Complemen a i ies be ween Ma hema ical
and Compu a ional Modeling: The Case
o he Repea ed P isone s’ Dilemma
Xa ie Vilà∗
Uni e si a Au ònoma de Ba celona
Abs ac
We s udy he p ope ies o he well known Replica o Dynamics when applied o a ini ely e-
pea ed e sion o he P isone s’ Dilemma game. We cha ac e ize he beha io o such dynamics
unde s ongly simpli ying assump ions (i.e. only 3 s a egies a e a ailable) and show ha he
basin o a ac ion o de ec ion sh inks as he numbe o epe i ions inc eases. A e discussing
he di icul ies in ol ed in ying o elax he “s ongly simpli ying assump ions” abo e, we ap-
p oach he same model by means o simula ions based on gene ic algo i hms. The esul ing
simula ions desc ibe a beha io o he sys em e y close o he one p edic ed by he eplica o
dynamics wi hou imposing any o he assump ions o he ma hema ical model. Ou main con-
clusion is ha ma hema ical and compu a ional models a e good complemen s o esea ch in
social sciences. Indeed, while compu a ional models a e ex emely use ul o ex end he scope o
he analysis o complex scena ios ha d o analyze ma hema ically, o mal models can be use ul
o e i y and o explain he ou comes o compu a ional models.
Keywo ds: Compu a ional Economics, Model-To-Model Analysis, Gene ic Algo-
i hms, E olu iona y Game Theo y, P isone s’ Dilemma
JEL-Codes: C63, C73
1 In oduc ion
In he g owing ield o Agen -Based compu e simula ions applied o social sciences,
model eplica ion is conside ed a key issue. Indeed, asse ing whe he he obse ed e-
sul s o a pa icula simula ion o a model a e co ec o gene alizable is a di icul ask
when no o mal (i.e. ma hema ical) p oo is p o ided. Only eplica ion, compa ison,
alignmen , and o he ela ed echniques can shed some ligh on he alidi y o simu-
la ions. See Axel od (1997) o a me hodological mo i a ion on his and Hegselmann
& Will (2008), Will (2009), and Macy & Sa o (2010) o an inspi a ional deba e. The
∗Financial suppo om g an ECO2008-04756 (G upo Consolidado-C) om he Spanish Minis y o
Science and Inno a ion and om g an SGR2009-578 o he Gene ali a de Ca alunya a e g a e ully acknowl-
edged. Depa amen d’Economia i d’His ò ia Econòmica and MOVE. Uni e si a Au ònoma de Ba celona.
[email p o ec ed]
1
2 The Ma hema ical Model 2
wo k by Izquie do e al. (2009), o ins ance, shows ha he alignmen o compu a-
ional and ma hema ical models can “assis esea che s in unde s anding he dynamics
o simula ion models” (Izquie do e al., 2009). Ou wo k con ains one such compa i-
son. We pu side-by-side wo di e en analysis (ma hema ical and compu a ional) o
he same model: he e olu ion o s a egies in he epea ed p isone s’ dilemma.
We i s conside , based on Imho e al. (2005), he case in which he e olu iona y
sys em is desc ibed by a de e minis ic dynamic sys em ha uses expec ed alues. Using
s ong simpli ying assump ions he model can be sol ed, and a comple e desc ip ion o
how he p ocess beha es is p o ided.
The second app oach, based on Mille (1996), is a compu a ional simula ion in
which ini e au oma a a e used o ep esen he s a egies played, and a decen alized
adap i e p ocess based on he models o gene ic algo i hms simula es he s ochas ic
e olu iona y p ocess. Wi h his echnique we can elax some o he s ong assump ions
used in he i s app oach and s ill ob ain he same basic esul s.
