Complex dynamics on a Monopoly Market with Discrete Choices and Network Externality.

1
Complex dynamics on a Monopoly Marketwith
Discrete Choices and Network Externality.

• Denis Phan1, Jean Pierre Nadal2,
• 1 ENST de Bretagne, Département ESH ICI -
  Université de Bretagne Occidentale, Brest
• 1 Laboratoire de Physique Statistique, Ecole
  Normale Supérieure, Paris.
• denis.phan_at_enst-bretagne.fr - nadal_at_tournesol.lps.
  ens.fr

Approches Connexionnistes en Sciences
Economiques et de Gestion10 ème Rencontre
Internationale Nantes, 20 et 21 novembre
2
Complex dynamics on a Monopoly Marketwith
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

3
In this paper, we use Agent-based Computational
Economicsand mathematical theorising as
complementary toolsOutline 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
  Sethnas inner hysteresis
• Long range (static) monopolists optimal
  position and the networks structure

4
The demand side I - modelling the individual
choice in a social contextDiscrete choice model
with social influence (1) Idiosyncratic
heterogeneity

• 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,?)
• Physicists quenched disorder (e.g. Random Field
  ) ? used in this paper
•  Thurstone  (psychological) Hi(t) H ??i
  (t) for all t ??i (t) Logistic(0,?)
• Physicists 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 )

5
The demand side I - modelling the individual
choice in a social contextDiscrete choice model
with social influence(2) Interactive (social)
heterogeneity
Willingness to pay (2) ? Interactive (social)
heterogeneity St(?-i)

• 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)

6
The demand side II - Local dynamics and the
network structure 1 - Direct versus indirect
adoption,chain effect and avalanche process
7
The demand side II - Local dynamics and the
network structure 2 - From regular network
towards small world structure matters (a)

• 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

8
The demand side II - Local dynamics and the
network structure 2 - From regular network
towards small world structure matters (b)
Source Phan, Pajot, Nadal, 2003
9
III -  Classical  issues in the  global 
externality case 1 - Simplest cases

• Profit per unit (? ? ?/N) with H1 c 0
• If only agents H2 buy ?(p2) N .?2 .p2p2 H2
  J.? 2   ? ?2
• If all agents buy ?(p1) N.p1 p1 H1 J  ?
  1

10
III -  Classical  issues in the  global 
externality case 1 - Analytical results in the
simplest caseglobal externality / full
connectivity (main field)

• H gt 0 only one solution
• H lt 0 two solutions results depends on ?.J
  ?

11
III -  Classical  issues in the  global 
externality case 2 - Inverse curve of supply and
demand comparative static
12
III -  Classical  issues in the  global 
externality case3 - 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)
13
IV - Exploration by ACE of more complex dynamics
at the global level 1 - Chain effect, avalanches
and hysteresis
14
IV - Exploration by ACE of more complex dynamics
at the global level 2 - hysteresis in the demand
curve connectivity effect
15
IV - Exploration by ACE of more complex dynamics
at the global level (3) hysteresis in the demand
curve Sethna inner hystersis
16
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
  Hlt0 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 ?