Paradigm shifts

Guy Maugis, Pie e-And é Wo king Pape Pa adigm shi s Economics Discussion Pape s, No. 2017-92 P o ided in Coope a ion wi h: Kiel Ins i u e o he Wo ld Economy – Leibniz Cen e o Resea ch on Global Economic Challenges Sugges ed Ci a ion: Guy Maugis, Pie e-And é (2017) : Pa adigm shi s, Economics Discussion Pape s, No. 2017-92, Kiel Ins i u e o he Wo ld Economy (I W), Kiel This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/171212 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. 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Licensed unde he C ea i e Commons License - A ibu ion 4.0 In e na ional (CC BY 4.0) Discussion Pape No. 2017-92 | No embe 08, 2017 | h p://www.economics-ejou nal.o g/economics/discussionpape s/2017-92 Pa adigm shi s Pie e-And é Guy Maugis Abs ac The au ho s udies he e olu ion o he numbe o coexis ing belie s in a inancial ma ke in a amewo k whe e he pa adigm d i ing he agen s' beha io is le unspeci ied. The o e eaching aim is o gain insigh s ega ding he dynamic o he a ie y o belie s in an auc ion based inancial ma ke independen ly o any assump ions on agen s' beha io s. The esul ing amewo k may be seen as an abs ac agen based model. In a compu e expe imen he exhibi s a cycle be ween wo s a es, so ha ei he all agen s ac acco ding o he same belie , o he e is no leading belie ; i.e., he e is one domina ing belie , o none. Fu he , he inds ha he equency o his cycle is posi i ely linked o he quali y o he in o ma ion a ailable o he agen s. (Published in Special Issue Agen -based modelling and complexi y economics) JEL G40 Keywo ds agen based model; in o ma ion cascade; he ding beha io Au ho s Pie e-And é Guy Maugis, Uni e si y College London, [email p o ec ed] Ci a ion Pie e-And é Guy Maugis (2017). Pa adigm shi s. Economics Discussion Pape s, No 2017-92, Kiel Ins i u e o he Wo ld Economy. h p://www.economics- ejou nal.o g/economics/discussionpape s/2017-92 1 In oduc ion We p o ide an assump ion ee amewo k o s udy he undamen al dynamic o he he e ogenei y o belie s in inancial ma ke . We do so by building upon he agen based modeling amewo k. F om a gene al abs ac pe spec i e, he heu is ic o agen based modeling is o i s c ea e a copy o a inancial ma ke , and hen o use his copy as p oxy o in e p ope ies o a eal inancial ma ke . The copy is buil on wo main componen s: a se o agen s and a ma ke s uc u e. Then, as in a eal inancial ma ke , agen s a e allowed o in e ac wi h each o he and he ma ke s uc u e, allowing o expe imen s can be pe o med, and phenomenon o in e es obse ed. The inali y o his p ocess is ha , assuming ha he cons uc ed copy is a sa is ying eplica e o a eal inancial ma ke , we can d aw in e ence on how e en s occu in eal ma ke s. See o ins ance A hu e al. (1996) and Huang e al. (2005), and LeBa on (2006) o a e iew. The ma ke s uc u e componen o an agen based model usually eplica es a s uc u e commonly ound in inancial ma ke s, and as such does no aise modeling issues; we will he e wo k on a ba ch auc ion ma ke . The agen componen and especially how and why agen s ac and in e ac is on he o he hand a modeling issue. Ou iew is ha as ew assump ions o pa ame ic es ic ions should be made on his ega d as hey migh be iola ed by un o eseen beha io and mo i a ions on he pa o he agen s (Shille ,1999). 