1 The Monopolist's Market with Discrete Choices and Network Externality Revisited: Small-Worlds, Phase Transition and Avalanches in an ACE Framework. Denis Phan 1, Stephane Pajot 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. - Ninth annual meeting of the Society of Computational Economics University of Washington, Seattle, USA, July , 2003

9 th Society of Computational Economics, Seattle 2 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 Sethnas inner hysteresis Long range (static) monopolists optimal position and the networks structure

9 th Society of Computational Economics, Seattle 3 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, ) Physicists 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, ) 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 )

9 th Society of Computational Economics, Seattle 4 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)

9 th Society of Computational Economics, Seattle 5 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:

9 th Society of Computational Economics, Seattle 6 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

9 th Society of Computational Economics, Seattle 7 The demand side: II - Local dynamics and the network structure 2 - From regular network towards small world : structure matters (b)

9 th Society of Computational Economics, Seattle 8 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 two fixed point for high low ; ( here = 20) Optimum / implicit derivation gives (inverse) supply curve :

9 th Society of Computational Economics, Seattle 9 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

9 th Society of Computational Economics, Seattle 10 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., > P + P -

9 th Society of Computational Economics, Seattle 11 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 = 20

9 th Society of Computational Economics, Seattle 12 IV - Exploration by ACE of more complex dynamics at the global level 2 - hysteresis in the demand curve : connectivity effect

9 th Society of Computational Economics, Seattle 13 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

9 th Society of Computational Economics, Seattle 14 IV - Exploration by ACE of more complex dynamics at the global level Optimal long run (static) pricing by a monopolist: the influence of local network structure optimal static (long run) monopoly prices increase with connectivity and small world parameter q ; higher with scale free than WS. J=0 Global externality

9 th Society of Computational Economics, Seattle 15 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. In the future: looking for cognitive agents …. Dynamic pricing & monopolists Bayesian learning process in the case of repeated buying Dynamic pricing & agents learning process in the case of durable good ( Coase conjecture) Dynamic network and monopolists learning about the network ….

9 th Society of Computational Economics, Seattle 16 References 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 Phan D. (2003) From Agent-based Computational Economics towards Cognitive Economics, in Bourgine, Nadal (eds.), Towards a Cognitive Economy, Springer Verlag, Forthcoming. www-eco.enst-bretagne.fr/~phan/moduleco Phan D. Gordon M.B. Nadal J.P. (2003) Social interactions in economic theory: a statistical mechanics insight, in Bourgine, Nadal (eds.), Towards a Cognitive Economy, Springer Verlag, Forthcoming. 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, Kiel. Any Questions ? (please speak slowly)