Small Worlds and Phase Transition in Agent Based Models with Binary Choices.

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

2. For Axtell (2000a) there are threedistinct 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 ABS4 - Denis.phan@enst-bretagne.fr

3. Small Worlds and Phase Transition in Agent Based Models with Binary ChoicesOverview • Aim : to study the effect of localisedsocial 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 ABS4 - Denis.phan@enst-bretagne.fr

4. 3,65 18,7 2,65 Total connectivity Small world (Watts Stogatz) Regular network (lattice) Random network Kevin Bacon G. W.S.Power Grid n number of vertices (agents) C.Elegans Graph k average connectivity 225 226 4941 282 L characteristic path length 61 267 14 What is « Small world » ? • Milgram (1967) the “six degrees of separation” > Watts and Strogatz (1998) ABS4 - Denis.phan@enst-bretagne.fr

5. 91 X> 6 :the whole population turns to defection 176 > X  92 :defection iscontained in a "frozen zone" « Phase transition » in a simple evolutionary game: the spatial prisoner dilemma Two strategies – states- « phases » S1 : cooperation - S2 : defection Phase transition at X<92  Revision rule : At each period of time, agents updatetheir strategy, given the payoff of their neighbours. The simplest rule is to adopt the strategy of the last neighbourhoodbest(cumulated) payoff. ABS4 - Denis.phan@enst-bretagne.fr

6. 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 • S1 : cooperation S2 : defection • High payoff for cooperationX = 170 • But the whole populationturns to defection ABS4 - Denis.phan@enst-bretagne.fr

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

8. A market case : discrete choice model with social influence (1) • Jikare non-unequivoqual parameters (several possible interpretations) • Two special case : • McFaden (econometric) : i = 0 for all i ; hi~ Logistic(h,) • Thurstone (psychological) : hi = 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 Vi = willingness to pay – price • willingness to pay (1) Idiosyncratic heterogeneity : hi + i • willingness to pay (2) Interactive (social) heterogeneity: S(-i) ABS4 - Denis.phan@enst-bretagne.fr

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

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

11. 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] ABS4 - Denis.phan@enst-bretagne.fr

12. 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) ABS4 - Denis.phan@enst-bretagne.fr

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

14. 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 the9°Meet. Society of Computational Economics, Seattle USA july) Any Questions ? ABS4 - Denis.phan@enst-bretagne.fr