sharply peaked at φ j = −1 and φ j = 1. As an example, we will consider Gaussian distribution of the form P (φ) = 1√ 8πσφ { exp [ − (φ−1)2 2σφ ] +exp [ − (φ+1)2 2σφ ]} , (2.1) where σφ characterises the effective stiffness of the distribution. When σφ → 0, the distribution of φ jin (2.1) becomes the sum of two delta-functions, just like the one for si .Further on, we suppose that our system comprising rigid and soft spins experiences an action of an external bias of intensity h, which favours the rigid spins to be aligned in the direction of the bias, while the soft spins tend to be aligned in the opposite to the bias direction. Then, the probability P ({si ,φ j }) of finding a particular configuration {si ,φ j } of the spin variables is given by P ({si ,φ j }) = ∏N2 j=1 P (φ j ) Z (A,B) exp [ −H (A,B) 0 [{si ,φ j }] T + h T ( N1∑ i=1 si − N2∑ j=1 φ j )] , (2.2) whereH (A,B) 0 [{si ,φ j }] is the Hamiltonian in the absence of an external bias. We will distinguish between two cases with respect to the coupling constants of the spins: in the case A, we suppose that the interac- tions between the soft and the rigid spins, as well as between the rigid-rigid and the soft-soft spins are the same, and are described by a positive coupling constant J , so that H (A) 0 [{si ,φ j }] =− J 2N ( N1∑ i , j=1; i, j si s j + N2∑ i , j=1; i, j φiφ j +2 N1∑ i=1 N2∑ j=1 siφ j ) , (2.3) 1We emphasize that in [29, 30] the contrarians are at odds with ordinary voters, which represent the majority. In our settings, they rather contradict the external bias but are not in a conflict with ordinary voters. 13801-2 Negative response to an excessive bias In the case B, we consider a bit more complicated situationwhen the interactions between the soft and the rigid spins are different from the interactions between the soft-soft and the rigid-rigid ones. In the latter case, we suppose that the interactions between the soft-rigid spins are described by a coupling constant Is ,while the interactions between the rigid-rigid and the soft-soft spins are still described by the coupling constant J . We will concentrate here on situations when Is > J > 0. In the case B, the Hamiltonian of a mixture of coupled rigid and soft spins in the absence of an external bias is defined by H (B) 0 [{si ,φ j }] =− J 2N ( N1∑ i , j=1; i, j si s j + N2∑ i , j=1; i, j φiφ j ) − Is N N1∑ i=1 N2∑ j=1 siφ j . (2.4) Both H (A) 0 [{si ,φ j }] and H (B) 0 [{si ,φ j }] are the Hamiltonians of some ferromagnets so that for h = 0 and for sufficiently strong couplings, the spins will tend to have the same sign. Lastly, Z (A,B) in (2.2) are the partition functions Z (A,B) = N1∏ i=1 ∑ si=±1 exp ( h T si ) ∞∫ −∞ . . . ∞∫ −∞ [ N2∏ j=1 dφ j P (φ j ) exp ( −h T φ j )] exp { −H (A,B) 0 [{si ,φ j }] T } . (2.5) Note that Z (A,B) depend on T , J , Is and h only in the combination J/T , Is /T and h/T , so that we may safely set T = 1 supposing that the dependence on the temperature is adsorbed in J , Is and h (or, in other words, that J , Is and h are measured in units of T ). In what follows we will be interested in the h-dependence of two “order” parameters: m(A,B) s = 1 N1 N1∑ i=1 〈si 〉 , m(A,B) φ = 1 N2 N2∑ j=1 〈φ j 〉 , (2.6) where the angle brackets denote averaging over the whole society, and also of their linear combination M (A,B) = 1 N ( N1∑ i=1 〈si 〉+ N2∑ j=1 〈φ j 〉 ) = (1−ρ)m(A,B) s +ρm(A,B) φ . (2.7) While m(A,B) s and m(A,B) φ define the average “opinion” of the ordinary voters and of the contrarians, re- spectively, the parameterM (A,B) determines the overall outcome of the vote. 3. General results for the cases A and B We proceed with the calculation of the partition function Z (A,B) in (2.5) focusing first on the case A. 3.1. The case A Since our eventual goal is the calculation of the order parameters, it is expedient to study a little bit more general object rather thanZ (A) itself, letting the biases acting on si and φ j be different. To this end,we consider an auxiliary partition function Z (A)(h1,h2) = N1∏ i=1 ∑ si=±1 exp ( h1 T si ) ∞∫ −∞ . . . ∞∫ −∞ [ N2∏ j=1 