Kinetic exchange models for social opinion formation
aa r X i v : . [ phy s i c s . s o c - ph ] J u l Kinetic exchange models for social opinion formation
Mehdi Lallouache ∗ D´epartement de Physique, ´Ecole Normale Sup´erieure de Cachan, 94230 Cachan, France andChaire de Finance Quantitative, Laboratoire de Math´ematiques Appliqu´ees aux Syst`emes,´Ecole Centrale Paris, 92290 Chˆatenay-Malabry, France
Anirban Chakraborti † Chaire de Finance Quantitative, Laboratoire de Math´ematiques Appliqu´ees aux Syst`emes,´Ecole Centrale Paris, 92290 Chˆatenay-Malabry, France
Bikas K. Chakrabarti ‡ Centre for Applied Mathematics and Computational Science,Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700 064, IndiaEconomic Research Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700 018, India andChaire de Finance Quantitative, Laboratoire de Math´ematiques Appliqu´ees aux Syst`emes,´Ecole Centrale Paris, 92290 Chˆatenay-Malabry, France (Dated: April 20, 2018)We propose a minimal model for the collective dynamics of opinion formation in the society, bymodifying kinetic exchange dynamics studied in the context of income, money or wealth distributionsin a society. This model has an intriguing spontaneous symmetry breaking transition.
PACS numbers: 87.23.Ge 02.50.-rKeywords: Econophysics; Sociophysics; kinetic theory
I. INTRODUCTION
A very interesting problem in studying society and so-cial dynamics is the one of “opinion formation”, whichis a collective dynamical phenomenon, and as such areclosely related to problems of competing cultures or lan-guages [1–3]. It deals with a “measurable” response ofthe society to e.g., political issues, acceptances of in-novations, etc. A number of models of competing op-tions have been introduced to study it, e.g., the “voter”model (which has a binary opinion variable with the opin-ion alignment proceeding by a random choice of neigh-bors) [4], or the Sznajd-Weron discrete opinion formationmodel (where more than just a pair of spins is associatedwith the decision making procedure) [5]. There have beenstudies of systems with more than just two possible opin-ions [6], or where the opinion of individuals is representedby a “continuous” variable [7–9]. Since opinion formationin a human society is mediated by social interactions be-tween individuals, such social dynamics was consideredto take place on a network of relationships by Holme andNewman [10]. Several other significant studies have fol-lowed, which we not not mention here.A two body exchange dynamics has already been de-veloped in the context of modelling income, money orwealth distributions in a society [11–14]. Detailed ana-lytical structure of the collective dynamics in these mod-els are now considerably well-developed [15, 16]. Here, ∗ [email protected] † [email protected] ‡ [email protected] we propose a minimally modified version of those modelsfor the collective dynamics of opinion formation in thesociety. II. KINETIC EXCHANGE MODELS OFMARKET
Recently physicists and mathematicians have been in-terested in studying the wealth distributions in a closedeconomy using kinetic exchange mechanism, which hasled to new insights into this field (see Refs. [11–14]).The general aim was to study a many-agent statisticalmodel of closed economy (analogous to the kinetic the-ory model of ideal gases) [17–19], where N agents ex-change a quantity x , that may be defined as wealth.The states of agents are characterized by the wealth { x i } , i = 1 , , . . . , N , and the total wealth W = P i x i is conserved. The question of interest is: “What is theequilibrium distribution of wealth f ( x ), such that f ( x ) dx is the probability that in the steady state of the system,a randomly chosen agent will be found to have wealthbetween x and x + dx ?”The evolution of the system is carried out accordingto a prescription, which defines the trading rule betweenagents, where the agents interact with each other througha pair-wise interaction characterized by a saving param-eter λ , with 0 ≤ λ ≤
