Early Adoption in a Dynamic Social Network

Transcription

1 Early Adopion in a Dynamic Social Nework David Goldbaum Universiy of Technology Sydney * Absrac An agen based model is developed in which a social hierarchy of leaders and followers emerges from a uniform or random social nework. The formaion of he social srucure is driven by he desire o be an early adoper of a subsequenly popular rend. The environmen is relaed o a maoriy game bu inroduces he imporance of he iming of adopion. The proposed environmen is relevan o a number of seings in which leadership and iming of decisions are imporan or being perceived as a rend seer is rewarded. The leadership posiion is aained from reinforcing behavior. Keywords: Dynamic Nework Social Ineracion Consumer Choice * School of Finance and Economics PO Box 123 Broadway NSW 2007 Ausralia david.goldbaum@us.edu.au 1

2 1. Inroducion This exercise models decisions by a populaion for a produc in which ases are uncerain and possibly endogenous o he process. Opinion leaders (Gurus) arise naurally from he populaion and conribue o he agens decisions possibly by shaping ases. Preferences for cerain producs are subecive. For many such producs he consumer may be able o recognize difference bu may seek guidance in ranking hese differeniaed producs. This proec focuses on issues concerning he social phenomenon by which an agen in he populaion becomes a leader influencing he choice of ohers in he populaion. Wihin his examinaion is he quesion of wheher he leadership posiion and srucure of he social nework of he followers is sable. Subsequen examinaions will explore feedback relaionship beween a leader and his or her followers and oher social phenomenon relaed o his environmen. Social neworks have recenly received considerable aenion in economics for explaining aggregae social behavior. In hese models individuals may be influenced globally by he aggregae behavior of he populaion or locally by heir neighbors a subse of he populaion who serve as a reference group. This proec incorporaes aspecs of boh. Economic incorporaion of local ineracions in decision making include Schelling (1971) and (1973) and Kaz and Shapiro (1985). The nework is a social srucure by which discree decisions are made or new producs or echnologies are adoped. The uiliy of a choice depends on he choices of hose in an agen s peer group. Because of social ineracions producs ha garner no paricular consumer preference and may even be perceived o be inferior can grow o dominae choice. The reward srucure for his proec moves away from he simple maoriy or minoriy payoff o ake iming ino accoun. Agens are rewarded for being early adopers of a rend ha subsequenly becomes popular. Because he agens seek reward hrough early adopion iming of an agen s decision mus be modeled as an endogenous componen of he social nework a clear deviaion from he radiional mean field examinaions of local ineracion. A componen of he reward also comes from conformiy o he norm as in Brock and Durlauf (2001) Dynamic social nework models such as Was (2001) Bala and Goyal (2000) Jackson and Was (2002) and Kirman e al (2007) are concerned wih evoluion and convergence. Bala and Goyal (2000) examine he evoluion of a nework in which agens creae links wih individuals wih whom he benefi exceeds he cos of mainaining he link. 2. Model The populaion consiss of L agens. In each period each agen i { L} chooses from among K available opions capured in noaion by k i. The choice represens he acion of adoping a produc fad or fashion saemen for he given period. The process is repeaed each period wih a new se of K opions. Each period sars wih each agen choosing wheher o ac independenly or hrough imiaion. Wih probabiliy i 0 i 1 agen i chooses o ac independenly. Agen i chooses among he available opions wih equal probabiliy k Pr( k k) p K k wih pi 1. k 1 i i Wih probabiliy 1 i agen i choose o imiae one of he oher agens in he populaion. Agen i mainains d non-redundan links o oher agens of he populaion. If agen i mainains a link o agen hen is one of agen i s friends. The links are one-way in ha agen i who 2

