1ICTP 27/05/02
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
ICTP 27/05/02
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
ICTP 27/05/02
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
ICTP 27/05/02
[[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
ICTP 27/05/02
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
ICTP 27/05/02
Relative agreementRelative agreement
j
i
hij
hij-ui
xj
xi
Relative agreement :Relative agreement :
ICTP 27/05/02
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
ICTP 27/05/02
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))
ICTP 27/05/02
Same uncertainty for all agents (0.5)Same uncertainty for all agents (0.5)
ICTP 27/05/02
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
ICTP 27/05/02
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
ICTP 27/05/02
Central convergence (U=0.4, pe=0.2)Central convergence (U=0.4, pe=0.2)
ICTP 27/05/02
Both extremes convergenceBoth extremes convergence(U=1.4, pe=0.2)(U=1.4, pe=0.2)
ICTP 27/05/02
Single extreme convergence Single extreme convergence (U=1.4, pe=0.05)(U=1.4, pe=0.05)
ICTP 27/05/02
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
ICTP 27/05/02
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
ICTP 27/05/02
ICTP 27/05/02
ICTP 27/05/02
ICTP 27/05/02
ICTP 27/05/02
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