How to choose under social influence?

Transcription

1 How to choose under socal nfluence? Mrta B. Gordon () Jean-Perre Nadal (2) Dens Phan (,3) - Vktorya Semeshenko () () Laboratore Lebnz - IMAG - Grenoble (2) Laboratore de Physque Statstque ENS - Pars (3) CREM Unversté de Rennes I Project ELICCIR supported by the program «Complex Systems n Socal and Human Scences» (CNRS-French Mnstry of Research)

2 motvaton n a smple market model a sngle good heterogeneous customers socal nteractons prce fxed by a monopolst many equlbra may exst, due to the socal effects whch equlbrum s reached depends on the customers coordnaton monopolst : needs to antcpate the customers equlbrum to fx the prce that optmzes hs proft queston : can the agents reach collectvely the Pareto-optmal equlbrum? 2

3 plan market model wth a sngle good and externaltes S.N. Durlauf (2) A framework for the study of ndvdual behavour and socal nteractons W.A. Brock & S.N. Durlauf (2) Interactons-based models J.-P. Nadal, D. Phan, M.B. Gordon, J. Vannmenus (23) Multple equlbra n a monopoly market wth heterogeneous agents and externaltes equlbrum propertes and collectve states multple solutons customer s phase dagram repeated game framework : learnng by the customers Experence Weghted Attracton (EWA) learnng scheme (Camerer, 23) results for dfferent EWA learnng models 3

4 the market model a sngle good at a prce P fxed by a monopolst a populaton of N agents each agent has to make a bnary choce : to buy (ω =) or not (ω =) one unt each customer's wllngness to pay s the sum of two terms : H = H + θ an dosyncratc term randomly dstrbuted n the populaton H : mean value of the dstrbuton θ : devaton wth respect to the mean, of pdf f(θ ) a socal nfluence term : a weghted sum of the choces of other agents ϑ k ϑ J k ω k weght gven by agent to the choces of hs neghbours choce of neghbour k ( or ) number of «neghbours» of «neghbourhood» of 4

5 smplfyng hypothess strategc complementarty : J k > makng the same choce as the others s advantageous homogeneous socal nfluence (J k =J): global neghbourhood and large N : η = N weght of neghbours' choces N k= (k ) ω k N N k= ω k η J ϑ k ϑ ω k = J η fracton of s neghbours that adopt = fracton of buyers η nsenstve to fluctuatons: sngle agents cannot nfluence ndvdually the collectve term Jη 5

6 collectve effects of the socal nfluence ndvdual s choce : buy f V P = H + θ + J η P > fracton of buyers fracton of buyers : f f η = η = for for H + θ > P : ω = H + θ < P J : ω = f(h+θ ) non buyers buyers P-J H P-Jη P H =H+θ Nash equlbrum : η = f P Jη ( H + θ) dθ 6

7 plan market model wth a sngle good and externaltes S.N. Durlauf, A framework for the study of ndvdual behavour and socal nteractons (2) W.A. Brock & S.N. Durlauf, Interactons-based models (2) J.P. Nadal, D. Phan, M.B. Gordon, J. Vannmenus (23) equlbrum propertes and collectve states multple solutons customer s phase dagram 7

8 logstc dstrbuton f( θ) β = 2 2cosh βθ fracton of buyers J H - P f(βθ) η,5,4,3,2, logstc pdf β f( βθ) = 2 2cosh βθ, h- p βθ customers phase dagram under blue=low h, above red=hgh h sngle soluton H > P H < P -3 two stable solutons j βj B 8

9 trangular dstrbuton f(h ) 2/(3b).4 dstrbuton : customers phase dagram :.2 h-p h h+2bx h-b fracton of buyers b -b/2 2-2 <η< η= -2b -4 η= b j B 2b 3b j coexstence of 2 solutons 9

10 plan market model wth a sngle good and externaltes S.N. Durlauf (2) A framework for the study of ndvdual behavour and socal nteractons W.A. Brock & S.N. Durlauf (2) Interactons-based models J.-P. Nadal, D. Phan, M.B. Gordon, J. Vannmenus (23) Multple equlbra n a monopoly market wth heterogeneous agents and externaltes equlbrum propertes and collectve states multple solutons customer s phase dagram repeated game framework : learnng by the customers Experence Weghted Attracton (EWA) learnng scheme (Camerer, 23) results wth dfferent EWA learnng models

11 hypothess the demand adapts to the prce faster than the tme scale of prce revson: prce s assumed to be fxed durng the customers' learnng. each agent must decde whether to buy or not under mperfect nformaton (he doesn't know the decsons of the others) and ncomplete nformaton (he doesn't know hs actual payoff) based on the "attracton" of buyng or not buyng attracton values are updated (learned) based on the actual fractons of buyers

12 modelsaton of the learnng dynamcs At each tme step each agent makes a bnary decson : Proba[ ω ( t) = ] = f ( t), ( ) wth ( t) A ( t) A ( t) relatve attracton for buyng best response: tremblng hand: f ω ( t) = f P If logstc : P ( t) P > ( t) P < [ ω ( t) = ] = ε ( ( t) P) [ ω () t = ] = + exp β [ () t P] Learns hs relatve attracton [Camerer: Experence Weghted Attractons] from the observaton of the actual fracton of buyers η(t): ˆ η () t = H + Jηˆ () t ηˆ ( t) = ( t + ) = ˆ η ( t) + µ ( t + ) {[ δ + ( δ ) ω ( t) ] η( t) ˆ η ( )} agent s estmate of η t µ ( t) µ ( t + ) = ( κ) + κ( φ) µ ( t) + φ 2

13 Experence Weghted Attracton updatng attractons no n the EWA cube: learnng (Camerer, 23) φ { [ ] η ( t ) η ˆ ( t )} ˆ η ( t + ) = ˆ η ( t ) + µ ( t + ) δ + ( δ ) ω ( t ) µ ( t + ) = ( κ ) µ µ ( t ) ( t ) + φ + κ ( φ ) unlearnng myopc fcttous play myopc renforcement κ type of learnng rate tme-averaged fcttous play δ = weghted belef φ weght played vs. not played fcttous play δ t φ tme-averaged renforc t no memory t δ = 3 renforcement memory of past payoffs

14 smulatons phase dagram for the trangular dstrbuton : h-p b -b/2-2b 2-2 <η< η= P -4 η= b j B 2b 3b j P 2 J=4 type of learnng rate κ φ memory of past payoffs δ weght played vs. not played myopc fcttous play µ ( t + ) = [ ] ˆη ( t + ) = ˆη ( t) + µ ( t + ) η( t) ˆη ( t) = η( t) 4

15 myopc fcttous play fracton of buyers. j=4, κ=, φ=, δ= ω ()= ω ()=.8.6 ω () = η.4 ω () = prce p p 2 5

16 myopc fcttous play fracton of buyers. j=4, κ=, φ=, δ=.8 t= ω () expectatons ratonal chosen at random.6 η prce p p 2 6

17 dscusson asymptotcs for φ < f δ > buyers : for non buyers: dscontnuty at = P ˆ η η ˆ η δ η < η attractons are underestmated asymptotcs for φ > the tme decay of µ(t) may hnder the learnng process work n progress: decson through a tremblng hand: nterference between ε and µ analytc results the learnng process as a specal random walk learnng drectly the attractons (estmatons of H +Jη-P) leads to very dfferent results 7