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1 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. - Approches Connexionnistes en Sciences Economiques et de Gestion 10 ème Rencontre Internationale Nantes, 20 et 21 novembre

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ACSEG 10, Nantes 2 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 , 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 Bourgine, Nadal, (Eds.) 2004, Cognitive Economics, An Interdisciplinary Approach, Springer Verlag forthcoming january, 7th Phan D. (2004) "From Agent-Based Computational 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)

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ACSEG 10, Nantes 3 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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ACSEG 10, Nantes 4 The demand side: I - modelling the individual choice in a social context Discrete choice model with social influence : (1) Idiosyncratic heterogeneity Agents make a discrete (binary) choice i in the set : {0, 1} Surplus : V i ( i ) = willingness to pay – price repeated buying willingness to pay (1) Idiosyncratic heterogeneity : H i (t) Two special cases (Anderson, de Palma, Thisse 1992) : « McFaden » (econometric) : H i (t) = H + i for all t ; i ~ L ogistic( 0, ) Physicist’s quenched disorder (e.g. Random Field ) used in this paper « Thurstone » (psychological): H i (t) = H + i (t) for all t ; i (t) ~ L ogistic( 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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ACSEG 10, Nantes 5 Myopic agents (reactive) : no expectations : each agent observes his neighbourhood J ik measures the effect of the agent k ’s choice on the agent i ’s willingness to pay: 0 (if k = 0 ) or J ik (if k = 1 ) J ik 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) heterogeneity Willingness to pay (2) Interactive (social) heterogeneity : S t ( - i ) In this paper, social influence is assumed to be positive, homogeneous, symmetric and normalized across the neighbourhood)

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ACSEG 10, Nantes 6 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 » effect Variation in price ( P 1 P 2 ) Change of agent i Change of agent k Variation in price ( P 1 P 2 ) Change of agent i Change of agent j Direct effect of prices An avalanche carry on as long as:

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ACSEG 10, Nantes 7 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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ACSEG 10, Nantes 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

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ACSEG 10, Nantes 9 III - « Classical » issues in the « global » externality case 1 - Simplest cases Profit per unit ( / N ) with H 1 = c = 0 If only agents H 2 buy: (p 2 ) = N. 2.p 2 p 2 = H 2 + J. 2 ; = 2 If all agents buy: (p 1 ) = N.p 1 p 1 = H 1 + J ; = 1 P = H + J P = H A -Homogeneous population:B two class of agents: H 2 > J > H 2 / 1 H 2 < J (p 1 ) = p 1 = J p 2 = H 2 + J. 2 (p 2 ) = 2 (H 2 + J. 2 ) J2J2 J1J1 H 2 < J. 1

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ACSEG 10, Nantes 10 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 Supply Side Optimal pricing by a monopolist in situation of risk Demand Side In this case, each agent observes only the aggregate rate of adoption, Let m the marginal consumer: V m = 0 for large populations. With F logistic : Aggregate demand may have 2 (3) fixed point for high low ; ( here = 20) Optimum / implicit derivation gives (inverse) supply curve :

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ACSEG 10, Nantes 11 J = 4 J = 0J = 0 H = 0 PsPs PdPd III - « Classical » issues in the « global » externality case 2 - Inverse curve of supply and demand: comparative static J = 4 J = 0 H = 2 PsPs PdPd = 1 (one single Fixed point) Dashed lines J = 0 no externality H = 1.9 J = 4 PsPs PdPd Low / high P J = 4 H = 1 J = 0 PsPs PdPd

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ACSEG 10, Nantes 12 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+ P -P -

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ACSEG 10, Nantes 13 IV - Exploration by ACE of more complex dynamics at the global level 1 - Chain effect, avalanches and hysteresis Chronology and sizes of induced adoptions in the avalanche when decrease from to First order transition (strong connectivity) P = H + J P = H Homogeneous population: H i = H i = 5 = 5 = 20

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ACSEG 10, Nantes 14 IV - Exploration by ACE of more complex dynamics at the global level 2 - hysteresis in the demand curve : connectivity effect

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ACSEG 10, Nantes 15 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] A B

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ACSEG 10, Nantes 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 ?

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