Simulation of influence of an “ opinion source “ over an electorate
CARLO ARTEMI
Scientific High School Spoleto
Vicolo San Matteo 8 , 06049 , Spoleto
ITALY
Abstract
This work tries to simulate the behaviour of a group of voters when subject to an “ external opinion source “ ,; this kind of influence has been widely discussed in many countries . .
The group of voters has been modelling using a version of a financial markets model that has been able to reproduce with good precision the performance of S&P 500 index and of Mibtel index in 1992-2002 .
This model represent people’s political feeling by a real number ranging form 0 to 100 that may change because (a) it becomes more similar to prevailing nearby feeling , (b) is subject to intrinsic random changes, and , above all , is influenced by external agents ( mass-media )..
These changes are realised because of some mathematical rules that work in a computer simulation and the distribution and the average value of this feeling are observed varying several parameters that characterised both the voters and the external source .
As result it can be seen that , in some cases effectively the external source is able to influence or overall to dominate people’s feeling but , in more realistic cases ,it isn’t able
Key-words : Simulation , social system , cellular models , electorate’s behaviour , mass-media influence econophysics
1 Introduction
In the last years in Italy and in other countries , there was a great discussion , stimulated also by important political events , about the possibility of an “ external opinion source “ ( for example a group of TV with the same owner ), to influence the behaviour of voters and then the results of elections .
Obviously “ external “ means that this source can influence the voters but voters cannot influence it .
This discussion there was both in public opinion and among social science scholars
In the same period several physical-mathematical models has been developed with the aim to realise computer simulations to describe behaviour of social-economic and cultural systems [1],[2] ,[3], [4], .and , in many cases , results of these simulations are very similar to real behaviour .
The author has realised a model [ 5 ] which has simulated with high precision the daily performance of Standard & Poor index during 1-1-1992 , 31-12-2002 and also Italian Mibtel index in the same period .
Now the author has inserted in his model an “ external opinion source “( from now “ external source “ or more simply “ source “ ) able to change the traders risk attitude and to create an irrational exuberance [ 6 ] causing a speculative bubble
Given this aspect of his model , the author has had the idea to adapt it .such a way to make it able to describe voters’ behaviour and to try to give an answer to above-mentioned question
In the following a description of model will be given and results of simulation in three cases ( better described below ) named “ perfect dictatorship “ , “ defective dictatorship “ , and “ democracy “ will be showed .
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2 The model
We start from a model as that used by other authors [1],[2] to study the diffusion of a culture . We imagine people that live in a territory ; this territory is a square 30 unit side where there are 400 points ( every point represents one person ).
At the beginning of the simulation every point are assigned two randomly generated real positive numbers ,that represent their co-ordinates , ranging from 0 to 30 ..
Hence all points are randomly distributed inside. the square .
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In a second step the simulation fixes a positive number ( called “ radius “ (ra) from now ) much smaller than square side and , for every point , finds the points “ friends “ or “neighbours “ .
Two points are considered friends if their distance ( the usually distance of two points of a plane ) is smaller or equal than ra .
These points are “ friends “ because they influence reciprocally their opinion according to the rules below explained
It is important to note that in our model , differently from models of other authors . the number of “neighbours “ is not a constant but is changeable from point to point .
This fact represents the differences that there are among people about the quantity and the intensity of social relations
In a third step every point is assigned a randomly generated real positive number which represents his , or her , opinion . This number ranges from 0 to 100
In original model this number was the risk attitude , in this model can indicate the political sentiment or an opinion about a law , about a news , about a government , ranging from a deep aversion ( 0 ) to an enthusiastic approval ( 100 ) .
For comparison in other models every point is assigned a randomly generated string of 6 bits which represents the culture of the person living in the point
Hence , in our model , variable that represents opinions isn’t a “ quantized “ number
but it can assume every value between two extreme values . . The author has done this choice to take account of every possible behaviour ..
Hence , in the beginning phase the program assigns to every point
its ( initial ) opinion ( ir from now )
the co-ordinates of all “ friends “ points
In the simulation the opinion of every point changes because it is influenced by the opinions of neighbouring people. Every points are updated synchronously for a given number of cycles.
