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

Complex dynamics on a Monopoly Market with Discrete Choices and Network Externality. Denis Phan 1 , Jean Pierre Nadal 2 , 1 ENST de Bretagne, Département ESH & ICI - Université de Bretagne Occidentale, Brest 1 Laboratoire de Physique Statistique, Ecole Normale Supérieure, Paris.

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

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  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@enst-bretagne.fr - nadal@tournesol.lps.ens.fr Approches Connexionnistes en Sciences Economiques et de Gestion10 ème Rencontre Internationale Nantes, 20 et 21 novembre

  2. Bourgine, Nadal, (Eds.) 2004, Cognitive Economics,An Interdisciplinary Approach,Springer Verlagforthcoming january, 7th Phan D. (2004) "From Agent-BasedComputational Economics towards Cognitive 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) Complex dynamics on a Monopoly Marketwith Discrete Choices and Network ExternalityRelated 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. VannimenusJ. (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 ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  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, andSethna’s inner hysteresis • Long range (static) monopolist’s optimal position and the network’s structure ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  4. Agents make a discrete (binary) choice i in the set :{0, 1} • Surplus : Vi(i) = willingness to pay – price • repeated buying The demand side: I - modelling the individual choice in a social contextDiscrete choice model with social influence : (1) Idiosyncraticheterogeneity • willingness to pay (1) Idiosyncratic heterogeneity : Hi(t) • Two special cases (Anderson, de Palma, Thisse 1992) : • « McFaden » (econometric) :Hi(t) = H + ifor allt ; i~ Logistic(0,) • Physicist’squenched disorder (e.g. Random Field)  used in this paper • « Thurstone » (psychological):Hi(t) = H + i (t)for allt ; 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 fieldfor this model) ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  5. Myopic agents (reactive) : • no expectations :each agent observes his neighbourhood • In this paper, social influence is assumed to be positive, homogeneous, symmetric and normalized across the neighbourhood) 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 ’schoice on the agent i ’swillingness to pay: 0 (if k=0) orJik(if k=1) • Jikare non-equivocal parameters of social influence • (several possible interpretations) ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  6. Indirect effect of prices: « chain » or « dominoes » effect Variation in price Variation in price Direct effect of prices ( P1P2) ( P1P2) Change ofagenti Change of agenti Change of agentj Change ofagentk Anavalanche carry on as long as: The demand side: II - Local dynamics and the network structure1 - Direct versus indirect adoption,chain effect and avalanche process ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  7. Regular network (lattice) Total connectivity Small world 1 (Watts Strogatz) Random network The demand side: II - Local dynamics and the network structure2 - 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 ” ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  8. The demand side: II - Local dynamics and the network structure2 - From regular network towards small world :structure matters (b) Source : Phan, Pajot, Nadal, 2003 ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  9. A -Homogeneous population: B two class of agents: (p1) = p1 = J P = H + J P = H (p2) = 2 (H2 + J.2 ) H2 < J.1 H2 < J H2 > J > H2 /1 p2 = H2+ J. 2 J2 J1 III - « Classical » issues in the « global » externality case1 - 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.p1p1= H1 + J ;  = 1 ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  10. Demand Side In this case, each agent observes only the aggregate rate of adoption,  Let mthe marginal consumer: Vm= 0 Supply Side Optimal pricing by a monopolist in situation of risk Optimum / implicit derivation gives (inverse) supply curve : for large populations. With F logistic : Aggregate demandmayhave 2 (3) fixed point for high   low  ; (here = 20) 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 ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  11.  = 1 (one single Fixed point) Ps Pd H = 2 Ps H = 0 Pd J = 4 J = 0 J = 4 J=0 Dashed lines J = 0 no externality Ps Ps Pd H = 1.9 H = 1 Pd J = 4 J = 4 J= 0 Low  / high P III - « Classical » issues in the « global » externality case 2 -Inverse curve of supply and demand: comparative static ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  12. P- + P+ +> - +> - + - - III - « Classical » issues in the « global » externality case3 - Phase diagram & profit regime transition Full discussion of phasediagram in the plane.J,  .h, and numerically calculated solutions arepresented in: Nadal et al., 2003 (WEHIA) ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  13. Homogeneous population:Hi = H i P = H + J P = H First order transition (strong connectivity) Chronology and sizes of induced adoptions in the avalanche when decrease from 1.2408 to 1.2407 =5 =20 IV - Exploration by ACEof more complex dynamics at the global level1 - Chain effect, avalanches and hysteresis ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  14. IV - Exploration by ACEof more complex dynamics at the global level2 - hysteresis in the demand curve : connectivity effect ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  15. A B (neighbourhood = 8, H = 1, J = 0.5, = 10) - Sub trajectory : [1,18-1,29] IV - Exploration by ACEof more complex dynamics at the global level(3) hysteresis in the demand curve :Sethna inner hystersis ACSEG 10, Nantes denis.phan@enst-bretagne.fr

  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 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 ? ACSEG 10, Nantes denis.phan@enst-bretagne.fr

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