dφ j P (φ j ) exp ( −h2 T φ j )] exp { −H (A) 0 [{si ,φ j }] T } , (3.1) which reduces to the expression in (2.5) for h1 = h2 = h. OnceZ (A)(h1,h2) is known, the order parameters can be simply obtained by differentiation of Z (A)(h1,h2) as m(A) s = 1 N1 d dh1 lnZ (A)(h1,h2) ∣∣∣∣ h1=h2=h , m(A) φ =− 1 N2 d dh2 lnZ (A)(h1,h2) ∣∣∣∣ h1=h2=h . (3.2) 13801-3 V.S. Dotsenko, C. Mejía-Monasterio, G. Oshanin Now, we formally rewrite the Hamiltonian in (2.3) as H (A) 0 [{si ,φ j }] =− J 2N [( N1∑ i=1 si + N2∑ j=1 φ j )2 −N1 − N2∑ j=1 φ2 j ] , (3.3) where we used the condition that s2 i ≡ 1. Taking advantage of the integral identity exp ( b2 4a ) = √ a π ∞∫ −∞ dx exp (−ax2 +bx ) , a > 0, (3.4) we cast the auxiliary partition function in (3.1) into a factorised, with respect to si and φ j , form, andperform straightforwardly averaging over the spin variables. In doing so, we get Z (A)(h1,h2) = √ N 2πJ exp [ − J 2 (1−ρ)+N (1−ρ) ln2 ] ∞∫ −∞ dx exp [−N Fh1,h2 (x) ]≡ exp [−N f (h1,h2) ] , (3.5) in which the “free energy density” F (x) is defined explicitly by Fh1,h2 (x) = x2 2J − (1−ρ) lncosh(x +h1)−ρ ln  ∞∫ −∞ dφP (φ)exp [ (x −h2)φ− J 2N φ2 ] , (3.6) and f (h1,h2) in the right-hand-side of (3.5) is the desired free energy of an auxiliary system with fields h1 and h2 acting on the spin variables si and φ j , respectively. Correspondingly, the order parametersm(A) s andm(A) φ in (2.6) can be written as m(A) s = ∫ ∞ −∞ dx tanh(x +h)exp [−N Fh1=h2=h(x) ]∫ ∞ −∞ dx exp [−N Fh1=h2=h(x) ] , m(A) φ = ∫ ∞ −∞ dxL (A)(x)exp [−N Fh1=h2=h(x) ]∫ ∞ −∞ dx exp [−N Fh1=h2=h(x) ] , (3.7) with L (A)(x) = ∫ ∞ −∞φP (φ)dφexp [ (x −h)φ− J 2N φ 2 ]∫ ∞ −∞ P (φ)dφexp [ (x −h)φ− J 2N φ 2 ] . (3.8) One notices next that for x → ±∞, the dominant contribution to F (x) comes from the first term in the right-hand-side of (3.6), which diverges in proportion to x2. This signifies that F (x) has at least one minimum. More thorough analysis shows that F (x) has a single minimum for sufficiently small values of the coupling constant J . On the other hand, for larger values of J , F (x) attains a double-well shape. For a particular example of the distribution of the soft spin variables in (2.1), the integral over dφ in the right-hand-side of (3.6) can be performed exactly, which permits us to illustrate the typical behavior of F (x) as a function of x in different regimes (see figure 1). We focus on the limit of large N , N À 1, when the integrals in (3.5) become concentrated in the small vicinity of the minimum x = x∗ [or two minima, when F (x) has a double-well form] of F (x), defined by dFh1=h2=h(x∗) dx∗ = 0. (3.9) Differentiating (3.6), we find the following transcendental equation, which defines x∗ implicitly, x∗ = J { (1−ρ) tanh(x∗+h)+ρ ∫ ∞ −∞φdφP (φ)exp [ (x∗−h)φ− J 2N φ 2 ]∫ ∞ −∞ dφP (φ)exp [ (x∗−h)φ− J 2N φ 2 ] } . (3.10) Further on, performing the integrals in (3.5) we find thatm(A) s andm(A) φ obey m(A) s = tanh(x∗+h) , m(A) φ =L (A)(x∗) . (3.11) Lastly, comparing (3.10) with m(A) s and m(A) φ defined in (3.11) and the definition of the order parameter M (A) in (2.7), we conclude that x∗ has quite a lucid physical meaning: indeed, x∗ = J M (A) and hence, x∗ defines the outcome of the vote multiplied by J . 13801-4 Negative response to an excessive bias -40 -20 0 20 40 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 h 0 x F(x) Fmax -40 -20 0 20 40 -8 -6 -4 -2 0 h 2 x F(x) Fmax -40 -20 0 20 40 -4 -3 -2 -1 0 1 h 5 x F(x) Fmax Figure 1. (Color online) Function F (x) in (3.6) (for convenience divided by its maximal value attained at x = 50) for h1 = h2 = h, ρ = 0.1 and N = 103, and P (φ) defined in (2.1) with σφ = 1. Solid curves correspond to J = 1, dashed— to J = 5 and dotted— to J = 7. 3.2. The case B The derivation of transcendental equations, which implicitly define the order parameters m(B) s and m(B) φ is only slightly more involved. Since it follows essentially the same line of thought, we present these equations here without a derivation. We have that in the case B: m(B) s = tanh [ J (1−ρ)m(B) s + Isρm(B) φ +h ] , m(B) φ =L (B) ( m(B) s ,m(B) φ ,h ) , (3.12) where L (B) ( m(B) s ,m(B) φ ,h ) = ∫ ∞ −∞φP (φ)dφexp {[ Jρm(B) φ + Is (1−ρ)m(B) s −h ] φ } ∫ ∞ −∞ P (φ)dφexp {[ Jρm(B) φ + Is (1−ρ)m(B) s −h ] φ } . (3.13) The expressions in (3.12) reduce to the ones in (3.11) for J = Is . Below we will discuss the behavior of theorder parameters in the cases A and B for the distribution P (φ) in (2.1). 