1 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]
1 Small Worlds and Phase Transition in Agent Based Models with Binary Choices. Denis Phan ENST de Bretagne, Département Économie et Sciences Humaines &
Mar 31, 2015
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
Small Worlds and Phase Transition in Agent Based
Models with Binary Choices.
Denis PhanENST de Bretagne, Département Économie et Sciences Humaines & ICI (Université de Bretagne Occidentale)
ABS4 - [email protected] 2
For Axtell (2000a) there are threedistinct 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
ABS4 - [email protected] 3
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?
By the way of Interrelated behaviours and chain reaction
What is « small world » ? A simple example with an evolutionary game of
prisoner dilemma on a one dimensional periodic network (circle)
A market case : discrete choice with social influence Key concept : phase transition and demand hysteresis
ABS4 - [email protected] 4
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)
3,65 18,7 2,65
Kevin Bacon
G. W.S.Power
Grid C.Elegans
Graph
61 267
225 226 4941 282
14 k average
connectivity
n number of vertices (agents)
L characteristic path length
ABS4 - [email protected] 5
« Phase transition » in a simple evolutionary game: the spatial
prisoner dilemma
(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
Revision rule :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.
176 > X 92 :defection is contained
in a "frozen zone"
91 X > 6 :the whole population
turns to defectionPhase transition at X<92
ABS4 - [email protected] 6
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
ABS4 - [email protected] 7
Making the network robust againstdefectors' invasion by rewiring one link
New defectors
defectors
16,80%
10,20%11,80%
26,60%
0,40% 1,00% 0,80% 0,40%
32,00%
0,00%
5,00%
10,00%
15,00%
20,00%
25,00%
30,00%
35,00%
cycl
es
2 de
fect
ors
3 de
fect
ors
4 de
fect
ors
6 de
fect
ors
8 de
fect
ors
17 d
efec
tors
22 d
efec
tors
36 d
efec
tors
Small World : strength against accidental defection1,4% links rewired (% on 500 simulations)
Statistical results
for 500 simulations
ABS4 - [email protected] 8
A market case : discrete choice model with social influence (1)
Jik are non-unequivoqual parameters (several possible interpretations) Two special case :
McFaden (econometric) : i = 0 for all i ; hi ~ Logistic(h,) Thurstone (psychological) : hi = h for all i ; i ~ Logistic(0,)
Social influence is assumed to be homogeneous, symmetric and normalized across the neighbourhood
ii i i i i i i ik k
0,1 k
max V h S( ) p with : S( ) J .
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)
ik kiJ
J J J 0N
ABS4 - [email protected] 9
A market case : discrete choice model with social influence (2) Chain effect, avalanches and
hysteresis
0
200
400
600
800
1000
1200
1400
1 1,1 1,2 1,3 1,4 1,5
First order transiton (strong connectivity)
0
10
20
30
40
50
60
70
80
90
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33
Chronology and sizes of induced adoptions in the avalanche when decrease from 1.2408 to 1.2407
0
200
400
600
800
1000
1200
1400
1 1,1 1,2 1,3 1,4 1,5
P=h P=h+J
i i i kk
V h J . p
ABS4 - [email protected] 10
A market case : discrete choice model with social influence (3) hysteresis in the demand curve :
connectivity effectprices-customers hysteresis neighbours = 2
0
200
400
600
800
1000
1200
1400
0,9 1 1,1 1,2 1,3 1,4 1,5 1,6
prices
customers
prices-customers hysteresis neighbours = 4
0
200
400
600
800
1000
1200
1400
0,9 1 1,1 1,2 1,3 1,4 1,5 1,6
prices
customers
prices-customers hysteresis neighbours = 8
0
200
400
600
800
1000
1200
1400
1 1,1 1,2 1,3 1,4 1,5 1,6
prices
customers
prices-customers hysteresis neighbours = world
0
200
400
600
800
1000
1200
1400
1 1,1 1,2 1,3 1,4 1,5
prices
customers
ABS4 - [email protected] 11
A market case : discrete choice model with social influence (3) hysteresis in the demand curve :
Sethna inner hystersis
0
200
400
600
800
1000
1200
1400
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]
ABS4 - [email protected] 12
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
0.5 1 1.5 2
0.5
1
1.5
2
1296 Agents optimal prices
adoptors profit Adoption rate
q
no externality 0,8087 1135 917,91 87,58%
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%
Neighbour 8 1,1009 1255 1 381,89 96,84%
Neighbour 8 260 x 2 1,1169 1249 1 395,43 96,37% 5%
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)
p
max (p) p. 1 F p h j. (p)
ABS4 - [email protected] 13
A market case : discrete choice model with social influence (5)
demonstration : straight phase transition under “world” activation
regime
ABS4 - [email protected] 14
References
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 ?
Related Documents