3 mainains a link o agen may choose o imiae agen. Associaed wih each link is a weigh w. The weigh w is he probabiliy ha agen i imiaes agen condiional on having chosen i i o imiae so ha (1) Pr(agen i imiaes agen ) = ( 1 (1) i ) w i wih w 1. The subscrip on i and he link weighs reflec ha he probabiliies are i adused over ime by agen i employing a process o be developed below. Each period is subdivided ino rounds indexed r = 1 R. Disseminaion of paricular choices hrough he populaion occur wih he progression of rounds wihin a period. Prior o he firs round each agen chooses eiher o ac independenly or has chosen a friend o imiae for he period. In round 1 of period hose who inend o ac independenly make heir choice. In round wo hose who have chosen o imiae hose who aced in he previous round do so. If agen i has chosen o imiae agen in period hen agen i is a fan of agen. The process coninues for R rounds as rends disseminae hrough he populaion via he nework of imiaors. The rounds are indexed r = 1 R In he final round any agens who have no ye made a R decision consider he populaion and choose opion k wih a probabiliy p k ha depends on he r populariy of he opion observed in he previous round. Le N k be he number of agens who have adoped choice k by round r. Then R1 exp( N 1 ) R R k (2) Pr( xi k xi unassinged). R1 exp( N ) (2) The parameer ρ is he inensiy of choice capuring how sensiive he residual populaion is o he relaive populariy. Seing produces he fully raional adopion of he mos popular choice. Seing can be inerpreed as bounded raionaliy or as he resul of noise in he percepion of relaive populariy of he choice among he populaion. The agens choose wihou considering he populaion for ρ = 0. The uiliy payoff o agen i derived from he adopion of k depends on hree componens J T (3) i J( Nki ) T( Nki ). (3) The J( N J ki ) componen capures he social ineracion erm rewarding he agen for he J populariy of a choice. Here N ki represens he oal number of agens in ime o have chosen J R he paricular opion k he same as agen i Nki Nk. The funcion J has J( x) 0 J (1) 0 and J'( x) 0. Blume and Durlauf (2001) refers o his as he communiy effec. For he simulaions a simple linear funcion is employed wih J J (4) J( N ) a ( N 1). (4) ki J ki The T T ) elemen of he payoff rewards he agen for being an early adopion. Here ( N ki T N ki represens o oal number of agens in ime who have adoped choice k ha coincides wih ha of agen i in rounds subsequen o agen i s adopion N T R r ki Nk Nk. In he hierarchical srucure his would be members of he populaion occupying iers below agen i ha is iers T i 1 hrough. Agen s inclusion in N ki does no require ha agen be an imiaor of i eiher direcly or indirecly hrough a chain of links us ha agen makes he same decision as K K 3

4 agen i in a subsequen round. The simulaions are based on T T (5) T( Nki ) at Nki. (5) The coefficiens a T and a J differeniae he payoff o agen for being an early adoper from he payoff for making a popular choice. Meaningful incenives require 0 aj at. Normalizing at 1 leaves a J as he down-weighing applied o he social payoff relaive o he early adopion payoff. Afer he las round of he period each agen aduss i and w i based on he percepion of how successful each opion would have been during he us compleed period. For each of he links mainained le z i represen he payoff earned by agen i by imiaing agen in period. For hose friends no seleced for imiaion in period he agen mus form a hypoheical payoff. This is done by compuing he payoff ha would have been achieved had agen i made he same choice as agen in he round following agen s selecion. This is an imperfec measure since i ignores he endogenous componen of agen i s own acion on his or her own payoff as a resul of he direc and indirec fans of agen i. These agens oo would have also swiched choices if agen i had swiched. In addiion o he compuaional complexiy involved for he agens par of he reason for ignoring his componen is ha agen i does no know who his or her fans are. The weighs are adused according o he algorihm derived from he Experience-Weighed aracion (EWA) learning rule suggesed by Camerer and Ho (1999). Le A i be a cumulaive performance measure associaed wih agen i s imiaion of agen. A separae performance measure is mainained for each friend of agen i. The value of A i is updaed according o (6) A i ( Ai 1N 1 ( (1 ) I( si )) ( zi )) / N. (6) Briefly ( z i ) is he period realizaion of performance associaed wih agen i selecing o imiae agen. The erm ( (1 ) I( si )) is a mechanism for down-weighing he unobserved speculaive payoff associaed wih he friends who were no seleced by i in he curren period. Thus he updae of A i ges full weigh if was acually chosen by i in ha period and has he diminished weigh of δ 0 1 if no. Seing < 1 decreases he weigh of an observaion wih he passage of ime. Finally N follows a process (7) N N 1 1. (7) Various seing of N 0 ρ and can be used o achieve a variey of weighed combinaions of he previously observed ( z i ). See Camerer and Ho (1999) for more deail. The weigh ha agen i assigns o his link o agen is based on he relaive performance measure. The power disribuion is one opion suggesed by Camerer and Ho and is one of wo primary mechanisms examined in his paper 1 (8) w i Ai d A 1 i. (8) 1 Oher opions explored were he exponenial disribuion k w A ) / exp( Ak ) exp( i i k i. 4

