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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. denis.phan@enst-bretagne.fr - nadal@tournesol.lps.ens.fr Approches Connexionnistes en Sciences Economiques et de Gestion 10 ème Rencontre Internationale Nantes, 20 et 21 novembre

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ACSEG 10, Nantes denis.phan@enst-bretagne.fr 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 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-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 denis.phan@enst-bretagne.fr 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 denis.phan@enst-bretagne.fr 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 denis.phan@enst-bretagne.fr 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 denis.phan@enst-bretagne.fr 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 denis.phan@enst-bretagne.fr 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 denis.phan@enst-bretagne.fr 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 denis.phan@enst-bretagne.fr 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 denis.phan@enst-bretagne.fr 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 denis.phan@enst-bretagne.fr 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 denis.phan@enst-bretagne.fr 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 denis.phan@enst-bretagne.fr 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 1.2408 to 1.2407 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 denis.phan@enst-bretagne.fr 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 denis.phan@enst-bretagne.fr 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 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 ?

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