Block size dependence of coarse graining in discrete opinion dynamics model: Application to the US presidential elections
aa r X i v : . [ phy s i c s . s o c - ph ] D ec Block size dependence of coarse graining in discrete opinion dynamicsmodel: Application to the US presidential elections
Kathakali Biswas , Soumyajyoti Biswas , Parongama Sen Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India. Department of Physics, SRM University - AP, Andhra Pradesh - 522502, India
Abstract
The electoral college of voting system for the US presidential election is analogous to a coarsegraining procedure commonly used to study phase transitions in physical systems. In a recentpaper, opinion dynamics models manifesting a phase transition, were shown to be able to explainthe cases when a candidate winning more number of popular votes could still lose the generalelection on the basis of the electoral college system. We explore the dependence of such possibilitieson various factors like the number of states and total population (i.e., system sizes) and get aninteresting scaling behavior. In comparison with the real data, it is shown that the probabilityof the minority win, calculated within the model assumptions, is indeed near the highest possiblevalue. In addition, we also implement a two step coarse graining procedure, relevant for bothopinion dynamics and information theory.
1. Introduction
The concept that human society with a large number of mutually interacting agents can beregarded as a complex system consisting of an ensemble of basic units following simple rules, hasled to the investigation of several social phenomena using models commonly studied in statisticalphysics. One of the most studied social phenomena has been the formation of opinion, whereinteraction between agents are considered at a microscopic level in the models, with or withoutexternal fields. The aim is often to find out the mechanism by which it is possible to achieve anagreement or consensus. The first step in such modeling approaches is to quantify the opinionsof the agents involved. This task is easier [1, 2, 3, 4] when the choice of the opinion is binary.Examples of such cases are yes/no referenda, a two party voting etc., where the said quantificationis straight-forward with the use of a binary variable to represent the support for either side. Itis often done with ± Preprint submitted to Physica A December 22, 2020 rocess of the US president-ship [13]. This is not a direct election process for the post of thepresident; an intermediate body, namely the electoral college delegates, ultimately decide who willbe the president. In the electoral college system, in most cases, all the delegates from a particularstate are assigned to the winner from that state and the overall winning candidate is then decidedon the basis of the support of the majority of those delegates. This process has led to the ratherrare cases when the popular candidate (PC henceforth), i.e., the one with the maximum number ofindividual votes could turn out to be the loser after the electoral college system is applied. Clearly,this is a result of strong local fluctuations in the voting pattern of the different states and morelikely to happen in closely fought elections. If there are two candidates only, say A and B, the votersmay be assigned the state values ± ± y ea r √ M Figure 1: In the US, the number of states ( M ) has changed over the years. As discussed in the text, while calculatingthe minority win probability, the relevant scaling variable turns out to be b/L , which is 1 / √ M assuming uniformpopulation density. The figure shows the variation of 1 / √ M with time in the US. The variation is non-monotonic intime during the period of the civil war, due to the confederate states. Other than that 1 / √ M varies from 1 / √
11 in1788 to 1 / √
50 in 1959 and remained constant thereafter. first ratification of admissions during 1787-88, the number of states in the US continued to increase,until the period of the civil war (1861-65), during which the confederate states were in existence,reducing the total number of states in the US. Since the end of the civil war, the total numberof states kept increasing, until the present tally of 50 was reached in 1959 with the admission ofHawaii.In this work, we therefore generalize the problem and ask the question: how does the probabilitythat the PC turns out to be the loser after the so called coarse graining, depend on the numberof states. We may thus predict what would be the probability of a PC losing if some states arejoined/split in future. We have used the Kinetic exchange model (KEM) and the Ising model(IM) (both show an order disorder phase transition) on two dimensional square lattices for differentsystem sizes N = L and applied coarse graining with different scale factors b . We note b/L emergesas the relevant scaling variable which is identical to p /M where M is the number of states.The entire procedure can also be regarded as a problem in information theory. Particularly,the time series of the sign of the opinion favoring the PC can be thought of as the input stringand that of the GW the output string for the electoral college or the coarse graining process. Therelative mutual information (see Eq. (4) in Ref. [8]) is then a measure of the probability of faithfultranslation of the popular opinion by the electoral college, which is close to unity in the orderedstate and decreases sharply near the critical point.Hence the estimate in which we are interested, the probability that the coarse grained result ofthe order parameter is opposite in sign, is generally termed the error ǫ ( b, L ), studied as a function of b and L . b , the area of a block is an integer, b itself can be irrational. For a given L , the maximumvalue of b is L/ √
2. Two limiting cases are immediately identified. If b is trivially 1, then eachindividual represents a state and essentially no coarse graining is required and ǫ (1 , L ) = 0. On theother hand, when b = L , we are taking the whole population as a block and the coarse graining willmerely yield the same value of the order parameter making ǫ ( L, L ) = 0. Intermediate values of b will show a point of maximum error i.e., the highest probability for the minority win. Given thatthe number of states in the US has varied over the years, it is interesting to identify the proximity3 -1 -0.5 0 0.5 1 b=200 m -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 b=100 -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 b=50 -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 b=40 m -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 b=20 -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 b=10 -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 b=4 m m -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 b=2 m -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 b=1 m Figure 2: In the Ising model near critical point, the magnetization of the original lattice ( m ) is plotted against that ofthe course grained lattice ( m ) for different values of b , the coarse graining box length. The cases where the signs ofthese two quantities are different from each other (second and fourth quadrant; points marked in purple), the minoritycandidate wins. Clearly, this is not possible in the limits b = 1 and b = L (here L = 200). However, in other casessuch situations exist (see text for further quantification). of the b/L value to the point where the maximum error is predicted from the model.