5 Noice ha he weighs are based on (a non-linear funcion of) relaive performance. Beer performing friends have a higher weigh bu all friends wih a non-zero performance measure have a non-zero probabiliy of being chosen (for finie λ). Simulaions ha se weighs according o (8) will be referred o as EWA simulaions The alernaive adusmen process employed is a Replicaor Dynamic ype process by which a higher performing friend aracs greaer weigh a he expense of poorly performing friends. The version of he Replicaor Dynamics developed for his paper is modified from he K- W choice Replicaor Dynamics of Branch and McGough (2008). Le A i be he average performance across agen i s friends weighed by curren link weighs A W w A. Furher i i i le a friend s performance fall ino he ses good or bad according o d ˆ {1 d } ˆ W Gd { d Ai Ai } ˆ W Bd { d Ai Ai } The weighs evolved according o he process (9) W r( Ai Ai ) wi for Bd w w. i i W xi r( Ai Ai ) wi for Gd Bd (9) wih W / Gd Ai Ai xi W ( A A ) Gd i i and rx ( ) anh( RD x/ 2). Each period hose friends falling ino he bad se are down weighed according o heir performance relaive o he weighed mean. The quaniy of he down weighing is hen disribued among hose friends considered good. The modificaion from Branch and W McGough is o use he weighed average A i raher han he simple un-weighed average performance. Wihou his modificaion here is no mechanism for he highes performing friend o arac weigh from oher good friends (paricularly once all of he bad friends reach zero weigh). The performance measure ( z i ) can reasonably be incorporaed ino he process of adapaion eiher as a sraigh forward measure of profis wih ( z i ) zi or hrough some nonlinear or disconinuous measure based on a variey of crieria. 2 The employed mehod was o normalize profis by he highes paying opion k (10) ( zi ) zi / max k ( zi ). (10) For example o give equal credi o all above average friends 1for z i zi (11) ( zi ). (11) 0 for zi zi The agen also aduss he probabiliy of acing independenly i. For his he agen compares he payoff o acing independenly o he bes ha could have been achieved hrough 2 Also examined was o employ a binary measure awarding 1 o all friends realizing average performance among friends or above zero oherwise. 5

6 ind imiaion. Le z i be he payoff o acing independenly. If agen i acs independenly in period ind hen z i is he acual realizaion achieved by agen i. If agen i imiaes in period hen he agen has o use a hypoheical payoff. Like hose who acually did ac independenly agen i selecs from he available opions wih equal probabiliy and uses he payoff ha would have 0 ind been earned had i been adoped in he firs round of he period. Le y i zi indicaes he 1 max( ) h profis from acing independenly and le yi zi max ( zi ). h The agen uses a flexible indirec measure of cumulaive performance B ( B N ( (1 ) I( s )) ( y ))/ N (12) (12) i 2 i i 2 i 2 = 01 and N 2 2 N The probabiliy associaed wih acing independenly evolves according o a Replicaor Dynamics process (13) anh( 2 ( Bi Bi ) / 2)(1 i ) for Bi Bi 1. i i (13) anh( 1 2 ( Bi Bi ) / 2) i for Bi Bi Here relaive performance deermines he innovaion in he probabiliies raher han he level. If one sraegy consisenly underperforms he oher i will be chosen wih zero probabiliy in he limi. 3. Analysis of he social nework Analysis of he seady sae and sabiliy properies of he dynamic social nework can be found in Goldbaum (2009). The efficiency of a hierarchy is measured based on he cumulaive disance of individual * agens from heir efficien ier. Le i represen he ier occupied by agen i in period. Le i indicae he efficien ier based on he shores roue o link o he curren leader a ime. Efficiency E is hus measured as * (14) E i i (14) i wih E 0 represening a fully efficien hierarchy and values of E 0 represen deviaions from efficiency. A hierarchy is considered opimal if he agen wih he greaes poenial o serve as he leader does acually emerge as leader. The hierarchy is considered opimal if he agen wih he greaes number of incoming links emerges as he leader. 3 The opimaliy of he leader is measured by he difference in he number of poenial fans of he curren leader relaive o he maximum number of poenial fans of he individuals in he populaion. Le f i be he number of agens wih links o agen i and le f 0 be he number of links o he leader a ime. The opimaliy of he leader O is hus (15) O max( f i ) f0 (15) 4. Simulaions Each simulaion sars wih a populaion of individuals linked hrough a social nework. The links are unidirecional represening a connecion by which an agen can choose o imiae 3 Oher designaion of a naural leader could be considered. The proposed definiion is boh reasonable and readily verifiable. 6