4. Results for the distribution in (2.1). Case A In the particular case when the distribution P (φ) is given by (2.1),L (x∗) in the right-hand-side of (3.8) can be calculated exactly. In the leading in the limit N →∞ order, we have L (x∗) = (x∗−h)σφ+ tanh(x∗−h) , (4.1) so that (3.10) becomes x∗ = J [ (1−ρ) tanh(x∗+h)+ρ(x∗−h)σφ+ρ tanh(x∗−h) ] . (4.2) Two latter coupled transcendental equations, together with (3.11), implicitly define the order parameters in the case A. These equations, which havemultivalued solutions for sufficiently large values of J , are quite difficult to study analytically in the whole parameter range (numerical solution is possible, of course). Nonethe- less, they permit us to get some general understanding of the behavior of the order parameters in the limiting cases of small and large h. We consider first the limit h ¿ 1, which corresponds to the limit of a linear response of a system to an external bias. In this limit, for sufficiently small J , we have that in the leading in h order the order parameterM (A) obeys M (A) = [ 1−ρ(2+σφ) ] 1− J (1+ρσφ) h . (4.3) 13801-5 V.S. Dotsenko, C. Mejía-Monasterio, G. Oshanin Since the prefactor in this dependence is positive definite for sufficiently small J and ρ, the order param- eter is an increasing function of h for small h. Further on, we find that in this limit, the order parameter of the soft and of the spins obey m(A) φ =− 1−2J (1−ρ) 1− J (1+ρσφ) (1+σφ)h , m(A) s = 1−2Jρ(1+σφ) 1− J (1+ρσφ) h . (4.4) We notice that for sufficiently small J , the order parameterm(A) φ is a decreasing function of h, whilem(A) sincreases with an increase of h. Within the opposite limit h →∞, we find from (4.2) that in the leading in h order M (A) =− ρσφ 1− Jρσφ h , (4.5) implying that M (A) →−∞ as h →∞. This means, in turn, that M (A) is a non-monotonous function of h and that for a strong (excessive) bias, the outcome of the vote is always against the preferential candidate. Regarding the behavior of two other order parameters, we notice that since (x∗+h) is always positive definite (for Jρσφ < 1), the order parameter of the rigid spinsm(A) s → 1 as h →∞, which means that the whole subgroup of ordinary voters is in favor of the preferential candidate. On the contrary, the order parameter of the contrarians behaves, in the leading in h order, as m(A) φ =− σφ 1− Jρσφ h , (4.6) i.e., is negative and is growing indefinitely by an absolute value with h so that the order parameter m(A) φ for sufficiently small J is always a monotonously decreasing function of h. In conclusion, we have that in the case A, an excessive bias breaks the ties within the community, letting the ordinary voters to have an opinion in favor of the preferential candidate, and the contrarians to have an opinion against this candidate. However, overall at large h, the contrarians dominate by increasing the strength of their opinion. We proceed with the numerical analysis, which permits us to have a broader look at the behavior of such a mixture of mutually-interacting rigid and soft spins. In particular, it permits us to determine the order parameters at arbitrary, not necessarily small or big values of h, to consider the situations with arbitrary values of the coupling J and also to analyse the effect of the initial configuration. In our numerical Monte Carlo simulations, we first fix an initial configuration of the spin variables, (e.g., stipu- lating that initially all spins are pointing downwards, or upwards, or are in some mixed configuration), fix J and h, and then let the system equilibrate via standard Metropolis algorithm for a sufficiently long time (typically, for 108 time steps) until it reaches an equilibrium configuration, which