2. Models and quantities calculated
As mentioned earlier, we study the Ising model (IM) and a kinetic exchange model (KEM)in capturing the dynamics of essentially two-party voting systems. The Ising model represents abinary opinion case with s i = ±
1, the opinion held by the i th agent at the site i . The two valuesare used to define the support for the two candidates. The Ising model with the Hamiltonian H = − J P
1, i.e., | o i | ≤ µ ij is an annealed variable assuming the values ±
1, andis negative with probability p . An order disorder transition occurs close to p c ≈ .
11 here [16, 17].The normalized order parameter m in the two models are given by P i s i /N and P i o i /N where N is the total number of agents; N = L in a L × L lattice. The initial state is chosen to be totallyrandom for both models corresponding to a high noise value. The system gradually relaxes towards4quilibrium state, which is still disordered but having large spatial correlation. The order parameteroscillates about zero close to the critical point as a function of time. For each time step, if the orderparameter m is not exactly zero, the coarse graining is applied with different scale factors b . Themagnetization m , after coarse graining, is calculated. We estimate the probability ǫ ( b, L ) that mm <
0. The variation of this probability with b for different values of L are then studied.Another quantity that is estimated is how the error ǫ propagates in successive coarse graining.For this we have considered a two step procedure. If the one step coarse graining is done with scalefactor b , we now consider successive coarse graining procedures by factors b and b respectivelywith b b = b . It is of course well known in critical phenomena that the the two procedures leadto the same behavior as far as normalization of the parameters are concerned [14]. However, thequestion we ask here is somewhat different. In an election procedure, the winner is usually decidedon a majority vote basis. Subsequently, the issues to be resolved are decided on the basis of thevotes by the elected representatives. The outcome, however, could be different if all the individualswere allowed to vote on these issues. So the two step procedure measures how far the people’sdecision could be propagated to a higher level. In the context of information theory, it could beregarded as a two step coding.In the simulations on the square lattices, periodic boundary condition has been used and severalconfigurations are run to get the average values.
3. Results
As mentioned before, the electoral college system in the US presidential election resembles asingle step coarse graining process, with each coarse graining block representing a state. Here wereport the results of a single step coarse graining when the number of coarse graining blocks coveringthe system is varied i.e., the block sizes are varied. In Fig. 2 the magnetization of the original latticeis plotted against that of the coarse grained lattice for the Ising model. For different values of b , thepoints where the signs of these two quantities are different, varies. In the extreme limits b = 1 and b = L such points do not exist. We now go on to further quantify the variations in probabilities ofsuch cases.Although we are interested in the case when the system is close to criticality, one can calculatethe errors for higher noise values as well. We calculate ǫ ( b, L ) for a fixed value of L for differentnoise values for the Ising model and the KEM, shown in Fig. 3a and 4a. As expected, ǫ increaseswith the noise factor. We also note that ǫ , which is zero at the extreme limits, increases with b and shows a shallow peak at b ∼ b p only very close to the maximum value of b = L/ √
2. For largervalues of the noise, the results are almost independent of b as the system approaches a random case,also shown for comparison.Next, the results for different system sizes very close to criticality are presented in Figs 3b and4b where ǫ is plotted for different system sizes. We note that the data collapse when they areplotted against b/L . It may be added that the same happens for the random case, and thereforepresumably for all values of the noise factor above criticality.Hence b/L is identified as the scaling variable. We note that the scaling collapse becomes betteras the system sizes are increased, in general there are quite strong finite size effects. Specifically, theerror ǫ ( b, L ) decreases for smaller system