Small Worlds and Phase
Transition in Agent Based Models
with Binary Choices.
Denis Phan
ENST de Bretagne, Département Économie et Sciences Humaines
& ICI (Université de Bretagne Occidentale)
[email protected]
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For Axtell (2000a) there are three
distinct uses of Agent-based
Computational Economics (ACE)
(1) « classical » simulations
 A friendly and powerful tool for presenting processes
or results
 To provide numerical computation
(2) as complementary to mathematical theorising
 Analytical results may be possible for simple case only
 Exploration of more complex dynamics
(3) as a substitute for mathematical theorising
Intractable models, specially designed for
computational simulations
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Small Worlds and Phase Transition in Agent Based Models with Binary Choices
Overview


Aim : to study the effect of localised social networks (non
market interactions, social influence) on dynamics and
equilibrium selection (weak emergence).
Question : how topology of interactions can change the
collective dynamics in social networks?



What is « small world » ?
A simple example with an evolutionary game of prisoner
dilemma


By the way of Interrelated behaviours and chain reaction
on a one dimensional periodic network (circle)
A market case : discrete choice with social influence

Key concept : phase transition and demand hysteresis
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What is « Small world » ?
Total
connectivity
Regular network
(lattice)
Small world
(Watts Stogatz)
Random network
• Milgram (1967) the “six degrees of separation” > Watts and Strogatz (1998)
n number of vertices (agents)
k average connectivity
L characteristic path length
Kevin Bacon G.
W.S.Power Grid
C.Elegans Graph
225 226
282
61
4941
267
3,65
18,7
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2,65
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« Phase transition » in a simple evolutionary
game: the spatial prisoner dilemma
176 > X  92 :
defection is contained
in a "frozen zone"
(J1,J2)
J1/S1
J1/S2
J2/S1
(X , X)
(176 , 0)
J2/S2
(0 , 176)
(6 , 6)
Two strategies – states- « phases »
S1 : cooperation - S2 : defection
Phase transition at X<92 
Revision rule :
91  X > 6 :
the whole population
turns to defection
At each period of time, agents update their
strategy, given the payoff of their neighbours.
The simplest rule is to adopt the strategy of the
last neighbourhood best (cumulated) payoff.
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Symmetric introduction of defection in a
regular network of co-operators


to improve the strength of a network against
accidental defection
four temporary defectors are symmetrically introduced
into the network
(J1,J2)
J1/S1
J1/S2
J2/S1
170,170
(176,0)
J2/S2
(0,176)
(6,6)
S1 : cooperation S2 : defection
• High payoff for cooperation
X = 170
• But the whole population
turns to defection 
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Making the network robust against
defectors' invasion by rewiring one link
Statistical results
for 500 simulations
Small World : strength against accidental defection
1,4% links rewired (% on 500 sim ulations)
32,00%
35,00%
30,00%
defectors
26,60%
25,00%
20,00%
16,80%
15,00%
New
defectors
10,20%11,80%
10,00%
5,00%
0,40% 1,00% 0,80% 0,40%
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36 defectors
22 defectors
17 defectors
8 defectors
6 defectors
4 defectors
3 defectors
2 defectors
cycles
0,00%
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A market case : discrete choice model
with social influence (1)

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
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Agents make a discrete (binary) choice i in the set :{0, 1}
Surplus Vi = willingness to pay – price
willingness to pay (1)  Idiosyncratic heterogeneity : hi + i
willingness to pay (2)  Interactive (social) heterogeneity : S(-i)
max Vi  i   i  hi  i  S(i )  p 
 i 0,1


 Jik .k
k
Jik are non-unequivoqual parameters (several possible interpretations)
Two special case :
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

with : S(i ) 
McFaden (econometric)
Thurstone (psychological)
: i = 0 for all i ; hi ~ Logistic(h,)
: hi = h for all i ; i ~ Logistic(0,)
Social influence is assumed to be homogeneous, symmetric and
J
normalized across the neighbourhood
J ik  J ki  J  
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N
0
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A market case : discrete choice model with social influence (2)
Chain effect, avalanches and
hysteresis
First order transiton (strong connectivity)


