1 complex dynamics on a monopoly market with discrete choices and network externality. denis phan 1,...
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1
Complex dynamics on a Monopoly Market
with Discrete Choices and Network Externality.
Denis Phan1, Jean Pierre Nadal2,
1 ENST de Bretagne, Département ESH & ICI - Université de Bretagne Occidentale, Brest1 Laboratoire de Physique Statistique, Ecole Normale Supérieure, Paris.
[email protected] - [email protected]
Approches Connexionnistes en Sciences Economiques et de Gestion
10 ème Rencontre Internationale Nantes, 20 et 21 novembre
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Complex dynamics on a Monopoly Market
with Discrete Choices and Network Externality Related papers by the authors
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” Ninth annual meeting of the Society of Computational Economics University of Washington, Seattle, USA, July 11 - 13, 2003
Nadal J.P., Phan D., Gordon M. B. Vannimenus J. (2003) “Monopoly Market with Externality: An Analysis with Statistical Physics and ACE”. 8th Annual Workshop on Economics with Heterogeneous Interacting Agents (WEHIA), Kiel,May 29-31
Bourgine, Nadal, (Eds.) 2004, Cognitive Economics,
An Interdisciplinary Approach,Springer Verlag
forthcoming january, 7th
Phan D. (2004) "From Agent-BasedComputational Economics towardsCognitive Economics" in Bourgine P., Nadal J.P. eds.
Phan D., Gordon M.B, Nadal J.P.. (2004)“Social Interactions in Economic Theory:an Insight from Statistical Mechanic”in Bourgine, Nadal. eds. (2004)
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In this paper, we use Agent-based Computational
Economics
and mathematical theorising as complementary tools
Outline of this paper (first investigations)
1 - Modelling the individual choice in a social context Discrete choice with social influence: idiosyncratic and interactive
heterogeneity
2 - Local dynamics and the network structure (basic features) Direct vs indirect adoption, chain effect and avalanche process From regular network towards small world : structure matters
3 - « Classical » issues in the « global » externality case Analytical results in the simplest case (mean field) « Classical » supply and demand curves static equilibrium
4 - Exploration of more complex dynamics at the global level « Phase transition », demand hysteresis, and Sethna’s inner
hysteresis Long range (static) monopolist’s optimal position and the network’s
structure
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The demand side: I - modelling the individual choice in a social context
Discrete choice model with social influence :
(1) Idiosyncratic heterogeneity
ii i i i t i t
0,1max V H (t) S ( ) P
Agents make a discrete (binary) choice i in the set : {0, 1}
Surplus : Vi(i) = willingness to pay – price repeated buying
willingness to pay (1) Idiosyncratic heterogeneity : Hi(t) Two special cases (Anderson, de Palma, Thisse 1992) : « McFaden » (econometric) : Hi(t) = H + i for all t ; i ~ Logistic(0,)
Physicist’s quenched disorder (e.g. Random Field ) used in this paper
« Thurstone » (psychological): Hi(t) = H + i (t) for all t ; i (t) ~
Logistic(0,) Physicist’s annealed disorder (+ad. Assumptions : Markov Random
Field ) Also used by Durlauf, Blume, Brock among others…
Properties of this two cases generally differ (except in mean field for this model )
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t i ik kk
S ( ) J . (t)
Myopic agents (reactive) : no expectations :
each agent observes his neighbourhood Jik measures the effect of the agent k ’s choice on
the agent i ’s willingness to pay: 0 (if k = 0 ) or Jik (if k = 1 )
Jik are non-equivocal parameters of social influence (several possible interpretations)
The demand side: I - modelling the individual choice in a social context
Discrete choice model with social influence
(2) Interactive (social) heterogeneityWillingness to pay (2) Interactive (social) heterogeneity :
St(-i)
ik kiJ
J J J 0N
In this paper, social influence is assumed to be positive, homogeneous, symmetric and normalized across the neighbourhood)
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The demand side: II - Local dynamics and the network structure 1 - Direct versus indirect adoption,chain effect and avalanche process
Indirect effect of prices: « chain » or « dominoes »
effectVariation in price
( P1 P2 )
Change of agent i
Change of agent k
k t i 1 2
k t i 2 2
H S ( P ) P
H S ( P ) P
i t i 1 2H S ( P ) P
Variation in price
( P1 P2 )
Change of agent i
Change of agent j
Direct effect of prices
An avalanche carry on as long as:
k t 1 i 2 2
k t i 2 2
k / H S ( P ) P