1. The dynamics of the model (CC)is as follows [19]: x ′ i = λx i + ǫ (1 − λ )( x i + x j ) ,x ′ j = λx j + (1 − ǫ )(1 − λ )( x i + x j ) . (1)It can be noticed that in this way, the quantity x is con-served during the single transactions: x ′ i + x ′ j = x i + x j ,where x ′ i and x ′ j are the agent wealths after the transac-tion has taken place.This model for λ > x m > x .For λ = 0, the model reproduces the results of Yakovenko[18], where the equilibrium distribution is the Gibb’s dis-tribution. In general, the functional form for such dis-tributions was conjectured to be a Γ-distribution on thebasis of an analogy with the kinetic theory of gases: f ( x ) = 1Γ( n ) (cid:18) n h x i (cid:19) n x n − exp (cid:18) − nx h x i (cid:19) , (2)where n = D ( λ )2 = 1 + 3 λ − λ . (3)Indeed, starting from the Maxwell-Boltzmann distribu-tion for the particle velocity in a D dimensional gas, itcan be shown that the equilibrium kinetic energy distri-bution coincides with the Gamma-distribution (2) with n = D . This conjecture is remarkably consistent withthe fitting provided to numerical data [20, 21].As a further generalization, the agents could be as-signed different saving propensities λ i [22]. In particular,uniformly distributed λ i in the interval [0 ,
1) had beenstudied numerically in Refs. [22]. This model (CCM) isdescribed by the trading rule x ′ i = λ i x i + ǫ [(1 − λ i ) x i + (1 − λ j ) x j ] ,x ′ j = λ j x j + (1 − ǫ )[(1 − λ i ) x i + (1 − λ j ) x j ] . (4)One of the main features of this model, which is sup-ported by theoretical considerations [15, 23, 24], is thatthe wealth distribution exhibits a robust power-law atlarge values of x , f ( x ) ∝ x − α − , (5)with a Pareto exponent α = 1 largely independent ofthe details of the λ -distribution. Note that other val-ues of exponents can also be generated by modifying theexchange rules [12]. III. A KINETIC EXCHANGE MODEL FOROPINION FORMATION
Toscani [25] had recently introduced and discussed ki-netic models of (continuous) opinion formation involv-ing both exchange of opinion between individual agentsand diffusion of information. He showed that there areconditions which ensure that the kinetic model reachesnon-trivial stationary states in case of lack of diffusion incorrespondence of some opinion point, and obtained ana-lytical results by considering a suitable asymptotic limitof the model yielding a Fokker-Planck equation for thedistribution of opinion among individuals. Based on this
FIG. 1. Schematic diagram of the minimal model where ran-dom discussions/arguments between two persons i and j withopinions O i ( t ) and O j ( t ), respectively, cause the update ofopinions O i ( t + 1) and O j ( t + 1). model, During et al [26] proposed another mathematicalmodel for opinion formation in a society that is built oftwo groups, one group of ordinary people and one groupof strong opinion leaders. Starting from microscopic in-teractions among individuals, they arrived at a macro-scopic description of the opinion formation process thatis characterized by a system of FokkerPlanck-type equa-tions. They discussed the steady states of the system,and extended it to incorporate emergence and declineof opinion leaders. On a different approach, Iniguez et al[27] examined a situation in which these non-identical in-dividuals form their opinions in information-transferringinteractions with others. They developed a dynamic net-work model, where they consider short range interactionsfor direct discussions between pairs of individuals, longrange interactions for sensing the overall opinion modu-lated by the attitude of an individual, and external fieldfor outside influence.Following the CC and CCM models, described in theearlier section, we now propose a minimal model for thecollective dynamics of opinion O i ( t ) of the i -th person inthe society of N ( N −→ ∞ ) persons: O i ( t + 1) = λ i O i ( t ) + ǫλ j O j ( t ) ,O j ( t + 1) = λ j O j ( t ) + ǫ ′ λ i O i ( t ) , (6)where − ≤ O i ( t ) ≤ i and t , and 0 ≤ λ i ≤ quenched variables (do not change with time, butvary from person to person), and ǫ and ǫ ′ are annealed variables (change with time), that