7 his or her friends. The baseline iniiaion is o randomly assign he links bu symmerically srucured social neworks were examined as well. The baseline is o assign he iniial weighs w i.0 equally. Boh random assignmen and uniform assignmen of i0 o 0.5 were examined. The oher baseline parameers of he simulaion are repored in Table 1. Table 1: Exogenous parameer seing used in he simulaions L = 100 Number of agens P = 500 Number of periods R = 20 Rounds per period K = 12 Number of opions per period d = 6 Number of friends δ A = δ B = 1 Weigh on unried opions A = 4 Srengh of preference in choosing among friends A = 0.5 depreciaion rae in friendship B = 0.01 Srengh of innovaion in choosing beween imiaion and independence B = 0 Depreciaion rae in independence w 0 No friend selecion From Goldbaum (2009) he equilibrium social srucure for he populaion is a single leader wih a hierarchy of followers. For everyone in he populaion o paricipae in his social srucure he oal number of opions available o he individuals mach or exceed a lower bound so ha K K wih at ( L 2) (16) K 1. (16) aj ( L1) Two ypes of figures are employed o display he findings. The ime-series figure plos a ime-series of a number of model-generaed variables. N(ind) is he number of agens choosing o ac independenly MaxFans is he number of imiaors of he agen wih he highes number of fans ha period and N(r=r) is he number of agens who make heir selecion in he final round. The only reason o be caugh wihou making a choice unil he final round is for he agen o have been caugh in a closed circle of links for example when wo agens link wih each. In his siuaion no agen in he circle links ou o he leader. In addiion are he measures of opimaliy and efficiency. The hierarchical figure plos he social srucure as i exiss in he final period of he simulaion. Across he op of he figure are he K choices labeled using capial leers. Individual agens labeled using numbers appear in rows below he choices based on he round of adopion. An arrow from an individual direcly o one of he choices represens a choice by an individual who has chosen o ac independenly for he period. An arrow from one agen o anoher represens a pah of imiaion wih he arrow originaing from he imiaor. 4.1 Emergence of a leader The emergence of a leader and a hierarchy of followers from he iniially unsrucured social nework is quie robus. An example of he process of emergence is ploed in Figure 1. Figure 2 depics he resuling social hierarchy. In his example agen #2 emerges from he populaion as he leader and a hierarchy based on agen #2 s leadership forms. In round 1 of each 7