H S ( P ) P
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The demand side: II - Local dynamics and the network structure 2 - From regular network towards small
world : structure matters (a)
Total connectivity
Regular network (lattice)
Small world 1(Watts Strogatz)
Random network
• Milgram (1967)“ six degrees of separation”
• Watts and Strogatz (1998)• Barabasi and Albert, (1999)
“ scale free ” (all connectivity) multiplicative process power law blue agent is “hub ” or “gourou ”
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The demand side: II - Local dynamics and the network structure 2 - From regular network towards small
world : structure matters (b)
0
5
10
15
20
25
30
35
40
0,650,750,850,951,051,151,251,351,451,55
Price
Nu
mb
er
of
cu
sto
mers
Empty
Neighb2
Neighb4
World
Neighb2 + SW
Neighb4 + SW
World Empty
Neighb2
Neighb4
Neighb2+ SW
Neighb4 + SW
Source : Phan, Pajot, Nadal, 2003
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III - « Classical » issues in the « global » externality case
1 - Simplest cases i i i iV H J P
Profit per unit ( /N) with H1 = c = 0
If only agents H2 buy: (p2) = N .2 .p2
p2 = H2 + J. 2 ; = 2
If all agents buy: (p1) = N.p1
p1= H1 + J ; = 1
P = H + JP = H
A -Homogeneous population:
iH H
i 1, ..N
1
2 1
2 1
H > H 0B two class of agents:
H2 > J > H2 /1
H2 < J
2 2
1 2
η HJ <
η (1+ η )
(p1) = p1 = J
p2 = H2 + J. 2
(p2) = 2 (H2 + J.2 )
J2 J1 H2 < J.12 2
1 2
η HJ <
η (1+ η )
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III - « Classical » issues in the « global » externality case
1 - Analytical results in the simplest case:global externality / full connectivity (main
field)
• H > 0 : only one solution• H < 0 : two solutions ; results depends on .J
P
max (P) P. 1 F P H J. (P)
Supply SideOptimal pricing by a monopolist
in situation of risk
m
1 F(z);
with : z P H J.
Demand SideIn this case, each agent observes only
the aggregate rate of adoption, Let m the marginal consumer: Vm= 0
1
1 exp( .z)
for large populations. With F logistic :
Aggregate demandmay have 2 (3) fixedpoint for high low ; (here = 20)
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
Optimum / implicit derivation gives (inverse) supply curve :
d f (z)
dP 1 J.f (z) P
s 1p ( ) J.
.(1 )
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0.2 0.4 0.6 0.8 1
1
2
3
4
5
6
J = 4
J = 0
H = 0
Ps
Pd
III - « Classical » issues in the « global » externality case
2 - Inverse curve of supply and demand: comparative static
s 1p ( ) J.
.(1 )
d 1 1
p ( ) H J. . ln
0.2 0.4 0.6 0.8 1
1
2
3
4
5
6
J = 4
J = 0
H = 2 PsPd = 1(one singleFixed point)
Dashed linesJ = 0no
externality
0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5
3
H = 1.9
J = 4
PsPd
Low / high P0.2 0.4 0.6 0.8 1
1
2
3
4
5
6
J = 4
H = 1
J = 0
Ps
Pd
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III - « Classical » issues in the « global » externality case
3 - Phase diagram & profit regime transition
Full discussion of phase diagram in the plane .J, .h, and numerically calculated solutions are presented in:Nadal et al., 2003 (WEHIA)
+> -
+
-
-
+> -
+
P+
P -
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IV - Exploration by ACE of more complex dynamics at the global level
1 - Chain effect, avalanches and hysteresis
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
First order transition (strong connectivity)
i i i kk
V H J . P
P = H + JP = H
Homogeneous
population: Hi = H i
0
200
400
600
800
1000
1200
1400
1 1,1 1,2 1,3 1,4 1,5
= 5 = 20
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IV - Exploration by ACE of more complex dynamics at the global
level 2 - 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
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IV - Exploration by ACE of more complex dynamics at the global level
(3) hysteresis in the demand curve :Sethna inner hystersis
(neighbourhood = 8, H = 1, J = 0.5, = 10) - Sub trajectory : [1,18-1,29]
0
200
400
600
800
1000
1200
1400
1,1 1,15 1,2 1,25 1,3 1,35 1,4
AB
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Conclusion, extensions & future developments
Even with simplest assumptions (myopic customers, full connectivity, risky situation), complex dynamics may arise.
Actual extensions: long term equilibrium for scale free small world, and dynamic regimes with H<0; dynamic network
In the future: looking for cognitive agents & learning process ….
Anderson S.P., DePalma A, Thisse J.-F. (1992) Discrete Choice Theory of Product Differentiation, MIT Press, Cambridge MA.
Brock Durlauf (2001) “Interaction based models” in Heckman Leamer eds. Handbook of econometrics Vol 5 Elsevier, Amsterdam
Any Questions ?