are random numbersuniformly distributed between 0 and 1.The above described model dynamics, follows thetwo-body “discussions/arguments” modelled as scatter-ing processes and depicted schematically in Fig. 1. It isbased on the logic that during the discussion/argumentevent with any person j , the person i with high/low “con-viction” (parametrized by the λ i ), will retain his/her ownearlier opinion O i ( t ) proportional to the factor λ i , and be influenced to change the opinion by the j -th person’s in-fluence determined by a contribution which will dependon the j -th person’s conviction λ j (and not by the factor1 − λ j as in market dynamics Eq. 1 or 4). Also, as noconservation in opinion is possible (unlike in the marketmodels above), the annealed variables ǫ and ǫ ′ are nowconsidered to be uncorrelated . Additionally we assumethat | O i ( t ) | ≤
1, for all i and t . A. Homogeneous conviction factor case
When we assume λ i = λ for all i (equivalent to the CCmodel for market dynamics), the above equations reduceto O i ( t + 1) = λ ( O i ( t ) + ǫO j ( t )) ,O j ( t + 1) = λ ( O j ( t ) + ǫ ′ O i ( t )) . (7)This leads to an intriguing spontaneous symmetry break-ing transition beyond a threshold value of λ c = 2 / O i (0) (at t = 0) (“symmetric” state, when the orderparameter h O i ≡ (1 /N ) P i O i ( t = 0) = 0), leads thesystem to collectively evolving to two kinds of state:(i) “Para” or “indifferent” state, where O i ( t )’s are all zeros ( h O i = 0) after a “relaxation” time τ , for λ values less than λ c = 2 /
3; or(ii) “Symmetry broken” or “polarised” state, where O i ( t )’s are either all positive or all negative ( h O i 6 =0) after a “relaxation” time τ , for λ > / λ values less than 2 /
3, with h ǫ i = 1 /
2, the recursion relation for the order parameter h O i becomes a simple multipler equation with the valueof the multiplier less than unity, leading to h O i = 0 even-tually. For higher values of λ , the fluctuations in ǫ areimportant (and cannot be replaced by its simple average,as above) because of asymmetric contributions from thesecond term of both the above equations (if the contri-bution of the second term in Eq. 7 takes the value of | O ( t + 1) | to greater than unity, only partial contribu-tion of the second term is accepted, while for its lowervalues the acceptance is full). We find, this seeminglyleads to a discontinous or “first order” symmetry break-ing transition at λ c = 2 / B. Heterogeneous conviction factor case
Here, we assume λ i ’s to be uniformly spread in theinterval [0,1) (equivalent to the CCM model for market FIG. 2. The variation of the order parameter h O i ≡ (1 /N ) P i O i ( t ) against λ . dynamics). We study similarly, starting from “symmet-ric” states (with random positive and negative values of O i (0), the evolution of the system. The dynamics hereleads collectively to the “Polarized” or “Symmetry bro-ken” state ( O i ( t ) are either all positive or all negative,for all i , and times t > τ ) only. The “indifferent” states(with O i ( t ) = 0 for all i , for times t > τ ) disappear inthe large system size limit, although this is clearly a fixedpoint of the dynamics given by Eq. 6. We believe, this isalso a clear feature of the opinion dynamics model pro-posed by Iniguez et al [27], where also this state is surelya fixed point of their model.It may be noted that the above dynamics can be con-siderably modified by the presence of “polarizing field”terms h i (fixed over time t but dependent on person i ),added linearly to the dynamical equations Eq. 6 of O i ( t ).Such “fields” can be provided by the “influences” of themedia in the society. Detailed analyses of the field terms,etc. will be reported elsewhere [29]. IV. DISCUSSION AND SUMMARY
The appearance of spontaneous symmetry breaking inthis kinetic opinion exchange model is truely remarkable.It appears to be one of the simplest collective dynamicalmodel of many-body dynamics showing non-trivial phasetransition behaviour. The details of this transition isunder investigation and will be reported elsewhere.
ACKNOWLEDGMENTS