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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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**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**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**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**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**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**Indirect effect of prices: « chain » or « dominoes »**effect Variation in price Variation in price Direct effect of prices ( P1P2) ( P1P2) 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**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**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**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**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** = 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**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**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**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**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**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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