may, in general, be different from the initial one. The purpose of doing so is as follows: for sufficiently large J and for h = 0, the ferromagnetic system under study acquires a spontaneous “magnetization” so that the order parameters m(A) s (h = 0) and m(A) φ (h = 0) are no longer equal to zero, unlike the case of small J when both m(A) s (h = 0) = m(A) φ (h = 0) = 0. The order parameter m(A) s (h = 0), for example, attains two values m(A) s (h = 0) =±ms , where positive and negative values are chosen by the system with equal probability.In other words, for h = 0, it is equally probable that the whole system will have a negative or a positive magnetization. By choosing the initial state, i.e., by forcing the spins to point in some direction, we break the symmetry between the negative and positive value “helping” the system to arrive into the state with a prescribed sign of magnetization. In figures 2, we first depict the order parameters as functions of h for a small value of J , J = 0.75 and for an initial configuration in which all the spins are pointing upwards and which is let to equilibrate at a fixed J and h for 108 time steps. We observe that for such a value of J , upon an increase of h, the order parameter of the rigid spins grows monotonously with h and quite rapidly, already for h ≈ 2, attains the limiting value m(A) s = 1. The order parameter m(A) φ of the soft spins, on the contrary, is a monotonously decreasing function of h. This property also quite rapidly, already for h ≈ 2, converges to the asymptotic form in (4.6). The combined effect ofm(A) s andm(A) φ is such that the order parameterM (A) first increases, being dominated by the rigid spins, passes through amaximal value, attained at h ≈ 2, and then gradually 13801-6 Negative response to an excessive bias 0 2 4 6 8 10 h 0 0.2 0.4 0.6 0.8 1 M 0 2 4 6 8 10 h -2 -1.5 -1 -0.5 0 0.5 1 mφ 0 2 4 6 8 10 h 0 0.2 0.4 0.6 0.8 1 m s Figure 2. (Color online) Order parameters as functions of the external bias h for J = 0.75, σφ = 0.1, ρ = 0.1 and N = 1111. Initially all spins (both rigid and soft ones) are all set equal to +1 and the system is let to equilibrate for 108 time steps at fixed J and h. The dashed lines stand for the corresponding asymptotic results in (4.5) (left-hand panel), (4.6) (central panel) andm(A) s = 1 (right-hand panel). decreases with h, becoming dominated by the soft spins. Overall, we observe a very good agreement between the numerical results and our predictions for the asymptotic large-h behavior. In figures 3 we plot the order parameters as functions of h for a sufficiently big value of J , J = 4, (such that the system may acquire a spontaneous magnetization at h = 0), and also for two different initial configurations of spins. We start our discussion with the case when the initial configuration is most favorable for the preferential candidate — when all the rigid and the soft spins point upwards (curves depicted by pluses in figures 3) and then are let to equilibrate within 1010 time steps. In this situation, the order parameter m(A) s of the rigid spins does not show any appreciable variation with h and stays constant and equal to 1 for any h. The order parameter m(A) φ appears to be a monotonously decreasing function of h and quite rapidly, already at h ≈ 4, starts to converge to the asymptotic form in (4.6). Overall, the order parameterM (A) is also a monotonously decreasing function of h, which means that prompting such a community to vote for a preferential candidate appears to be counter-productive even at very small values of h. For the initial configuration when all the spins— the rigid and the soft ones— point downwards and are let to equilibrate afterwards, the behavior of the order parameters appears to be more interesting and rich, and exhibits a seemingly discontinuous variation with the value of the external bias resembling 0 2 4 6 8 10 h -1 -0.5 0 0.5 1 M 0 2 4 6 8 10 h -2 -1 0 1 2 mφ 0 2 4 6 8 10 h -1 -0.5 0 0.5 1 m s Figure 3. (Color online) Order parameters as functions of the external bias h for J = 4, σφ = 0.1, ρ = 0.1 with N = 1111. Symbols correspond to two different initial states: {si } = {φ j } = +1 (pluses) and {si } = {φ j } = −1 (crosses), which are subsequently let to equilibrate for 108 time steps. The dashed lines stand for the corresponding asymptotic results in (4.5) (left-hand panel), (4.6) (central panel) and m(A) s = 1 (right-hand panel). 