8 period subsequen o formaion agen #2 occupying ier 0 of he hierarchy randomly selecs one of he 12 opions available for he period. In round 2 hose in ier 1 of he hierarchy imiae agen #2. By round 5 agen #2 s choice for he period has disseminaed hrough he enire populaion. The condiion necessary o produce emergence is pah dependence in he social links. In he early periods of he simulaion success by he individual is he resul of random evens ha in he absence of any adusmen are ransiory. For a leader o emerge i mus be ha ohers in he populaion respond o he lucky individual s success by for example increasing he weigh w i if hey are linked o someone successful and decreasing heir own i since he imiaion offers a higher payoff. The process of observaion and adusmen allows he successful agen o become empowered by his or her followers evenually generaing a hierarchy. Success breeds furher success. Efficiency in he hierarchy depends on he individual s abiliy o find and sele on he friend offering he shores roue he leader. Given he emergence of a leader simulaions indicae ha asympoic efficiency in he hierarchy requires ha individuals asympoically sele on he bes friend offering he mos direc link o he leader (or friends if muliple friends offer equal lengh shores pahs). For his evenually he random componen of heir decision process converges o zero as accomplished when employing he replicaor dynamic process wih infinie memory. Alernaively allocaing weigh according o he EWA process fails o produce an efficien hierarchy. The non-zero weigh allows an individual follower o occasionally choose an inefficien friend. Figure 3 is he ime-series oupu from a EWA simulaion. The leadership posiion is sable bu hose in he lower iers of he hierarchy shuffle posiions each period. The resuling hierarchy ends o have a greaer number of iers as individuals end up employing inefficien pahs. Wih K K i is beer for a non-leader o imiae han o ac independenly. Wih infinie memory in he process governing i individuals learn his lesson once a leader has emerged. For he leader i 1 quickly while i 0 for all non-leaders making hem ino followers. Once a leader has emerged in simulaion ha individual will reain he leadership posiion hroughou. A finie memory in he i process ends o preven convergence of i 0 for followers bu sill allows i 1 for he leader. In he exreme wih 2 0 he individual is adapive using he previous period s payoffs as he basis for he curren period s i. Recall ha he follower evaluaes he value of acing independenly employing a naïve process based on considering a hypoheical choice hey would have made if acing independenly. While good for creaing efficien hierarchies he robusness of he leader generaed from he replicaor dynamic process means ha once a leader has been esablished no evens arise in simulaion o disrup ha posiion allowing a new leader o emerge. Opimaliy of he leader ends o be poor under replicaor dynamics. While poor a generaing efficiency he EWA process is beer able o occupy he leadership posiion wih a leader of greaer finess han is he replicaor dynamics. Look back a F4 and observe ha he emergen leader is a naural leader wih an opimaliy measure of zero. Incorporaing a shor memory in i by seing 2 0 combines wih he movemen in he social srucure under EWA creaes enough mixing o help he naural leader emerge as he 8

9 original leader or o depose an exising sub-opimal leader allowing a more fi leader o emerge. This can be seen in Figure 4 as agen #77 is replaced by agen #49 (in period = 116) who is replaced by agen #75 (in period = 239). 4.5 Non-emergence of a hierarchy A leader can sill fail o emerge from he populaion if agens fail o respond o an individual s successes. If here is inadequae adusmen o he social nework in response o a momen of success hen he opporuniy o build on individual success is los. In a process governed by replicaor dynamics he rae of adusmen o he social links is deermined by [0 ) RD. The greaer RD he more responsive is he individual in sifing here is no adusmen from he he agens shif all he probabiliy weigh away from he lowes probabiliy owards he higher performing friend. A 0 RD iniial weighs. For RD performing friend. In simulaion wih RD 0 a leader does emerge bu he hierarchy based upon he leader is iself ransien as individuals ofen fail o link o he leader direcly or indirecly. To build a nework of fans an agen needs o be visible. If an agen is lucky enough o choose a rend early he or she needs o be observed doing so in order o arac fans. Seing A 1 reduces he successful agen s exposure. Seing δ A = 0 or close o zero prevens he emergence of a leader and hus he formaion of a social srucure. Wihou exposure here is no response and hus no emergence. In order o preven emergence δ A had o be se very low. Though he process is slowed compared o he baseline simulaion a value of δ A = 0.1 sill generaes an emergen efficien hierarchy. Seing δ B = 0 or near zero leads o a sable pool of agens acing independenly. They do so subopimally seling on a sraegy based on insufficien informaion. Figure 5 resuls from seing δ B = Conclusions An efficien hierarchy emerges from he populaion. The emergen hierarchy is efficien when individuals respond and adus owards high performing sraegies driving he probabiliy of following an inefficien sraegy o zero. Such a process favors sabiliy as well as efficiency and as a resul ends o propel an individual from he populaion owards leadership based on luck raher han advanage. An adusmen process ha allows individuals o occasionally pursue inferior sraegies will produce less efficien hierarchies and fluidiy in he hierarchy. Through his insabiliy a more fi leader measured by he number of social links ends o emerge. The hierarchy emerges because early ransiory evens become embedded in he social nework o favor hose wih early success. An individual s good luck leads o adusmen in he social nework ha perpeuaes he success by aracing followers and followers lead o greaer success. Cerain environmens can hinder he process of hierarchy formaion. If individuals are less observan of he individual success around hem hen hey fail o respond adequaely hereby denying he successful individuals he opporuniy o perpeuae his or her success hrough social evoluion. Subbornness on he par of individuals when adusing sraegies will also undermine he social process ha produces hierarchies. Mindful of he Kirman e al (2007) criicism ha endogenous nework models end o require unreasonably high knowledge regarding he full social nework hierarchy formaion is achieved wih no knowledge of he exising or poenial social srucure. 9