2 F amewo k Elici a ion To make minimal assump ion ega ding agen s beha io , we make he ollowing easoning. We decons uc he exp ession o a beha io as a h ee s ep p ocess: i s in o ma ion is pe cei ed, hen his in o ma ion is p ocessed, and inally a beha io is exp essed. The ans o ma ion a he cen e o his p ocess, ha goes om in o ma ion o beha io , is wha could be loosely e e ed as an agen ’s belie , bu ha we de ine he e as an agen ’s pa adigm. Wi hin his decons uc ion, he e exis s an in ini e numbe o possible pa adigms. In i s b oades sense, a pa adigm may encompass he agen belie in one economic o inancial model, his expe ience o inancial ma ke s, his isk appe i e, and so on. Howe e , since he e exis s only a ini e numbe o agen s a a gi en ime, we can app oxima e all cu en ly used pa adigms a bi a ily well by npa adigms. 2 Likewise, his app oach allows o an in ini e numbe o di e en kind o in o ma ion o be pe cei ed; especially since he e he pe cei ed in o ma ion also con ains in o ma ion ha migh ha e been ob ained by in e ac ing wi h o he agen s. Howe e , o he same eason as abo e, we can app oxima e all cu en ly pe cei ed in o ma ion by lbi s o in o ma ion. Hence he e exis s in ou amewo k only nl possible beha io s a a gi en ime. Then, as in an agen based model, each o hese beha io s will ansla e in o placing an o de a a speci ic p ice, o by abs aining o do so. Thus, i we o de he nl beha io s, say h ough a andom pe mu a ion, we can ma ch he o de s in any o de book in ou amewo k, and simula e auc ions be ween ou agen s. As such auc ions ake place, agen migh change o pa adigm. This may be andom, o igge ed by he model ailing o p oduce a good p ice. Thus, i an agen ’s o de is a om he auc ion closing p ice, and he had sa is ac o y in o ma ion ( o his end we will o ganize he in o ma ion bi s as sa is ac o y ones and dissa is ac o y ones), he will ha e a p obabili y p1 o changing pa adigm, he o he wise has a p obabili y p2< p1o doing so. We now explo e using a compu e expe imen how he p opo ion o agen s using each pa adigm e ol es as mo e auc ions ake place. No e ha since ou app oach is based on an app oxima ion we will in he ollowing only epo quali a i e esul s. 3 Expe imen Design We now o malize he p ocess desc ibed abo e. To do so we will ha e o dis inguish bo h: be ween he n di e en pa adigms, and be ween he l di e en in o ma ion bi s conside ed. To his end, we assign o each a numbe , espec i ely om 1 o n and om 1 o l , and e e o hese numbe s as pa adigm index and in o ma ion index. The simula ion will equi e a ew a i ac s pa ame e s, which we will ind o ha e small o no e ec on ou esul s, as long as hey ake easonable alue, as in (Ki man,1993;Al a ano e al.,2005). Ini ializa ion: To each N agen s ( his numbe is s a ic h ough ime, and all N agen s emain ac o s in he ma ke h oughou he p ocess), we assign andomly a pa adigm index and an in o ma ion index. The pa adigm index is d awn uni o mly o e { 1 ,· · · , n} , while he in o ma ion index is d awn 3 o e { 1 ,· · · , l} using he p obabili ies {pi 1,· · · , pi l} . These las alues a e pa ame e s o he simula ion. We hen decide andomly which agen s will ake pa in he nex auc ion, each agen ha ing he same p obabili y p o pa icipa ing (pis also a pa ame e o he simula ion). Auc ion: As explained in he in oduc ion, he e exis s in ou app oxima- ion nl possible beha io s, which ansla es in o agen s ha ing he possibili y o pu ing o de s a nl di e en unknown possible alues 1,· · · , nl . By d awing a andom pe mu a ion s o e Snl , we can o de hese alues. We now assume ha i an agen ades, he ades only one uni . Then we can in e he p ice ha maximizes he numbe o ades and close he auc ion: ∀k≤nl we can compu e he numbe ok o o de a he alue k , so ha he closing p ice is he p ice s(k∗)wi h k∗such ha : k∗= a gmin 1≤k≤nl ( N 2− k X j=1 os(j)). I e a ion: A e he auc ion, agen a e able o de e mine whe he he in o ma ion hey possessed was sa is ac o y o no , his is made independen ly o he auc ion i sel . Igno ing he case whe e he sa is ac o y bi o in o ma ion is possibly di e en o each agen o pa adigm, wi hou loss o gene ali y, we