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1 Small Worlds and Phase Transition in Agent Based Models with Binary Choices. Denis Phan ENST de Bretagne, Département Économie et Sciences Humaines & ICI (Université de Bretagne Occidentale) denis.phan@enst-bretagne.fr

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ABS4 - Denis.phan@enst-bretagne.fr2 For Axtell (2000a) there are three distinct uses of Agent-based Computational Economics (ACE) (1) « classical » simulations A friendly and powerful tool for presenting processes or results To provide numerical computation (2) as complementary to mathematical theorising Analytical results may be possible for simple case only Exploration of more complex dynamics (3) as a substitute for mathematical theorising Intractable models, specially designed for computational simulations

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ABS4 - Denis.phan@enst-bretagne.fr3 Small Worlds and Phase Transition in Agent Based Models with Binary Choices Overview Aim : to study the effect of localised social networks (non market interactions, social influence) on dynamics and equilibrium selection (weak emergence). Question : how topology of interactions can change the collective dynamics in social networks? By the way of Interrelated behaviours and chain reaction What is « small world » ? A simple example with an evolutionary game of prisoner dilemma on a one dimensional periodic network (circle) A market case : discrete choice with social influence Key concept : phase transition and demand hysteresis

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ABS4 - Denis.phan@enst-bretagne.fr4 What is « Small world » ? Total connectivity Regular network (lattice) Small world (Watts Stogatz) Random network Milgram (1967) the six degrees of separation > Watts and Strogatz (1998) 3,6518,72,65 Kevin Bacon G.W.S.Power Grid C.Elegans Graph 61267 225 226 4941282 14 k average connectivity n number of vertices (agents) L characteristic path length

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ABS4 - Denis.phan@enst-bretagne.fr5 « Phase transition » in a simple evolutionary game: the spatial prisoner dilemma (J 1,J 2 )J 1 /S1J 1 /S2 J 2 /S1(X, X)(176, 0) J 2 /S2(0, 176)(6, 6)(6, 6) Two strategies – states- « phases » S1 : cooperation - S2 : defection Revision rule : At each period of time, agents update their strategy, given the payoff of their neighbours. The simplest rule is to adopt the strategy of the last neighbourhood best (cumulated) payoff. 176 > X 92 : defection is contained in a "frozen zone" 91 X > 6 : the whole population turns to defection Phase transition at X<92

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ABS4 - Denis.phan@enst-bretagne.fr6 Symmetric introduction of defection in a regular network of co-operators to improve the strength of a network against accidental defection four temporary defectors are symmetrically introduced into the network (J 1,J 2 )J 1 /S1J 1 /S2 J 2 /S1170,170(176,0) J 2 /S2(0,176)(6,6) S1 : cooperation S2 : defection High payoff for cooperation X = 170 But the whole population turns to defection

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ABS4 - Denis.phan@enst-bretagne.fr7 Making the network robust against defectors' invasion by rewiring one link New defectors defectors Statistical results for 500 simulations

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ABS4 - Denis.phan@enst-bretagne.fr8 A market case : discrete choice model with social influence (1) J ik are non-unequivoqual parameters (several possible interpretations) Two special case : McFaden (econometric) : i = 0 for all i ; h i ~ Logistic(h, ) Thurstone (psychological) : h i = h for all i ; i ~ Logistic(0, ) Social influence is assumed to be homogeneous, symmetric and normalized across the neighbourhood Agents make a discrete (binary) choice i in the set :{0, 1} Surplus V i = willingness to pay – price willingness to pay (1) Idiosyncratic heterogeneity : h i + i willingness to pay (2) Interactive (social) heterogeneity : S( - i )

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ABS4 - Denis.phan@enst-bretagne.fr9 A market case : discrete choice model with social influence (2) Chain effect, avalanches and hysteresis First order transiton (strong connectivity) Chronology and sizes of induced adoptions in the avalanche when decrease from 1.2408 to 1.2407 P=h P=h+J

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ABS4 - Denis.phan@enst-bretagne.fr10 A market case : discrete choice model with social influence (3) hysteresis in the demand curve : connectivity effect

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ABS4 - Denis.phan@enst-bretagne.fr11 A market case : discrete choice model with social influence (3) hysteresis in the demand curve : Sethna inner hystersis (voisinage = 8 seed 190 = 10) - Sous trajectoire : [1,18-1,29]

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ABS4 - Denis.phan@enst-bretagne.fr12 A market case : discrete choice model with social influence (4) Optimal pricing by a monopolist in situation of risk : analytical solution only in two extreme case h>0 : only one solution h<0 : two solutions ; result depends on.J optimal price increase with connectivity and q (small world parameter ; more with scale free)

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ABS4 - Denis.phan@enst-bretagne.fr13 A market case : discrete choice model with social influence (5) demonstration : straight phase transition under world activation regime

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ABS4 - Denis.phan@enst-bretagne.fr14 References Nadal J.P., Phan D., Gordon M.B. (2003), Network Structures and Social Learning in a Monopoly Market with Externality: the Contribution of Statistical Physics and Multi-Agents Simulations (accepted for WEIA, Kiel Germany, May) Phan D. (2003) From Agent-based Computational Economics towards Cognitive Economics, in Bourgine, Nadal (eds.), Towards a Cognitive Economy, Springer Verlag, Forthcoming. 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. 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 ( accepted for the 9°Meet. Society of Computational Economics, Seattle USA july) Any Questions ?

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