Information flow in political elections: a stochastic perspective
IInformation flow in political elections: a stochastic perspective
Santosh Kumar Radha
Department of Physics, Case Western Reserve University,10900 Euclid Avenue, Cleveland, OH-44106-7079
Often times, a candidate’s attractiveness is directly associated with his clear ideologies and opin-ions on various policies and social issues. Using the ideas of stochastic differential equations andOrnstein-Uhlenbeck Process, we develop a phenomenological model to understand the effect of(un)clearly communicating a candidate’s stance on policies to the voting public. We will show that,counter intuitively, there are quantifiable advantages to be vague on one’s stance.
Quantitative analysis of political elections have been astaple for many years. Models of election are many andvaried, each with its own focus. There is a vast litera-ture on methods to forecast the elections using tools likefundamental indicators[1], market indicators[2], Bayesianmethods[3] and even social media strategies [4] and othergeneral statistical tools [5–7]. Although there is extensivework on modeling of political dynamics between candi-dates [8–12], most of the study focus on global/macroperspective while either neglecting or including the com-plex effects of individual components ( e.g. stances on dif-ferent policies) as a mean field effect.In this letter, we develop a quantitative phenomeno-logical model to understand and analyze the effect of onesuch individual component - efficient information flow inan election/voting system. Information flow is a funda-mental notion in many areas from physics to quantitativefinance. Whenever a decision is being made on a can-didate, one typically encounters uncertainties about thecandidate’s stance on various policies. These uncertain-ties can occur due to various reasons from candidate’slack of effective communication to spread of deliberatemisinformation. While qualitatively, inefficient transferof information from candidate to public might seem likea negative effect, we show that there are certain caseswhere these inefficiencies are not only advantages, butalso necessary. We use Ornstein-Uhlenbeck equations[13]to model the time dependent information flow betweenthe candidate and public. Use of stochastic theory tomodel complex elections have also been used by Fenner etal. [14] to analyze the polls leading up to the UK 2016EU referendum. We will start by introducing the model,after which we will explore the properties and effects ofvarious parameters phenomenologically.
Model : Let us denote the stance of the candidate onpolicy p as µ with his own uncertainty on the stance be-ing σ , where −∞ < µ < ∞ with positive values being infavor for policy and negative values being against. Thiscould model a variety of situations from social issues toideologies. For example, in case of left vs right, a leftleaning moderate candidate might have µ ≈ . , σ ≈ . µ > µ < perceived stance of the candi-date’s policy at time t during the election to by a random variable X t . At the start of election ( t = 0) (or the startof announcing the candidacy), X t | t =0 = X = N ( µ , σ ) i.e. at t = 0, the public has a predisposed idea on whatthe candidate’s stance is with a distribution given by X .For instance, X = µ = 0 would mean that the publichas no idea of what the candidate’s stance is. As timeflows, the candidate’s opinion on p is made clearer thoughvarious information transfer processes like public speech,social media posts etc. and E [ X t | t →∞ ] → µ . This sys-tem is modeled as a solution to a stochastic differentialequation (SDE) given byd X t = r ( µ − X t ) d t + σ d W t , t > . (1)where r is the rate at which the candidate makes hisstance µ clear with a variance of σ ≥ W t is a standardBrownian Motion on R. Equation 1 can alternatively bewritten in terms of stochastic integral form as X t = µ (cid:0) − e − rt (cid:1) + σe − rt (cid:90) t e rs d W s + X e − rt , t ≥ t is given by E [ X t ] = µ (cid:0) − e − rt (cid:1) + e − rt E [ X ] , t ≥ , (3)which gives us the required asymptotic Gaussian behav-ior E [ X t | t →∞ ] → µ . These type of processes are oftenrefereed to as mean-reverting processes. Although thereare different generalizations of Ornstein-Uhlenbeck Pro-cess [16], we here choose the vanilla model to introducethe phenomenology.In general, the formal solution of (1) is given by X t = N ( E [ X t ] , Var[ X t ]) whereVar[ X t ] = σ r (cid:0) − e − rt (cid:1) + e − rt Var [ X ] (4)1 shows a numerical simulation of X t for values of µ = 2 (positive stance) and µ = − N (0 ,
2) in green and red respectively. a r X i v : . [ phy s i c s . s o c - ph ] S e p Black lines show the evolution of mean according to (3).As one can see, initial idea of candidate’s perspective onpolicy is quite spread out which then converges to µ astime proceeds. FIG. 1. Sample trajectories of X t starting from X ≈ N (0 , σ = 0 . r = 1, (green) µ = 2 and (red) µ = − In this model, as seen from (4), Var[ X t ] plays the vitalrole of how collectively confused the public is about thecandidate’s opinion at any given time t . This is mainlygoverned by the variable σ r . It is interesting to note thatthis confusion is not dependent on how strong a negativeor positive stance the candidate takes i.e. µ .With these notion in mind, we now propose that theprobability that the candidate wins ( P T ) Proposition .1
Probability ( P T ) that the candidate winsmaximum votes when the election is at time T is givenby Hellinger distance measure − H ( X T , ˜ X p ) where ˜ X isthe public collective public opinion on policy p . Hellinger distance H is given by[17] H ( f, g ) = 12 (cid:90) ( (cid:112) f ( t ) − (cid:112) g ( t )) dt (5)where f, g are two continuous probability density func-tions. Because Hellinger distance is a bounded metricon the space of probability distributions, we can directlyrelate it to the probability of winning. Prop..1 is noth-ing but a measure of how close the candidate’s perceivedopinion aligns with actual public onion at any given time,matching of which would dictate the winning.Based on (3) and 4, we have X T to be a Gaussiandensity function and if the public stance on p can bemodeled as Gaussian, (5) reduces to P T = (cid:115) σ T ˜ σσ T + ˜ σ exp (cid:40) −
14 ( µ T − ˜ µ ) σ T + ˜ σ (cid:41) , (6)where σ T , µ T are the variance and mean of the randomvariable X T and public stance = N (˜ µ, ˜ σ ). Using (6),we can now study the effect of various parameters in themodel and its result on the probability of winning.Based on (6), as a sanity check, we will first explorethe effect of candidates controversial take on policy p , i.e. | µ − ˜ µ | >>
0. Since σ T is unaffected by µ , we will assign α T = 2 σ T ˜ σ and β T = σ T + ˜ σ , then P T ( µ )= (cid:114) α T β T exp (cid:26) − β T (cid:0) µ (cid:0) − e − rT (cid:1) + e − rT E [ X ] − ˜ µ (cid:1) (cid:27) , (7)Which as T → ∞ is a Gaussian distribution with mean µ − ˜ µ , which essentially says that when every other vari-able = 1, the maximum likelihood of winning the electionis when the public and candidate have the same stanceon p . FIG. 2. P T for 0 < T < µ = 1 , ˜ σ = 1 , σ = 1 and1 < µ <
10 for starting distribution with (a) X = 0 (b) X = − Things get more interesting if we study the likelihoodof winning as a function of number of days to election.Figure 2 shows the evolution of P T as a function of num-ber of days from election for all quantities set to 1, except µ , which varies from 1 to 10. As one can clearly see from(a), for ˜ µ = 1 (the majority accepted stance on policy =1) and the candidate’s stance is far away from it µ = 10,at T → ∞ , the candidate has ≈ i.e. X = − confusioni.e. σ , for a given stance. We will focus on T → ∞ limitfor clarity as extending to finite T is straightforward.Figure 3 (a) shows the calculated P T for various µ asa function of σ , with ˜ µ = 5 and ˜ σ = 1, since we aretaking T → ∞ , one should note that the initial startingpoint ( X ) does not matter as we are interested in theasymptotic behavior. FIG. 3. P T for 0 < T < µ = 1 , ˜ σ = 1 , σ = 1 and1 < µ <