Vi  i   i  h  J  .  k  p 


k


1400
1200
1000
800
600
400
200
P=h+J
P=h
0
1
1,1
1,2
1,3
1,4
1,5
Chronology and sizes of induced adoptions in the
avalanche when decrease from 1.2408 to 1.2407
1400
90
1200
80
1000
70
800
60
50
600
40
400
30
200
20
10
0
1
1,1
1,2
1,3
1,4
1,5
0
1
3
5
7
9
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15 17 19
21 23
25 27
29 31 33
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A market case : discrete choice model with social influence (3)
hysteresis in the demand curve :
connectivity effect
p rices-customers hy steresis neighbours = 4
p rices-customers hysteresis neighbours = 2
1400
c ust ome r s
1400
c ust om e r s
1200
1200
1000
1000
800
800
600
600
400
400
200
200
pr ic e s
0
pr ic e s
0
0,9
1
1,1
1,2
1,3
1,4
1,5
1,6
0,9
1
1,1
1,2
1,3
1,4
1,5
1,6
p rices-customers hy steresis neighbours = world
p rices-customers hy steresis neighbours = 8
1400
c
ust om e r s
1400
c ust om e r s
1200
1200
1000
1000
800
800
600
600
400
400
200
200
pr ic e s
pr ic e s
0
0
1
1,1
1,2
1,3
1,4
1,5
1,6
1
1,1
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1,2
1,3
1,4
1,5
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A market case : discrete choice model with social influence (3)
hysteresis in the demand curve :
Sethna inner hystersis
1400
1200
1000
800
600
400
200
0
1,1
1,15
1,2
1,25
1,3
1,35
1,4
(voisinage = 8 seed 190  = 10) - Sous trajectoire : [1,18-1,29]
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A market case : discrete choice model with social influence (4)
Optimal pricing by a monopolist in situation
of risk : analytical solution only in two
extreme case
1296 Agents
no externality
optimal adoptors
prices
0,8087
1135
profit
Adoption
rate
917,91 87,58%
q
max (p)  p. 1  F  p  h  j.(p) 
p
Neighbour2
1,0259
1239 1 271,17 95,60%
Neighbour 4
1,0602
1254 1 329,06 96,76%
Neighbour 4_130x2
1,0725
1250 1 340,10 96,45%
5%
Neighbour 4_260x2
1,0810
1244 1 344,66 95,99%
10%
Neighbour 4_520x2
1,0935
1243 1 358,86 95,91%
20%
Neighbour 4_1296x2 1,1017
1237 1 362,35 95,45%
50%
Neighbour 6
1,0836
1257 1 361,48 96,99%
Neighbour 6_260x2
1,0997
1252 1 376,78 96,60%
7%
Neighbour 6_520x2
1,1144
1247 1 389,05 96,22%
13%
Neighbour 6_1296x2 1,1308
1241 1 403,03 95,76%
33%
Neighbour 6_1296x4 1,1319
1240 1 403,02 95,68%
66%
1.5
Neighbour 8
1,1009
1255 1 381,89 96,84%
Neighbour 8 260 x 2
1,1169
1249 1 395,43 96,37%
5%
1
Neighbour 8 520 x 2
1,1306
1245 1 407,20 96,06%
10%
Neighbour 8 1296x2
1,1461
1238 1 419,28 95,52%
25%
Neighbour 8 1296x4
1,1474
1239 1 421,97 95,60%
50%
Neighbour 8 1296x6
1,1498
1238 1 423,84 95,52%
75%
world
1,1952
1224 1 462,79 94,44%
• h>0 : only one solution
• h<0 : two solutions ; result depends on .J
• optimal price increase with connectivity and
q (small world parameter ; more with scale free)
2
0.5
0.5
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1
1.5
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2
A market case : discrete choice model with social influence (5)
demonstration : straight phase transition
under “world” activation regime
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References
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Nadal J.P., Phan D., Gordon M.B. (2003), “Network Structures and Social
Learning in a Monopoly Market with Externality: the Contribution of Statistical
Physics and Multi-Agents Simulations” (accepted for WEIA, Kiel Germany, May)
Phan D. (2003) “From Agent-based Computational Economics towards
Cognitive Economics”, in Bourgine, Nadal (eds.), Towards a Cognitive
Economy, Springer Verlag, Forthcoming.
Phan D. Gordon M.B. Nadal J.P. (2003) “Social interactions in economic
theory: a statistical mechanics insight”, in Bourgine, Nadal (eds.), Towards a
Cognitive Economy, Springer Verlag, Forthcoming.
Phan D., Pajot S., Nadal J.P. (2003) “The Monopolist's Market with Discrete
Choices and Network Externality Revisited: Small-Worlds, Phase Transition and
Avalanches in an ACE Framework” (accepted for the 9°Meet. Society of
Computational Economics, Seattle USA july)
Any Questions ?
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