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How can extremism prevail ? A How can extremism prevail ? A study based on the Relative study based on the Relative

agreement modelagreement model

G. Deffuant, G. Weisbuch, F. Amblard, T. Faure

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Influence of extremistsInfluence of extremists

• The extremists win :– Majority to one extreme (Germany in the

thirties, islamic revolution…)– Bipolarisation (affaire Dreyfus, religion

wars…)

• The extremists remain marginal

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Bounded confidence Bounded confidence

An opinion An opinion xx has an uncertainty has an uncertainty uu..

First model : all agents have the same uncertaintyFirst model : all agents have the same uncertaintyif :if :

then :then :

No dynamics on the uncertaintyNo dynamics on the uncertainty

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[[ww/2/2uu]=1]=1 [[ww/2/2uu]=2]=2nb attractors approximately the integer part of nb attractors approximately the integer part of ww/2/2uu

With a uniform distribution of the opinions of width With a uniform distribution of the opinions of width ww

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New model with dynamics of New model with dynamics of uncertaintiesuncertainties

• Give more influence to more confident agents

• Avoid the discontinuity of the influence when the difference of opinions grows

• Explore the influence of extremists

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Relative agreementRelative agreement

j

i

hij

hij-ui

xj

xi

Relative agreement :Relative agreement :

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The modification of the opinion and the uncertainty are The modification of the opinion and the uncertainty are proportional to the relative agreement :proportional to the relative agreement :

if if

More certain agents are more influentialMore certain agents are more influential

Relative agreement dynamicsRelative agreement dynamics

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Variation of the relative agreementVariation of the relative agreement

xj xj+ ujxj- uj

xi

ui< uj

ui> uj

1

ui> 2uj

((ii influences influences jj))

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Same uncertainty for all agents (0.5)Same uncertainty for all agents (0.5)

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Same uncertainty for all agentsSame uncertainty for all agents

0

2

4

6

8

10

12

0 2 4 6 8 10 12

W/2U

clus

ters

' num

ber

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Population with extremistsPopulation with extremists

u

x-1 +1

U : initial uncertainty of moderate agents

ue : initial uncertainty of extremists

pe : initial proportion of extemists

: bias between positive and negative extremists

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Central convergence (U=0.4, pe=0.2)Central convergence (U=0.4, pe=0.2)

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Both extremes convergenceBoth extremes convergence(U=1.4, pe=0.2)(U=1.4, pe=0.2)

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Single extreme convergence Single extreme convergence (U=1.4, pe=0.05)(U=1.4, pe=0.05)

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Convergence indicatorConvergence indicator

• p’+ and p’- are the proportion of initially moderate agents which were attracted to the extreme opinion regions

• y = p’+2 + p’- 2

• central convergence : y close to 0

• both extreme convergence : y close to 0.5

• single extreme convergence : y close to 1

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Exploration of the parameter Exploration of the parameter spacespace

0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 20,025

0,05

0,1

0,15

0,2

0,25

0,3

Y

U

pe

ue = 0.1 mu = 0.5 delta = 0

0-0,15 0,15-0,3 0,3-0,45 0,45-0,6

0,6-0,75 0,75-0,9 0,9-1,05

0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 20,025

0,05

0,1

0,15

0,2

0,25

0,3

Y

U

pe

ue = 0.1 mu = 0.5 delta = 0.1

0-0,15 0,15-0,3 0,3-0,45 0,45-0,6

0,6-0,75 0,75-0,9 0,9-1,05

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ConclusionConclusion• In the model, the convergence to the extremes

takes place :– when the initially moderate agents hare very uncertain– by the action of the medium opinion attracted by the

extremes

• The convergence to a single extreme occurs when the uncertainty is even higher, and results of fluctuations of medium opinion agents