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Complex Behaviours in binary Choice Models with Global or Local Social Influence

AE2006 : A Symposium in Agent-based Computational Methods in Finance, Game Theory and their applications, Aalborg, Denmark, September 14-15. Complex Behaviours in binary Choice Models with Global or Local Social Influence. Denis Phan (1,2) - Stephane Pajot

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Complex Behaviours in binary Choice Models with Global or Local Social Influence

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  1. AE2006 : A Symposium in Agent-based Computational Methods in Finance, Game Theory and their applications, Aalborg, Denmark, September 14-15. Complex Behaviours in binary Choice Models with Global or Local Social Influence Denis Phan (1,2) - Stephane Pajot (1) CREM CNRS Université de Rennes I(2) ICI Université de Bretagne Occidentale This work is part of the project 'ELICCIR' supported by the joint program "Complex Systems in Human and Social Sciences“ of the French Ministry of Research and of the CNRS. M.B.G., J-P. N. and D.P. are CNRS members

  2. Related literature - two kind of heterogeneities (1) Consumption with externality • Social dependence & Bandwagon effect(Veblen, Leibenstein 1950) • Telecommunication and network effect(Rabeneau, Sthal 1974, Curien Gensollen 1987, Rohlf 2002) • Becker (1974):“Restaurant Pricing and Other Examples of Social Influences on Price”Granovetter, Soong (1986)”Threshold effects of Interpersonal effects in consumer demand” • Partial equilibrium approach # Random (walrassian) economies: Föllmer (1974), Horst (2) Discrete choice with social influence • Schelling (1973, 1978), Granovetter (1978), Galam, Gefen, Shapir (1982) • Becker (1974), Glaeser, Sacerdote, Scheinkman (1996) Glaeser, Scheinkman (2002) (3) BDD versus GNP modelBDD: (Durlauf 1997, Blume, Durlauf 2001, 2003, Brock Durlauf 2001a, 2001b…)GNP: (Nadal et al. 2003, Gordon et al. 2005; Nadal et al. 2005, Phan, Semeshenko 2006) • BBD: Random Utility Model(Thurstone 1927, Luce 1959) + Quantal Choice Analysis(Luce, Supes 1965, McFadden 1974) : idiosyncratic heterogeneity concerns the random term > “Classic Ising Model with annealed disorder”each alternative random term is i.i.d. double exponential (extreme value type I) distributed > probabilistic choices: the join distribution of choice is logistic • GNP: The heterogeneity concerns the fixed idiosyncratic willingness to pay (IWA) > “Quenched Random Field Ising Model” > deterministic maximization AE 2006 - denis.phan@univ-rennes1.fr

  3. Outline and motivation • Plurality of equilibria is a generic propriety of discrete choice modelswith heterogeneous, idiosyncratic preferences and social influence in case of sufficiently strong social influence, for a large class of mono-modal pdf (Gordon, Nadal, Phan, Semeshenko, 2006) Part I : Discrete choice with social influence :Agent’s choices and collective outcome • Uniqueness vs Multiplicity > hence, coordination plm. Part II : Multi-Agent Simulation analysis :Finite Size effects; Global or Local interactions • Avalanches et Intermediate equilibrium positions ( due to the irregular discrete distribution of IWA among Agents) • Multiplicity of path-dependant equilibrium positions due to local structures (clusters) Sethna’s hysteresis AE 2006 - denis.phan@univ-rennes1.fr

  4. I - Discrete choice with social influence :Agent’s choices and collective outcome.Uniqueness vs Multiplicity (coordination)

  5. weight given by agent i to the choices of his neighbours choice of neighbour k (0 or 1) « neighbourhood » of i number of « neighbours » of i the général framework of the GNP modelheterogeneous (fixed) IWA and social influence • A population of N agents • Each agent i has to make a binary choice : to adopt /buy (i=1) or not (i=0) • Each customer's willingness to adopt/ pay (IWP) is the sum of two terms: • An idiosyncratic term Hi randomly distributed in the population H : mean value of the distribution Si = H-Hi : deviation with respect to the mean, of pdf f(Si) • A social influence term: a weighted sum of the choices of other agents AE 2006 - denis.phan@univ-rennes1.fr

  6. Jik > 0 weight of neighbours' choices fraction of i’s neighbours that adopt = fraction of buyers Simplifying hypothesis for analytical purpose • strategic complementarity: making the same choice as the others is advantageous • homogeneous social influence (Jik=J) : • global neighbourhood and large N : • h insensitive to fluctuations : single agents cannot influence individually the collective term Jh • Agent choice to maximize: • Where P is an exogenous cost (Price) AE 2006 - denis.phan@univ-rennes1.fr