In each cycle the opinion is modified using the following rules:
Assimilation Rule: For each point the program calculates the mean value of opinion of the neighbours ( we call this number VA ) and , if the difference between VA and the value of point’s opinion is smaller of a parameter fixed called influence ( in from now ) , updates the attitude to a new value .
This new value is the mean between previous value and VA ,
If the difference is equal or greater than in nothing happens
Formally
if VA-opinion) in opinion(i)=opinion(i)+VA)/2
where opinion(i) is the opinion of i-thpoint
Mutation Rule: In each cycle the program changes the value of opinion . The amount of change is a number that is the product of a fixed parameter ( called variation ) for a randomly generated real number ranging from -0.5 to +0.5 ..
Formally
opinion(i)=opinion(i)+(rnd-0.5 )*variation
where rnd is the standard random number generator function
These rules are applied in order 2-1 and are applied in every cycles . The application of rule 2 in every cycles can sounds unrealistic because people don’t change opinion every days but alternative mutation rules , where there isn’t a change in every cycles ,give same results .
. It is important to note that the point doesn’t take the same opinion of “ friends “ points but “ it’s influenced “ from others’ ideas and the author’s job experience of financial planning teaches that things go in this way . .
In fig. 1 you can see the distribution of opinion after 500 cycles ..
Changing the number of cycles and values of characteristic parameters in , variation and ra, there are only few variations
As you can see the distribution is not uniform ,also if there are people for every value of opinion ; there is a structure where there are some groups of people having the same opinion .
Now we introduce “ external source “ This is a point whose opinion has been fixed to the beginning of simulation and doesn’t change during simulation . This point influences the opinion of people according rule below explained but isn’t influenced by people .
The formulation of a rule corresponding to external influence , or influences , had to be done with particular care because there are strong qualitatly differences between influence among people and influence of mass-media .
A person can influence the decisions of people which he ( or she ) is acquainted with , while mass-media can influence every people .
Some people can have opinion much different than external agent’s one and in this case the influence of mass-media is much less strong .
To take in account these and other facts the author has developed
External agent comparison rule :The program compares the opinion of point with agent’s one .and calculates the difference between these values ( dif ) . Then program calculates the interaction agent-person ( in1 from now ) according to following formula
in1=f*dif*(1-diff/100)
where f is a fixed parameter and the part of expression in parenthesis takes account of the fact that the external agent is “ stronger “ for people that have opinion nearer to it .
The in1 is
added to opinion value if ea> opinion(i)
subtracted to opinion value if ea<opinion(i)
where ea is the external agent’s opinion In both cases the voter becomes “ more similar “ to the external agent
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3 Perfect dictatorship
We are in the case in which there are one opinion source that influences every voter and exerts an attractive force .
Picture 2a shows the same of fig.1 with external source . There is a dramatically change because practically every people has the same opinion of source . Figure 2b shows the time evolution of the average of opinions ( that is 50 without external agent ) and it can be seen that is rises very speedily until a value near to source’s opinion .
This dramatic change is present also if we change the values of parameters as you can see in figure .3 that refers to a very high value of parameter variation (55 ) ... If there isn’t a flattening towards source’s opinion there is anyway a strong polarisation of public opinion towards the source .
The author has also tried to make a simulation with an external source that doesn’t act always but only in some iterations because of a person don’t read newspapers or sees TV every time but the effect above showed remains
The difference between situation with and without external source is so strong to put the question : “ But , is it possible that mass-media or other , can change a right-wing person in a fervent Communist or back ? or the model is unrealistic ?”
4Defective dictatorship
To give an answer to this question the rule 3 has been modified such a way to realise the “ defective dictatorship “
Defects can derive from two causes both present in real life
1) the voter can change his or her idea ,but within certain limit
2) the influence of external source can be repulsive if the difference between voter’s opinion and source’s opinion is too strong ( there are many politicians that persuade their supporter but aren’t pleasant to others )
Hence the author has made a first correction to rule 3 ( correction 1 from now ) to avoid that variation between present and initial opinion exceeds a fixed parameter called rov ( range of variation ) .
Limitation to the change of opinion.