We like o hink ha he limi a ions o he i s app oach (ma hema ical) p o ide
a good mo i a ion o he second app oach (Compu e -Based simula ions). Indeed,
al hough bo h app oaches add ess he same p oblem, we show ha he use o compu-
a ional echniques allows us o elax hypo hesis and o e come he limi a ions o he
ma hema ical app oach. On he o he hand, i is shown ha he ma hema ical model is
ex emely use ul in o de o explain he beha io and he causali y o he esul s o he
compu a ional model
The choice o he epea ed p isone s’ dilemma o conduc he expe imen desc ibed
abo e is no a bi a y. I is a well know and la gely s udied game, and many hings
abou i ha e been lea ned hanks o he ools o o mal game heo y. Bu when he
game is s udied om an e olu iona y pe spec i e, he esul s a e no always clea . The
indings in Boyd & Lo be baum (1987) and Binmo e & Samuelson (1992), o in-
s ance, show ha e olu iona y s able solu ions may ail o exis s in many e sions o
he game. On he o he hand, expe imen s and simula ions like hose epo ed in Axel-
od (1985), Axel od & Hamil on (1981), Nowak & Sigmund (1992), o Mille (1996),
seem o sugges ha Ti - o - a (and o he simila s a egies) p e ail in mos scena -
ios. Thus, he in e es o ou esea ch is pu ing hese wo app oaches, ma hema ical
and compu a ional, side-by-side o achie e a be e unde s anding o he e olu iona y
beha io o playe s in he epea ed p isone s’ dilemma
2 The Ma hema ical Model
The basic s age game (P isone s’ Dilemma) ha playe s will play epea edly is gi en
by
C D
C 3,3 0,5
D 5,0 1,1
We now conside he epea ed e sion o he game played a ini e numbe o ounds
R.In o de o ha e a ac able model, we only conside h ee possible s a egies (as in
Imho e al. (2005)):
2 The Ma hema ical Model 3
•D: Always de ec
•C: Always coope a e
•T: Ti - o -Ta
as hey a e he h ee s a egies ha ha e dese ed a highe a en ion in almos all he
li e a u e dealing wi h he Repea ed P isone ’s Dilemma om an e olu iona y poin
o iew. The ac ha we only conside 3 possible s a egies clea ly imposes a s ong
es ic ion o he analysis, as we will discuss la e . Gi en he abo e, he epea ed game
can be ep esen ed as ollows
C D T
C 3R,3R0,5R3R,3R
D 5R,0R,R5+(R−1),(R−1)
T 3R,3R(R−1),5+(R−1)3R,3R
Thus, o ins ance, when a D- ype s a egy mee s a T- ype s a egy, he o me ge s
5 in he i s ound and hen 1 in each subsequen ound (5+(R−1)in o al), while he
la e ge s 0 i s and hen 1 in each subsequen ound (R−1 in o al).
2.1 The Replica o Dynamics analysis
Le p (C)be he p obabili y ha , a ime , a playe in his popula ion is an “always
coope a e” ype, and he same o p (D)and p (T). We hus ha e ha p (C)+ p (D)+
p (T) = 1∀ .
The eplica o dynamics s a es ha he a e o change o such p obabili ies is a
unc ion o he ela i e pe o mance o each s a egy wi h espec o he a e age pe -
o mance o he popula ion. In his sense, gi en p (C),p (D),p (T), he expec ed
payo a ime o each s a egy (E π(·)) is:
E π(C) = 3Rp (C)+0p (D)+3Rp (T) = 3R(P
(C)+ p (T))
E π(D) = 5Rp (C)+Rp (D)+(5+(R−1))p (T)
E π(T) = 3Rp (C)+(R−1)p (D)+3Rp (T) = 3R(P
(C)+ p (T))+(R−1)p (D)
and hus he a e age payo will be:
E ¯
π=E π(C)p (C) + E π(D)p (D)+E π(T)p (T)
No ice ha since p (C)+ p (D)+ p (T) = 1∀ only wo dimensions ma e .
Hence, he eplica o dynamics in his case is gi en by:
∂p (C)
∂ =p (C)(E π(C)−E ¯
π)
2 The Ma hema ical Model 4
∂p (D)
∂ =p (D)(E π(D)−E ¯
π)
The co esponding ec o ield showing he ajec o ies o he sys em is ep esen ed
in Figu e 1, whe e he poin s aand bgi en by
a=2R−4
2R−3,b=R−2
R−1
Fig. 1: Vec o Field
The ho izon al and e ical axis in Figu e 1 co espond o p (C)and p (D) espec-
i ely. Thus, he h ee e exes o he iangle ((1,0),(0,1),and(0.0)), co espond o
he s a es p (C) = 1,p (D) = 1,and p (T) = 1 espec i ely.