sizes. Intuitively it is clear, as the effective ‘critical point’of smaller systems (say, the peak of the heat capacity in the Ising model) is higher than that ofthe thermodynamic limit, the order parameter at a given distance from the thermodynamic criticalpoint will increase with the decrease in the system size and hence will give lower values of ǫ .5 -3 -2 -1 (1876,1888)(2000,2016)(a) ε ( b , L ) b/L T=1.05T c T=1.1T c T=1.3T c T=1.5T c T=1.7T c T=1.9T c Random 10 -2 -1 -3 -2 -1 (1876,1888)(2000,2016) (b) ε ( b , L ) b/LL=975L=1071L=1197 Figure 3: Variation of the minority win probability ǫ ( b, L ) for the Ising model. (a) The variation of the minoritywin probability ǫ ( b, L ) is shown for the Ising model in two dimensions for L = 819 at different temperatures above T c . As the temperature is raised significantly higher than T c , the curves approach the random configuration limit, asexpected. The arrows indicate the points where the events of minority win did occur in the past. (b) The minoritywin probabilities are plotted for different system sizes near the critical point. A data collapse is seen for the scalingvariable b/L . As before, the arrows indicate the events of minority win. It is clear that for further rise in the numberof states would significantly reduce the minority win probability. Now, clearly the variable b/L = 1 / √ M , where M is the number of states. In the case of theUS, as shown in Fig. 1, the number of states in the US has varied considerably over the years.Particularly, the quantity 1 / √ M has changed between 1 / √ ≈ .
301 in 1788 to 1 / √ ≈ . b decreasedin the coarse graining. In the Figs 3 and 4, the arrows indicate the years of PC losing. We now discuss the two step coarse graining done with scale factors (block size) b and b andcompare the results with the one step process with b = b b . We also compare the results when theorder of the two step process is reversed, i.e., first with a scale factor b and then b . We use theconvention b > b .We first consider the random case which corresponds to infinite noise. Here it is found that thetwo step process significantly increases the error but the order hardly matters. Table 1 shows theerrors in the two step process.Let ǫ ′ ( b , b , L ) denote the error for the two step process, i.e., at the end of the two coarsegrainings. After the first step, the fraction which retains the original sign of the order parameter is(1 − ǫ ( b , L )). The probability the sign is changed in the second step for this fraction is ǫ ( b , L/b ).For the fraction ǫ ( b , L ) for which the sign did change after the first step, the contribution to ǫ ′ willcome if these configurations retain the signature in the second step. This happens with probability(1 − ǫ ( b , L/b )). Hence ǫ ′ ( b , b , L ) = ǫ ( b , L/b )(1 − ǫ ( b , L )) + (1 − ǫ ( b , L/b )) ǫ ( b , L ) . (2)In the scaling regime, ǫ ( b, L ) is a function of b/L only. So ǫ ′ ( b , b , L ) = ǫ ( b/L )(1 − ǫ ( b /L )) + (1 − ǫ ( b/L )) ǫ ( b /L ) . (3)6 -3 -2 -1 (1876,1888)(2000,2016)(a) ε ( b , L ) b/L p=0.12p=0.16p=0.20p=0.25p=0.30p=0.35Random 10 -2 -1 -3 -2 -1 (1876,1888)(2000,2016) (b) ε ( b , L ) b/LL=1035L=1155L=1275 Figure 4: The variation of the minority win probability ǫ ( b, L ) of the KEM. (a) The variation of the minority winprobability ǫ ( b, L ) is shown for the KEM in two dimensions for L = 819 at different values for the parameter p above p c . As the parameter p is raised significantly higher than p c , the curves approach the random configuration limit, asexpected. The arrows indicate the points where the events of minority win did occur in the past. (b) The minoritywin probabilities are plotted for different system sizes near the critical point. A data collapse is seen for the scalingvariable b/L . As before, the arrows indicate the events of minority win. It is clear that for further rise in the numberof states would significantly reduce the minority win probability. Let us call ǫ ( b/L ) = y and ǫ ( b /L ) = x . Then the RHS of above eq becomes: y (1 − x ) + x (4)This quantity will increase with x unless y is greater than 0.5 which is usually not the case. Onecan proceed further for the random case and obtain an upper bound for the error. Here, we havenoted that ǫ has a monotonic slow increase for small values of b and attains a constant value forlarger b values. So an upper bound on x is y (since b < b ). y has more or less a constant value ≈ . ǫ ′ ≤ y (1 − y ) ≈ .