  7. Idiosyncratic Willingness to pay distribution: hypothesesGordon, Nadal, Phan, Semeshenko (2006) Discrete Choices under Social Influence: Generic Properties H1. Modality: f is unimodal, that is it has a unique maximum. H2. Smoothness:f is non zero, continuous, and at least piecewise twice continuously differentiable inside its support, ]xm, xM[ , where xm and xM may be finite or equal to  ( in the latter case f is strictly monotonically decreasing towards zero as x) H3 Boundedness:the maximum of f, fB (that may be reached at xm or xM if these numbers are finite), is finite: prototypical cases A - Unbounded supports: The support of the distribution is the real axis; • Typical exemple: the logit distribution (we do not assume that the pdf is symmetric) • Supplementary hypothesis for extreme values: H 4-5 the pdf has a finite mean value and a finite variance B- Compact supports: the support of the distribution is some interval[xm, xM] with xm and xM finite; the pdf is continuous on [xm, xM] and continuously derivable on ]xm, xM[ with a unique maximum on [xm, xM]. AE 2006 - denis.phan@univ-rennes1.fr

  8. (3) i = 1 for  = e (1) i = 0  (2) i = 1  sm(1)= p h j >0 s(e) = p h j.e= 0 s(0)= p h < 0 For a given P, four categories of agents Ex 1 : logistic pdf Normalization by standard deviation p = P / ; h = H / ; j = J / (4) i = 0 for  = e AE 2006 - denis.phan@univ-rennes1.fr

  9. J = 3 JB J = 1 Uniqueness or multiplicity of equilibrium(s) for J {0,1,2,3} Relationship with the “classical” downward-slopping demand curve (Becker 1974) Uniqueness : Moderate Social Influence Assumption Glaeser, Scheinkman, (2002) Triangular distribution : Phan, Semeshenko(2006) AE 2006 - denis.phan@univ-rennes1.fr

  10. customers phase diagram with Logistic distribution (Nadal et al. 2005) hgreat h > p h-p fraction of buyers Inverse demand single solution h < p hsmall Two solutions j jB Unbounded distributions: the Logistic case Nadal J.P., Phan D., Gordon M.B., Vannimenus J. (2005) Quantitative Finance Gordon M.B. Nadal J.P., Phan D.,Semeshenko V (2006) Discrete Choices under Social Influence: Generic Properties AE 2006 - denis.phan@univ-rennes1.fr

  11. Symmetric triangular on [-a,+a] (Phan, Semeshenko 2006) symmetric triangular Asymmetric triangular coexistence of 2 solutions fraction of buyers Dmax(, j) +,  , +=1 hp = 0 = 0 += 1 + a  = 1  =1 a  J Dmin(,j) _ a2 + = 1 0 <  < 1  - a  =0  =0 jB j* j Bounded Distributions: triangular cases distributions: customers phase diagram : ² AE 2006 - denis.phan@univ-rennes1.fr

  12. II – Multi-Agent Simulation analysis :Finite Size effects& Global or Local interactions

  13. Indirect effect of prices: « chain » or « dominoes » effect Variation in price Variation in price Direct effect of prices ( P1P2) ( P1P2) Change ofagenti Change of agenti Change of agentj Change ofagentk Direct versus indirect adoption,chain effect and avalanche process AE 2006 - denis.phan@univ-rennes1.fr

  14. Homogeneous population:Hi = H i P = H + J P = H First order transition (strong connectivity) Chronology and sizes of induced adoptions in the avalanche when decrease from 1.2408 to 1.2407 =5 =20 Chain effect, avalanches and hysteresis AE 2006 - denis.phan@univ-rennes1.fr

  15. Global or Local interactions + FSE One dimensional, periodic two nearest neighbours Full connectivity Symmetric Triangular pdf(Phan, Semeshenko 2006) AE 2006 - denis.phan@univ-rennes1.fr

  16. Finite Size effects Simulated Symmetric Triangular distribution (Phan, Semeshenko 2006) Theoretical equilibrium plots with J = 0, 1, JB 2.5, J*, 4and upstream branch with J= 3, J*, 3.8, 4 AE 2006 - denis.phan@univ-rennes1.fr

  17. A B Sethna’s inner hystersis (neighbourhood = 8, H = 1, J = 0.5, = 10)Sub trajectory : [1,18-1,29] AE 2006 - denis.phan@univ-rennes1.fr

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