10 for starting distribution with (a) X = 0 (b) X = − We start with µ = 5 (red curve), where we have thepublic stance given by ˜ N (5 , σ = 1, giving us that max-imum likelihood is reached when X T | T →∞ = ˜ N (5 , i.e. µ (cid:54) = ˜ µ (red to green curve), his probability to win reduces.Despite this reduction, one can still tune σ to reach theoptimal probability. This essentially dictates that whenan unpopular stance is held by a candidate, there is anoptimal noise that can be added while communicating topublic which maximizes the chances of winning. From(a), one can also see that as µ − ˜ µ → ∞ , the effect of op-timizing σ does not create appreciable advantage. Thus,when a polarizing candidate is introduced, the best betis to be extremely confusing about the stance and verylittle is to be gained by optimizing σ . Polar opinions : One important use case of the aboveformulation is when there is a binary stance by the publicon a policy. For instance, there can be scenarios wherethe public opinion on a policy p is given by ˜ x ∈ {− , } while the candidate can have his stance x ∈ ( −∞ , ∞ ).We now have the random variable X ∈ ( −∞ , ∞ ), whichthe public perceive as the candidate’s stance, based on which at election time ( T ), public makes the choice ˜ X T based on candidate’s perceived stance X T , given by˜ X T = (cid:40) +1 if X T > , − X T < . (8)Equation 12 says that, at the election date, the pub-lic makes the decision based on the candidate’s posi-tive/negative stance on the policy, irrespective how howstrong or weak the stance is. Following the phonologicalmodel, the important quantity to calculate is the proba-bility kernel of (1), which is given by P ( t, x, y )= 1 (cid:112) πσ (1 − e − rt ) /r exp (cid:40) − ( y − µ − ( x − µ ) e − rt ) σ (1 − e − rt ) /r (cid:41) , (9)where P is the probability kernel for X to reach y from x in time t with µ, σ being the mean and spreadof candidate’s stance. Using (9), one can calculate theideal time to start the election[18]. We will illustratethis with an example where we know that the majorityfavors +1 i.e. ˜ µ = +1 and ˜ σ = 0 (this information can beextracted from public polls etc. ). In this case, we firstneed to calculate the probability ( P † ( t, x )) for a point x to reach y > t , P † ( t, x ) = (cid:90) ∞ P ( t, x, y ) dy = 1 −
12 erfc (cid:18) − √ r ( − µ − ( − µ + x ) e − rt ) σ √ − e − rt (cid:19) (10)Figure 4(a) shows P † ( t, x ) for µ = σ = r = 1. This isthe probability that a person who thinks that the can-didate’s stance is x would choose ˜ X t = 1, qt time t .Thus under the assumption that majority favors +1, thiswould guarantee that the person is voting for the candi-date. Using P † ( t, x ), we can now calculate exactly howlong to wait for the election for various starting points.For example, if we know that at this given moment t = 0,the public perceives the stance of candidate to be f ( x )where f ( x ) := N ( µ , σ ), then the probability that theyreach +1 at time t is given by (11) P (cid:48) ( t ) = (cid:90) + ∞−∞ f ( x ) P † ( t, x ) dx. This is easy to Figure 4(a), as we integrate out x ,sampling from the points where we start i.e. N ( µ , σ )(shown in black line (a)). We show P (cid:48) ( t ) in (b) for var-ious starting point µ . It is clear that when the publicperceives the candidate to be on the negative side, thereis a huge advantage to wait (assuming the public stance FIG. 4. P T for 0 < T < µ = 1 , ˜ σ = 1 , σ = 1 and1 < µ <