The ajec o ies ha ep esen he e olu ion o he sys em a e di ided in wo a eas o
basins o a ac ion, one o De ec ion and ano he o Coope a ion. Thus, depending
on he loca ion in he simplex o he ini ial p obabili ies a =0, (p0(C),p0(D)), he
sys em will e ol e acco ding o he co esponding pa h owa ds de ec ion ( he e ex
(0,1)) o coope a ion (somewhe e along he line (0,0)→(1,0)). I is clea ha he
basin o a ac ion o coope a ion g ows as Rbecomes la ge. Indeed, we ha e ha
a→1 and b→1 as R→∞
S a iona y poin s ( es poin s) o he sys em a e ma ked ed:
2 The Ma hema ical Model 5
•(0,1), ha co esponds o e e ybody playing always de ec p (D) = 1,
•(0,a), which is a singula poin ,
•all he poin s in he line ha goes om (0,0) o (1,0) ha co espond o poin s
wi h no de ec an s, ha is, p (D) = 0 and p (C)+ p (T) = 1.
No ice ha only he poin (0,1)co esponding o p (D) = 1 is asymp o ically s able in
he sense ha i he sys em is sligh ly pe u bed away om (0,1),any ajec o y will
b ing i back o he same poin . The singula poin (0,a), which is no asymp o ically
s able, can only be eached i he sys em s a s somewhe e in he line ha goes om
(0,a) o (b,0),which occu s wi h ze o p obabili y. Finally, poin s in he line (0,0)→
(1,0)a e s a iona y bu no s able.
An impo an esul is ha he ela i e size o hese basins o a ac ion depends
on he numbe o epe i ion R.Tha is, i he sys em s a s a andom, he p obabili y
o eaching he poin (0,1)(p(0,1), e e ybody de ec ing) o he line (0,0)→(1,0)
(p((0,0)→(1,0)), e e ybody coope a ing) depends on R.
p((0,0)→(1,0)) = (2R−4
2R−3)(R−2
R−1)
p(0,1) = 1−(2R−4
2R−3)(R−2
R−1)
Thus, we can compu e he expec ed pe - ound payo (E¯
π) as a unc ion o R.
E¯
π= (2R−4
2R−3)(R−2
R−1)·3+(1−(2R−4
2R−3)(R−2
R−1))·1
Figu e 2 shows he beha io o such expec ed pe - ound payo as a unc ion o R.
We obse e ha i g ows apidly as he numbe o epe i ions (R) inc eases. In ac ,
E¯
π→3 as R→∞

3 The Compu a ional Model 6
Fig. 2: Expec ed Payo as a unc ion o R
3 The Compu a ional Model
Gi en he analysis abo e, he dynamics seem o sugges ha he e is oom o coop-
e a ion. A leas o a b oad ange o ini ial condi ions, he ajec o ies lead o some
poin in he ho izon al axis co esponding o a popula ion consis ing o only C and T
s a egies.
Ne e heless, such analysis is ex emely pa ial since we a e only conside ing 3
s a egies a a ime, namelyC,D, and T. One can easily see ha ex ending his app oach
(ma hema ical) o a mo e gene al case (wi h mo e s a egies conside ed) is a di icul
ask as i would be ex emely di icul o s udy he beha io o a dynamic sys em wi h
mo e han 2 dimension
To o e come his limi a ion, we de elop a compu e simula ion1in which he s a e-
gies a e ep esen ed by ini e au oma a o size ou (encoded as bina y s ings o 0’s and
1’s) and a Gene ic Algo i hm ou ine is used o simula e he e olu iona y p ocess as in
Mille (1996). The algo i hm was un o 5000 gene a ions s a ing om an ini ial an-
dom popula ion o 100 s a egies using he s anda d single-cu c osso e ope a o and
wi h a p obabili y o pe -bi mu a ion o 0.005. In mos o he cases, he esul s o such
simula ions p oduce he ou come in Figu e 3, in which he e olu ion o he (pe ound)
a e age payo is displayed.