will assume ha only he bi o in o ma ion numbe ed 1 is sa is ac o y ( his is possible since in o ma ion indices a e a bi a y). Then, o each agen we dis inguish wo cases: ei he he possessed he sa is ac o y in o ma ion ha p oduced s(k∗) , bu pu an o de mo e han ank away om i , o no (i an agen placed an o de a s(k) , hen we say ha his o de is mo e han ank away om s(k∗) i |s ( k ) −s ( k∗ ) | ≥ ; is also a pa ame e o he simula ion). In he i s case he agen will change o pa adigm index wi h p obabili y p1 , in he o he he will change o pa adigm index wi h p obabili y p2 ( p2< p1 ). In bo h cases he new pa adigm index is d awn uni o mly o e all emaining pa adigm indexes. Finally, new in o ma ion indexes a e d awn o all agen s using he p obabili ies {pi 1· · · , pi l}as in he ini ializa ion. 4 Expe imen Resul s We will now p esen he esul s ob ained. To do so we will begin by p esen ing a ep esen a i e example o p oduced dynamic. Then, using his example, 4 0.0 0.2 0.4 0.6 0.8 1.0 Pa adigm 1 0.0 0.2 0.4 0.6 0.8 1.0 Pa adigm 2 0.0 0.2 0.4 0.6 0.8 1.0 Pa adigm 3 0 200 400 600 800 1000 I e a ion 0.0 0.2 0.4 0.6 0.8 1.0 Pa adigm 4 Figu e 1: P opo ion o agen s using each pa adigm in an example o 1000 i e a ions wi h n = 4 models, l = 5 bi s o in o ma ion, N = 500 agen s. The p obabili y o ha ing sa is ac o y in o ma ion is pi 1 = 0 . 6, he emaining pa ame e s a e p= 0.9, p1= 0.9, p2= 0.05, and = 1. we will in oduced in mo e de ails he wo s a es men ioned abo e: he homogeneous s a e whe e all agen s sha e he same pa adigm index and he he e ogeneous s a e whe e he e is no leading pa adigm index. Finally, we will compu e he p obabili ies o going om one s a e o he o he o a ied alues o he pa ame e s. Rep esen a i e dynamic example Ou ocus is he p opo ion o agen s using each o he n pa adigms h ough ime. Mo e p ecisely, using he p ocedu e epo ed abo e, we will epo he p opo ion o agen being assigned each pa adigm index a each i e a ion. In Figu e 1we plo hese p opo ions o a simula ion o ou amewo k (All compu a ions a e done in R (R De elopmen Co e Team,2011).) In his example he e a e N = 500 agen s, he numbe o models n is 4, he numbe o bi s o in o ma ion l is 5. The emaining pa ame e s a e as ollows: he p obabili y o ha ing sa is ac o y in o ma ion is pi 1 = 0 . 6, he p obabili ies o ha ing unsa is ac o y in o ma ion ( pi 2 o pi 5 ) a e 0 . 1, he p obabili y p o en e ing a ade is 0 . 9 and inally p1= 0.9, p2= 0.05, and = 1. In his example he p opo ion o agen s being assigned each pa adigm 5 label is beha ing di e en ly om i e a ion 1 o 200 han om i e a ion 200 o 300. In he i s window (1 o 200) he e is no leading pa adigm label, while in he second window (200 o 300), pa adigm label ou is consis en ly concen a ing abo e 80% o he popula ion. We will call he i s s a e he he e ogeneous s a e, and he o he he homogeneous s a e. In his example he sys em epea edly changes o s a e h ough ime: app oxima ely a i e a ions 200, 300, 500, 750 and 900. This ype o dynamic is ep esen a i e o wha we ob ained o a la ge numbe di e en pa ame e s. I should be no ed ha such oscilla ions be ween wo sa es co esponds o esul s ound in o he agen based models (Huang e al.,2005;Hube man and Glance,1993;Guesne ie, 2002;Ki man,1993). Homogeneous & He e ogeneous S a es As he epa i ion o agen s among pa adigms a a gi en i e a ion is quali a i ely well desc ibed by he s a e—homogeneous o he e ogeneous— he sys em is in, i is o in e es o us o e alua e he p obabili ies o he sys em mo ing om one s a e o he o he . We he e o e se po as he p obabili y o en e ing he homogeneous s a e om he he e ogeneous s a e. We likewise de ine pe as he p obabili y o en e ing he he e ogeneous s a e om he homogeneous s a e. To es