32. We indeed get that the two stepprocedure gives values less than this upper bound (see Table 1).According to Eq. (3), the results should depend on the order in which the 2 step coarse grainingis done. For the reverse order we will get Eq. (4) where x is replaced by x ′ = ǫ ( b /L ). However, forthe random case, as ǫ remains almost constant for a considerable range of values of b/L (see Figs3, 4), we get negligible difference when the order is changed.On the other hand, for both KEM and IM, close to criticality, we find that there is an appreciabledependence of ǫ on b/L and the results for the two step process are found to be sensitive to theorder (see Tables 2 and 3). In particular, we note that when the coarse graining is done with b < b first, the resultant errors are more or less same as that for the two step process. This might notbe expected from eq. 4, however, it must be remembered that the above analysis is made for thethermodynamic limit, L → ∞ ; the finite size effects mentioned earlier will be enhanced in a twostep process.When b is used first, the error increases. This is not difficult to explain: a larger value of b gives larger errors after the first step (unless it is beyond b p where a peak value occurs which hasnot been considered). Naturally, in the second step, when the system size has got reduced, evenwith b smaller, the error is increased.
4. Discussion and conclusions
The indirect nature of the election of the US president highlights the importance of the coarsegraining process while modeling the voting process using discrete Ising-symmetric models. The7 able 1: Two step coarse graining in the random case
L b , b b b /L ǫ ′ ( b , b , L ) ǫ ′ ( b , b , L ) ǫ ( b b , L )495 3,5 0.030 0.269 0.270 0.205495 3,15 0.091 0.272 0.272 0.204495 5,9 0.091 0.277 0.276 0.204495 3,55 0.333 0.262 0.264 0.191495 5,33 0.333 0.268 0.270 0.191495 11,15 0.333 0.271 0.270 0.191585 3,5 0.026 0.268 0.268 0.203585 3,39 0.200 0.268 0.270 0.201585 9,13 0.200 0.277 0.277 0.201585 3,65 0.333 0.263 0.265 0.193585 5,39 0.333 0.269 0.269 0.193585 13,15 0.333 0.274 0.272 0.193693 3,7 0.030 0.273 0.269 0.204693 3,33 0.143 0.273 0.274 0.204693 9,11 0.143 0.276 0.276 0.204693 3,77 0.333 0.266 0.264 0.194693 7,33 0.333 0.270 0.272 0.194693 11,21 0.333 0.270 0.272 0.194 Table 2: Two step coarse graining in the Ising model
L b , b b b /L ǫ ′ ( b , b , L ) ǫ ′ ( b , b , L ) ǫ ( b b , L )495 3,5 0.030 0.086 0.091 0.084495 3,15 0.091 0.145 0.165 0.142495 5,9 0.091 0.149 0.154 0.142495 3,55 0.333 0.177 0.222 0.175495 5,33 0.333 0.179 0.214 0.175495 11,15 0.333 0.188 0.195 0.175585 3,5 0.026 0.096 0.099 0.093585 3,39 0.200 0.186 0.236 0.185585 9,13 0.200 0.195 0.201 0.185585 3,65 0.333 0.187 0.238 0.186585 5,39 0.333 0.191 0.233 0.186585 13,15 0.333 0.203 0.209 0.186693 3,7 0.030 0.112 0.118 0.109693 3,33 0.143 0.180 0.218 0.178693 9,11 0.143 0.188 0.192 0.178693 3,77 0.333 0.181 0.241 0.179693 7,33 0.333 0.186 0.222 0.179693 11,21 0.333 0.193 0.213 0.1798 able 3: Two step coarse graining in the kinetic exchange model L b , b b b /L ǫ ′ ( b , b , L ) ǫ ′ ( b , b , L ) ǫ ( b b , L )495 3,5 0.030 0.063 0.065 0.063495 3,15 0.091 0.118 0.138 0.121495 5,9 0.091 0.117 0.122 0.121495 3,55 0.333 0.169 0.219 0.169495 5,33 0.333 0.168 0.203 0.169495 11,15 0.333 0.168 0.176 0.169585 3,5 0.026 0.0793 0.081 0.082585 3,39 0.200 0.181 0.228 0.179585 9,13 0.200 0.178 0.194 0.179585 3,65 0.333 0.170 0.238 0.169585 5,39 0.333 0.170 0.219 0.169585 13,15 0.333 0.177 0.184 0.169693 3,7 0.030 0.090 0.097 0.091693 3,33 0.143 0.190 0.210 0.192693 9,11 0.143 0.185 0.188 0.192693 3,77 0.333 0.183 0.234 0.183693 7,33 0.333 0.183 0.215 0.183693 11,21 0.333 0.187 0.204 0.183process of coarse-graining, particularly near the critical point of a spin system, is supposed to keepthe system invariant under a renormalization group sense. However, as was noted before (see e.g.Ref. [8]), the ‘invariance’ does not guarantee that the sign of the magnetization of the original andthe coarse grained lattice would remain the same. In fact, the probability of such events is known tovary systematically, showing finite size scaling behavior, near the critical point. Indeed, the processof coarse-graining is a loss of information that can have very significant effect where the final signof the order parameter matters. One such situations is the US presidential election. The electoralcollege system in the US, that assigns all delegates of the winning candidate in a state, is similarto a process of coarse-graining. In this context, a difference in the sign of the magnetization inthe original and coarse grained lattice would mean that the candidate winning most of the popularvotes did not win the overall election.So far as the dissimilarity of the final outcome of the election result and the popular vote isconcerned, it is expected that the effect will be most relevant when the elections