10 for starting distribution with (a) X = 0 (b) X = − on policy is positive ˜ µ = +1 ) while the inverse is true ifthey perceive him to be on the positive side. Althoughwe discussed the case for majority favors +1, this caneasily be extended to − et al. [19] showed that Fake news can be modeled by adding anoise with non-zero drift.Even though all previous discussions pertained to sin-gle policy, one can easily generalize this model to ac-commodate multiple policy by increasing the dimension-ality of the random variable. This is important be-cause public decide between candidates based on numberof different policies that are both quantifiable and nonquantifiable. Subconsciously, these stances are mappedinto an overall score, and the public at the end votesfor the candidate with the highest score. Similar to(1), one can define the n − dimensional OU process with X t = ( X t , X t , . . . , X nt ) which are random variables forpolicies p = ( p , p , . . . , p n ) satisfying d X it = (cid:88) k R ik (cid:0) M k − X kt (cid:1) d t + (cid:88) l Σ il d W lt , t > , (12)where R and Σ are n × n matrices and M is a n × W t is a vector of n independent Brownianmotions and repeated. Formal solution of (12) is givenby time dependent Gaussian vector[20] Summery : In this work, we have developed a quan-titative phenomenological model to understand the effectof uncertainty in information flow in elections. This offersa initial model upon which complexities can be adornedto quantitatively analyze the effects of isolated variables,which helps in quantitative strategizing of election cam-paigns. Further, we showed the existence of situationswhere ineffective information flow can be advantageous.Finally, we noted that this formalism, although devel-oped for single policy study, can be extended to multiplepolicies and is still analytically tractable. [1] P. Hummel and D. Rothschild, Fundamentalmodels for forecasting elections, ResearchDMR.com/HummelRothschild FundamentalModel (2013).[2] J. Berg, R. Forsythe, F. Nelson, and T. Rietz, Re-sults from a dozen years of election futures markets re-search, Handbook of experimental economics results ,742 (2008).[3] D. A. Linzer, Dynamic bayesian forecasting of presiden-tial elections in the states, Journal of the American Sta-tistical Association , 124 (2013).[4] D. Gayo-Avello, No, you cannot predict elections withtwitter, IEEE Internet Computing , 91 (2012).[5] P. Hummel and D. Rothschild, Fundamental models forforecasting elections at the state level, Electoral Studies , 123 (2014).[6] C. Klarner, Forecasting the 2008 us house, senate andpresidential elections at the district and state level, PS:Political Science and Politics , 723 (2008).[7] B. E. Lauderdale and D. Linzer, Under-performing, over-performing, or just performing? the limitations offundamentals-based presidential election forecasting, In-ternational Journal of Forecasting , 965 (2015).[8] L. B¨ottcher, H. J. Herrmann, and H. Gersbach, Clout,activists and budget: The road to presidency, PloS one , e0193199 (2018).[9] D. Braha and M. A. De Aguiar, Voting contagion: Mod-eling and analysis of a century of us presidential elections,PloS one , e0177970 (2017).[10] J. Fern´andez-Gracia, K. Suchecki, J. J. Ramasco,M. San Miguel, and V. M. Egu´ıluz, Is the voter modela model for voters?, Physical review letters , 158701(2014).[11] S. Galam, The dynamics of minority opinions in demo-cratic debate, Physica A: Statistical Mechanics and itsApplications , 56 (2004).[12] S. K. Radha, Stochastic differential theory of cricket(2019), arXiv:1908.07372 [physics.soc-ph]. [13] G. E. Uhlenbeck and L. S. Ornstein, On the theory ofthe brownian motion, Phys. Rev. , 823 (1930).[14] T. Fenner, M. Levene, and G. Loizou, A stochastic differ-ential equation approach to the analysis of the uk 2016eu referendum polls, Journal of Physics Communications , 055022 (2018).[15] P. E. Protter, Stochastic differential equations,in Stochastic integration and differential equations (Springer, 2005) pp. 249–361.[16] A. K. Dixit, R. K. Dixit, and R. S. Pindyck,
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