ima e po and pe , we will simula e a la ge numbe o dynamic and a e age he es ima es p oduced o each dynamic. Mo e p ecisely, o a dynamic such as he one p esen ed abo e, we cons uc om he se ies shown in Figu e 1an his o ic o he isi ed s a es (we cha ac e ize he homogeneous s a e by one pa adigm ga he ing mo e han 80% o he agen s o a leas wo i e a ions), and hen use his his o ic o es ima e o poand pe. Theo e ically po and pe a e unc ions o all he pa ame e s o he model. Howe e , we epo no quali a i ely signi ican link be ween any o he pa ame e s and hese p obabili ies bu o pi 1 , he p obabili y o ha ing sa is ac o y in o ma ion. In ou simula ions we exhibi a posi i e link be ween pi 1 and po and 1 −pe . In Figu e 2we plo ed po and 1 −pe as unc ions o pi 1 and N . We obse e ha o all N he p obabili ies a e quali a i ely simila , bu ha hey inc ease wi h pi 1 , he p obabili y o possessing sa is ying in o ma ion. 6 po1−pe 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 N: Numbe o agen s P obobili y o ue in o ma ion 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 N: Numbe o agen s P obobili y o ue in o ma ion Figu e 2: po and 1 −pe , each poin es ima ed wi h 500 simula ions o 1000 i e a ions. 5 Discussion We obse ed ha independen ly o all pa ame e s, he sys em oscilla es be- ween wo s a es: a homogeneous one, whe e all agen s use simila pa adigms; and he e ogeneous one, whe e he e is no leading pa adigm. By i s gene ali y, and he simplici y o he amewo k i is elici ed in, his esul p o ides a deepe pe spec i e o simila esul s ound in di e en agen based models and in labo a o y expe imen s; see Huang e al. (2005); Hube man and Glance (1993); Guesne ie (2002) and Heemeije e al. (2009). Fu he , we ind ha he quali y o he in o ma ion a ailable o he agen s is a key ac o ha de e mines he equency o he oscilla ions be ween he wo s a es. We a gue ha his is su p ising, as i implies ha be e in o med agen s (agen s ha ing a highe p obabili y o possessing sa is ac o y in o ma ion), as a g oup, con e ge owa d he iskie equilib ium. We say he homogenous s a e is isky, as hen he s abili y o ma ke is almos solely dependen on he used pa adigm. Indeed, he leading pa adigm is d i ing he ma ke : i o ins ance he leading pa adigm was a di e en one, say he pa adigm labeled 1 ins ead o 3, he p ice dynamic gene a ed migh be comple ely di e en . On he o he hand he he e ogeneous s a e can be seen as less isky. As he g oup has a di e si ied se o used pa adigms, heu is ically he isk in ol ed in using each pa adigm is limi ed (since i is used by a small po ion o he popula ion) and migh compensa e each o he (see Hube man 7 and Glance (1993); Easley and O’Ha a (2010)). A possible explana ion is ha he homogenous s a e allows o sha e in o ma ion e icien ly: in he homogeneous s a e, he sys em is much simple (only one pa adigm a he han many), and allows agen s o communica e mo e e icien ly (as hey sha e he common g ound o he ha ing he same pa adigm, see Shille (1995)). P ac ically, as o ins ance h ough he use o implied ola ili y Dumas e al. (1998), i ‘p ice=pa adigm(in o ma ion)’, and pa adigm is known, a p ice communica es in o ma ion in as much as pa adigm is in e ible; i.e., w i ing ‘in o ma ion = pa adigm−1(p ice)’. Re e ences Al a ano, S., Lux, T., Wagne , F., 2005. Es ima ion o agen -based models: he case o an asymme ic he ding model. Compu a ional Economics 26, 19–49. A hu , B., Holland, J., LeBa on, B., Palme , R., Tayle , P., 1996. Asse p icing unde endogenous expec a ions in an a i icial s ock ma ke . San a Fe Ins i u e Publica ion. Dumas, B., Fleming, J., Whaley, R. E., 1998. Implied ola ili y unc ions: Empi ical es s. 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