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.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h p://c ea i ecommons.o g/licenses/by/4.0/
Recei ed Oc obe 26, 2017 Accep ed as Economics Discussion Pape No embe 1, 2017
Published No embe 8, 2017
© Au ho (s) 2017. 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. Jou nal o Finance 53, 2059–2106.
Easley, D., O’Ha a, M., 2010. Mic os uc u e and ambigui y. The Jou nal o
Finance 65, 1817–1846.
Guesne ie, R., 2002. Ancho ing economic p edic ions in common knowledge.
Econome ica 70, 439–480.
Heemeije , P., Hommes, C., Sonnemans, J., Tuins a, J., 2009. P ice s abili y
and ola ili y in ma ke s wi h posi i e and nega i e expec a ions eedback:
an expe imen al in es iga ion. Jou nal o Economic Dynamics and Con ol
33, 1052–1072.
Huang, H., Wang, D., Hommes, C., 2005. A obus a ional ou e o andom-
ness in a simple asse p icing model. Jou nal o Economic Dynamics and
Con ol 29, 1043–1076.
Hube man, B. A., Glance, N. S., 1993. Di e si y and collec i e ac ion. Xe ox
Resea ch Pa k Wo king Pape s.
8