are closely contestedand there is spatial fluctuation in the voting pattern. In that case, if the votes of one candidate isheavily concentrated in a few states, while the other candidate wins in more number of states eventhough marginally, the latter would win the election due to the effective single step coarse grainingcoming from the electoral college system.This then motivates the quantification of the probability of the minority win ǫ ( b, L ) as a functionof the system size L and the coarse graining block size b , in the models such as the Ising and theKEM. Interestingly, b/L emerges as the relevant scaling variable, at least in the limit of the largesystem sizes (see Fig. 3, 4). However, it is obvious that in the two extreme limits b = 1 and b = L , the result of the coarse graining has no significance and ǫ is exactly zero (see Fig. 2). Inthe intermediate range, therefore, ǫ ( b/L ) will show a point of maximum. It is not obvious at whichpoint the maximum would occur, but given that b/L is the scaling variable, it is determined by thecritical fluctuation of the model and not the system or block sizes.9ow, given that the number of states in the US has varied over the years and that the minoritywin probability ǫ ( b/L ) has a non-monotonic variation, it is interesting to check according to theprediction of this model, how has ǫ ( b/L ) changed over the years for the US presidential election.Indeed, as is indicated in Fig. 1, the effective b/L values, estimated as 1 / √ M ( M is the number ofstates), is such that ǫ ( b/L ) has been close to its maximum. In fact, it can also be seen from Figs3 and 4 that for larger values of M i.e., by splitting up larger states, the minority win probabilitycan be sharply reduced. However, it should also be mentioned here that it is an idealized situationand in practice there are some mechanisms in place to counter the effect introduced by the electoralcollege viz., the variation in the number of delegates according to the sizes of the states, which wedid not consider here to keep things simple.Finally, if the coarse graining process is repeated a second time, the scale factors being b and b in the first and second steps respectively, the errors ǫ ′ ( b , b , L ) are higher for the random cases.Here, one can estimate an upper bound based on the numerical results. For the two models usedhere, the error depends on the order in which the two step process is implemented. In general, ascan be seen from the tables, systematically ǫ ′ ( b , b , L ) > ǫ ′ ( b , b , L ) if b > b near the criticalpoints of the models in two dimensions. The results when the smaller of b , b is taken first in factyields an error not much different from the one step case with b = b b . This, however, is nottrue for random cases (or when the noise is far above the critical value), where the coarse grainingprocesses commute. Therefore, this observation can also be attributed to the critical fluctuations ofthe model. The two step coarse graining is relevant in the real world situations where the electedmembers of a legislative body can further form coalitions among themselves. ǫ ′ measures the chancesof their decisions not being aligned with that of their electorates. It is also relevant for coding ininformation theory, the error is expected to increase if the coding is done in two steps.The above mentioned results remain qualitatively true in the mean field limit i.e., a fully con-nected topology (see also [8]). However, a more realistic topology for the interaction domains ofthe agents would be a network structure that can more closely resembles social connectivity [18].Furthermore, the assumptions of equal sizes for each coarse grainin box (each state) could be mademore realistic and the implicit assumption of equal population densities in the states could also bevaried according to data.In conclusion, we have reported the effect of coarse graining in the elections where an interme-diate body is present between the population and the winner, for example in the US presidentialelection. We showed, using finite size scaling of the Ising model and kinetic exchange opinion modelsnear criticality that the probability of a candidate winning the election without winning the popularvote depends non-monotonically with the coarse graining block size. Furthermore, using the datafor the number of states in the US, we show that according to the model studied here, the minoritywin probability is near to the maximum value and could sharply decrease, provided the number ofstates are increased further.Acknowledgments: PS is grateful to the late Dietrich Stauffer who inspired novel research ideasand application of statistical physics in social phenomena. Financial support from SERB schemeEMR/2016/005429 (Government of India) is also acknowledged. References [1] D. Stauffer,
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