13801-7 V.S. Dotsenko, C. Mejía-Monasterio, G. Oshanin true phase transitions in physical systems. We observe that here the order parameter m(A) s stays equal to its value at h = 0, m(A) s (h) ≈ −1, up to a certain threshold value of h, (which only slightly exceeds h = 2), and then rather abruptly jumps tom(A) s (h) ≈+1, when the bias becomes strong enough to pay the penalty set by the interaction energy with the soft spins. The order parameterm(A) φ first slightly decreases with an increase of h and then abruptly jumps upwards, being turned over by the interactions with the rigid spins. In a way, one may claim that here the opinion of the contrarians gets inverted by the opinion of the ordinary voters. Further, m(A) φ monotonously decreases from the peak value and ultimately, for h ≈ 4, converges to the asymptotic form in (4.6). The global order parameter M (A), which represents the combined effect of the rigid and the soft spins, follows mostly the behavior of the order parameter of the soft spins and exhibits a discontinuous jump from M (A) ≈ −1 to M (A) ≈ +1 at h ≈ 2, and then gradually decreases upon an increase of the external bias. Here, aswell, we observe a very good agreement between the numerical data and our predictions for the asymptotic large-h behavior. 5. Results for the distribution in (2.1). Case B In this case, we have that the order parameterm(B) s obeys a transcendental (3.12), which couples it to the order parameterm(B) φ . For the latter, performing the integrals in (3.13) with the distribution in (2.1), we find the following equation m(B) φ =σφ [ Jρm(B) φ + Is (1−ρ)m(B) s −h ] + tanh [ Jρm(B) φ + Is (1−ρ)m(B) s −h ] . (5.1) As in the previous section, we start here with the solution of the coupled transcendental equations (3.12) and (5.1) in the limits h ¿ 1 and h →∞. In the limit of a linear response, we have m(B) s = 1−ρ (J + Is ) (1+σφ)[ 1− J (1+ρσφ)+ ( J 2 − I 2 s ) ρ(1−ρ)(1+σφ) ] h , (5.2) m(B) φ =− 1− (J + Is ) (1−ρ)[ 1− J (1+ρσφ)+ ( J 2 − I 2 s ) ρ(1−ρ)(1+σφ) ] (1+σφ)h , (5.3) and M (B) = 1−ρ(2+σφ)[ 1− J (1+ρσφ)+ ( J 2 − I 2 s ) ρ(1−ρ)(1+σφ) ] (1+σφ)h . (5.4) These expressions simplify to the ones given by (4.3) and (4.4) in which we set J = Is and, in principle,show a very similar behavior. On the contrary, the behavior in the limit h → ∞ appears to be quite different. At the same time, the order parametersm(B) φ and M (B) have the same linear dependence on h [compare with (4.5) and (4.6)]; that being m(B) φ =− σφ 1− Jρσφ h , M (B) =− ρσφ 1− Jρσφ h , (5.5) and hence, both tend to −∞when h →∞, the order parameterm(B) s exhibits some novel and interesting features. The point is that, unlike the case A, here the argument in the hyperbolic tangent in (3.12) can be positive or negative depending on the value of the coupling parameter Is . More specifically, we have thatfor Is < I cs , where I cs ≈ 1 ρσφ − J , (5.6) the order parameter m(B) s → 1 as h →∞, while for Is > I cs the order parameter m(B) s →−1. This means that in the case B, these are the soft spins which may turn over the rigid ones, present in majority. This is an opposite effect, as compared to the phenomenon which we observed for the case A. 13801-8 Negative response to an excessive bias We illustrate the inversion of the rigid spins by the soft ones by solving the coupled transcendental equations (3.12) and (5.1) numerically, which also permits us to highlight some interesting features of the behavior at the intermediate values of h and Is . In figures 4 we depictm(B) s andm(B) φ as functions of Is for fixed J = 1 and h = 32. We observe thatm(B) φ is a growing function of Is for Is É 12.5. At Is = 12.5, the order parameter abruptly jumps downwards and starts to