1The so wa e used o his simula ion consis s o a se o ou ines w i en in ANSI C. I is a ailable om
h p://ideas. epec.o g/c/aub/g ecss/001.05.h ml
3 The Compu a ional Model 7
Fig. 3: E olu ion o he a e age pe - ound payo when Coope a ion is he esul
Because he inal a e age payo is 3 we can conclude ha all playe s ollow a
coope a i e s a egy.
In o he cases, hough, coope a ion is no he inal esul as he e olu ion o he
a e age payo esul s as in Figu e 4, which co esponds o he case o all he playe s
de ec ing.
3 The Compu a ional Model 8
Fig. 4: E olu ion o he a e age pe - ound payo when De ec ion is he esul
In bo h cases, hough, he esemblance be ween he ec o ield in Figu e 1 and he
e olu ion o payo s in Figu es 3 and 4 is e y appealing:
•When he inal esul is coope a ion (as in Figu e 3), bo h in he eplica o dy-
namic analysis and in he simula ions, he e olu iona y p ocess seems o a o
he g ow h o De ec an s a egies a i s , and hen hese disappea and Coope -
a i e s a egies s a o eplica e o end up wi h he payo co esponding o he
coope a i e beha io .
•On he con a y, when he inal esul is de ec ion (as in Figu e 4), he e olu ion
goes “mono onically” owa ds ha poin .
How o en each o hese wo esul s occu s in he simula ions ? Gi en ha in he
ma hema ical model we ha e ound ha he answe o his ques ion depends upon he
numbe o epe i ions R, we check whe he Ralso has an e ec in he compu a ional
model. In his sense, Figu e 5 complemen s Figu e 2 by showing how he inal ob-
se ed a e age payo o he simula ions2( he payo o gene a ion 5000) depends on
R. Fo obus ness, we pe o m such exe cise wi h wo di e en c osso e ules ( he
“canonical” single-cu c osso e and a i y- i y c osso e 3) and wi h no c osso e
2Fo each alue o Rwe un 1000 simula ions and compu e he a e age payo o he las gene a ion
(5000)
3Fi y- i y c osso e consis s o gene a ing new bina y s ings in such a way ha each “locus” o he new
s ing has 0.5 p obabili y o being a copy o he co esponding locus o he “ a he ” s ing and 0.5 p obabili y
o ha o he “ma he ’s” s ing
4 Conclusions 9
Fig. 5: Expec ed and obse ed payo s as a unc ion o R
Figu e 5 shows o wha ex en he beha io o he simula ions (cu es ma ked “x”
and “*”) esembles wha we ob ained ma hema ically in he p e ious sec ion (“Theo-
e ical” cu e). We obse e ha , as he numbe o epe i ions Rg ows, he highe is
he p obabili y o eaching coope a ion a he end and hence, he highe is he a e age
payo , bo h heo e ical and empi ically.
In his sense, i seems ha he use o Gene ic Algo i hms o simula e he e olu ion-
a y p ocess closely ma ches he beha io p edic ed by he eplica o dynamics while
a oiding he s ong limi a ion o conside ing only 3 possible s a egies.
To es “how close” hese esul s (ma hema ical and compu a ional) a e, Figu es 6
and 7 show how s a is ically signi ican is he hypo hesis ha he mean o he a e age
payo s o he compu a ional model equals he heo e ical expec ed payo o he ma h-
ema ical model. To his pu pose he do ed line co esponds o he lowe end o a 95%
con idence in e al
We obse e ha , specially in he case o he “canonical” single-cu c osso e and
excep o a ew a ypical obse a ions, he esul s a e signi ican ly close.
4 Conclusions
We ha e s udied he e olu ion o s a egies in he well known Repea ed P isone ’s
Dilemma using wo di e en app oaches: one ma hema ical based on he eplica o
dynamics and one compu a ional based on gene ic algo i hms. We show ha he esul s
ob ained om he wo app oaches coincide o a g ea ex en in he sense ha ,