If the difference between initial and present value of opinion exceeds rov the value of opinion is forced to the extreme value of range of variation
Formally
if ir(i)-opinion(i)rov
Fig..4 above shows new situation and we see a more realistic situation because of the opinion of population is “ pointed towards “ the external source value but there is a large “ tail “ of voters whose opinions ranging from a value near to source to opposite position
Fig 4 below shows the time evolution of the average of opinions and this goes always nearer to source
In brief , if we limit change of opinion , there isn’t a sway of source but there is still its influence .
Now we go on with a second correction ( correction 2 from now ) that takes account of second cause of imperfection and modify role 3 in this way
in1=dif*((1-dif/100)-k)
where k is a parameter , we have chosen value 0.5 , that , if difference of opinion between voter and source is too high , can brings an interaction value less than zero .
Fig 5 shows the same of fig 4 with both corrections inserted and it can be seen an interesting change because of repulsive effect of correction. 2
Now source “ creates its opposition “ ; an opposition very far from source’s opinion and “ more compact “ than source’s supporters .
And there isn’t only this , in fact the average of opinions doesn’t change appreciably respect to situation without source and then the aim of owner’s source to influence the voters in favour of himself is failed .
In this failure the role of correction 2 is fundamental because these effects are more emphasised if only correction 2 is applied
5 Democracy
.What happens if sources are more than one ?
The answer to this question isn’t trivial because of it depends both on number of sources and on their kind ( attractive , repulsive and so on ) and on parameters of model .
Hence the problem is very complex and the author has only examined two cases that , despite their simplicity , describe real situations .
In first case we have two sources of opposite opinion , both with k=0 , and voters have limitation to opinion’s change ..
Fig 6 shows that effect of sources vanish each other and voters remain in intermediate positions
The second is analogous to first one but with k=0.5 , and , as it can be seen in fig. 7 , there is a deep division of voters between supporters of source’s opinion and its antagonists .
6 Concluding remarks
Also if it can have doubts about the capability of this model to describe really a public opinion ,(after all voters don’t coincide with traders especially in countries as Italy) , simulation results seem interesting because they closely remember real life facts .
It’s interesting that , also if source influences always electorate , we cannot take for granted the overall result of this influence because of is strongly dependent by source
characteristics , particularly its capability to be pleasant to every person , and by voters characteristics.
References
[1] Antinucci, F., Cecconi, F., Natale, F., and Parisi, D. Simulation of Agricultural Expansion Through a Cellular Automaton. Cognitive Sciences and technology Institute National Council of Research Rome., 2002
[5] Artemi C. Simulation of daily performance of S&P index by an cultural-economic model submitted for publication to JEMED , 2003
[2] Axelrod, R. . The dissemination of culture: a model with local convergence and global polarization. Journal of Conflict Resolution, Vol. 41:1997 pp.203-226.
[6] Alan Greenspan has used this expression in a meeting with congressional commission the 5 of December 1996 .
[3] Daniel’s M.G. , Farmer J.D. (2003) Quantitative model of price diffusion and market friction based on trading as a mechanistic random process Phs. Rev. Letters Vol.90 ,no. 10 , 2003 pp. 108-110
[4] Schweltzer F. Brownian agents and active particles, .Springer ed. ,2003
1 Opinion distribution without external source . Simulation parameters r=1.5 , variation = 5 . in = 20 , number of iterations=500
Fig.2 Above Opinion distribution with external source ea=15 other parameters as fig.1 .Below time evolution of average value of opinions , parameters as above
Fig.3 Above Opinion distribution with external source variation=55 other parameters as fig.2 . Below time evolution of average value of opinions , parameters as above
Fig.4 Above Opinions distribution with external source and limitation to opinion change rov=16.5 other parameters as fig. 2 .Below time evolution of average of opinions , parameters as above
Fig.5 Above Opinions distribution with external source k=0.5 other parameters same of fig.2 .. Below time evolution of average of opinions ; parameters as above
Fig.6 Above Opinions distribution with two sources , ea respectively 15 and 85 Both the sources have k=0 , rov=16.5 . Other parameters same of fig.1 : Below time evolution of average of opinions ; parameters as above
Fig.7 Above Opinions distribution with two sources ; ea respectively 15 and 85 Both the sources have k=0.5 , rov=16.5 . Other parameters same of fig.1 . Below time evolution of average value of opinions ; parameters as above
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