decreasewith a further increase of Is . The downwardjump ofm(B) φ provokes all the rigid spins to turn over their direction, so thatm(B) s discontinuously changes from m(B) s = 1 to m(B) s = −1. Further, in figures 5 we plot m(B) s and m(B) φ as functions of h for fixed J = 1 and Is = 12.5. We notice thatm(B) φ is a decreasing function of h with a discontinuous downward jump at h ≈ 36. The order parameter m(B) s first is an increasing function of h, up to the value of h at which m(B) φ jumps down: here, as well,m(B) s exhibits a discontinuous jump to −1 and stays constant further. Lastly, in figure 6, we depict the global order parameterM (B) as a function of Is (left-hand panel) and as a functionof h (right-hand panel). We observe that its behavior is entirely dominated by the behavior of the order parameter of the soft spins. Figure 4. Numerical solution of (3.12) and (5.1) for J = 1, σφ = 1, ρ = 0.1 and h = 32. Left-hand panel presentsm(B) s , while the right-hand one presentsm(B) φ as functions of Is . Figure 5. Numerical solution of (3.12) and (5.1) for J = 1, σφ = 1, ρ = 0.1 and Is = 12.5. Left-hand panel presentsm(B) s , while the right-hand one presentsm(B) φ as functions of h. Figure 6. Numerical solution of (3.12) and (5.1) for J = 1, σφ = 1, ρ = 0.1 and Is = 12.5. Left-hand panel presents the global order parameter M (B) vs Is for h = 32, while the right-hand one presents M (B) as a function of h for Is = 12.5. 13801-9 V.S. Dotsenko, C. Mejía-Monasterio, G. Oshanin Conclusions To recap, we have studied here a toy model of opinion formation in a mixed society consisting of two groups of individuals, interacting all with all across both groups, and that differ in their response to an ex- ternal bias favoring one of two potential candidates in elections. While one group of individuals, present in majority, is characterized by their eagerness to accept the externally favored candidate, the minority group— the contrarians— are antagonistic to the bias and tend to have an opinion different from the ex- ternally imposed one. Modelling such a society as a system of appropriately defined interacting, effective spin variables, we arrive at a general conclusion that pushing too strongly for the preferential candidate appears to be counter-productive and leads to a smaller effect than the one achieved with a modest bias. We show, as well, that depending on how the interactions between the two groups and within each of the groups are defined, interesting phenomena of an opinion turn-over can take place, when either of two groups can abruptly change its opinion in favor of the opinion of another group. Acknowledgements We express our warmest thanks to Dr. Olesya Mryglod for her interest in this work and stimulating communications. 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Популяцiя складається зi звичайних iндивiдуумiв, якi становлять бiльшiсть i чия думка має тенденцiю до вирiвнювання пiд дiєю зовнiшнього змiщення, i з деякого числа протилежно настроєних iндивiдуумiв, якi завжди є противниками змiщення, але не знахо- дяться в конфлiктi з простими виборцями. Виборцi взаємодiють мiж собою як всi зi всiма, намагаючись прийти до спiльної думки спiльноти як цiлого. Ми показуємо, що для достатньо слабого зовнiшнього змiщення, думка звичайних iндивiдуумiв є завжди вирiшальною, i результат голосування є в користь преференцiйного кандидата. Навпаки, для надто сильного змiщення протилежно настроєнi iндивiдууми домiнують, приводячи вцiлому до негативного вiдгуку на прикладене змiщення.Ми також показуємо,що для достатньо сильних взаємодiй в межах спiльноти, будь-яка з двох пiдгруп може несподiвано змiнити думку iншої групи. Ключовi слова: нелiнiйний i негативний вiдгук, зовнiшнє змiщення, популяцiя виборцiв 13801-11 https://doi.org/10.1088/1751-8113/49/12/125001 https://doi.org/10.1119/1.1427088 https://doi.org/10.1016/j.physa.2003.10.041 Introduction The model General results for the cases A and B The case A The case B Results for the distribution in (2.1). Case A Results for the distribution in (2.1). Case B