10 Bibliography Arhur B Inducive reasoning and bounded raionaliy. American Economic Review (A.E.A. Papers and Proc.) Bala V. Goyal S A noncooperaive model of nework formaion. Economerica 68(5) Blume L.E. Durlauf S.N The ineracions-based approach o socioeconomic behavior in Social Dynamics Ed Durlauf S.N. Young H.P. MIT Press Cambridge MA Branch W. McGough B Replicaor dynamics in a cobweb model wih raionally heerogeneous expecaions. Journal of Economic Behavior and Organizaion forhcoming. Brock W. A. S. N. Durlauf 2001 Discree Choice wih Social Ineracions. Review of Economic Sudies 68(2) Challe D. A. Chessa M. Marsili and Y-C. Zhang 2001 From Minoriy Games o Real Markes Quaniaive Finance Camerer C. and T.-H. Ho 1999 Experience-weighed Aracion Learning in Normal form Games Economerica 67(4) Chang M. H. and J. E. Harringon Jr Innovaors Imiaors and he Evolving Archiecure of Social Neworks working paper. Ellison G. Fudenberg D Work-of-mouh communicaion and social learning. The Quarerly Journal of Economics 110(1) Goldbaum D Follow he Leader: Simulaions on a Dynamic Social Nework. Universiy of Technology School of Finance and Economics Working Paper No Hill S. Provos F. Volinsky C Nework-Based Markeing: Idenifying Likely Adopers via Consumer Neworks. Saisical Science. 21(2) Iacobucci D. Hopkins N Modeling dyadic ineracions and neworks in markeing. Journal of Markeing Research Jackson M.O. Was A The evoluion of social and economic neworks. Journal of Economic Theory Jackson M.O. Wolinsky A A sraegic model of social and economic neworks Journal of Economic Theory Kaz M. L. C. Shapiro 1985 Nework Exernaliies Compeiion and Compaibiliy American Economic Review 75(3) Kirman A. Markose S. Giansane S. Pin P Marginal conribuion reciprociy and equiy in segregaed groups: bounded raionaliy and self-organizaion in social neworks. Journal of Economic Dynamics and Conrol Mizrach B. S. Weers 2006 Expers Online: An Analysis of Trading Aciviy in a Public Inerne Cha Room Rugers Universiy Working Paper # Mongomery J Social neworks and labor marke oucomes American Economic. Reveiw Schelling T Dynamic Models of Segregaion. Journal of Mahemaical Sociology Schelling T Hockey Helmes Concealed Weapons and Dayligh Saving: A Sudy of Binary Choices wih Exernaliies. The Journal of Conflic Resoluion 17(3) pp Topa G Social ineracions local spillovers and unemploymen. Review of Economic Sudies Was A A dynamic model of nework formaion. Games and Economic Behavior

11 Figure 1: Time-series of populaion characerisics. Replicaor Dynamics K = 12 w 0 A 0.5 B 1 RD 1. The leader emerges as sole leader he only agen acing independenly. Soon afer he efficiency of he hierarchy increases o zero. The leader is no a naural leader. Once esablished here are no disrupions o he hierarchy. Figure 2: Srucure of he social hierarchy in he final period ( = 500). Replicaor Dynamics K = 12 λ = 1 w 0 A 0.5 B 1 RD 1. Efficien hierarchy based on Agen #2 who emerged from he populaion as he leader. 11

12 Figure 3: Time-series of populaion characerisics. Experience Weighed Aracor K = 12 w 0 A 0.5 B The leader emerges as sole leader he only agen acing independenly. Efficiency does no arise in he hierarchy. There is fluidiy in he posiion of he followers in he hierarchy. The leader is a naural leader. Figure 4: Time-series of populaion characerisics. Experience Weighed Aracor K = 12 w 0 A 0.5 B In mos periods here is one leader. There are wo changes in leadership (a =116 and =239). 12

13 Figure 5: Time-series of populaion characerisics. Replicaor Dynamics K = 12 w 0 A 0.5 B 1 RD 1 δ B = A lack of informaion leads a populaion of agens o subopimally ac independenly. 13