Distinguishing the opponents in the prisoner dilemma in well-mixed populations

0.16 MB·English

by  Lucas Wardil

Distinguishing the opponents in the prisoner dilemma in well-mixed populations
Lucas Wardil†∗ and Jafferson K. L. da Silva†
†Departamento de F´ısica,
Universidade Federal de Minas Gerais,
Caixa Postal 702, CEP 30161-970,
Belo Horizonte - MG, Brazil
(Dated: January 12, 2010)
Here we study the effects of adopting different strategies against different opponent instead of
adopting the same strategy against all of them in the prisoner dilemma structured in well-mixed
0 populations. We consider an evolutionary process in which strategies that provide reproductive
1 success are imitated and players replace one of their worst interactions by the new one. We set
0
individualsinawell-mixedpopulationsothatnetworkreciprocityeffectisexcludedandweanalyze
2
both synchronous and asynchronous updates. As a consequence of the replacement rule, we show

n that mutual cooperation is never destroyed and the initial fraction of mutual cooperation is a
a lower bound for the level of cooperation. We show by simulation and mean-field analysis that
J for synchronous update cooperation dominates while for asynchronous update only cooperations

9 associatedtotheinitialmutualcooperationsaremaintained. Asasideeffectofthereplacementrule,
an “implicit punishment” mechanism comes up in a way that exploitations are always neutralized

] providingevolutionary stability for cooperation.
h
p
PACSnumbers: 87.23.-n,89.65.-s,02.50.Le
- Keywords:
c
o
s
. 1. Introduction wayof reciprocityis networkreciprocity. Individuals are
s
c set on the vertices of a network and interact only with
i their neighbors. In this context, cooperators form clus-
s
y Cooperative dilemma was initially studied in the ters of mutual cooperationand this mutualism is viewed
h as reciprocity [9–14]. But human behavior is not so sim-
framework of classical game theory. Usually individu-
p ple: individualscanadoptreciprocalstrategiesbut,moti-
als have two strategies: cooperation and defection. A
[
vatedbyinternalemotion,likeangeragainstexploitation
cooperator provides a benefit to the opponent and pays
1 a cost for that. A defector receives the benefits if the [15], they can punish defectors [15–17]. This would not
v opponent is a cooperator. This defines a material pay- be so intriguing, as it is just another way of reciprocal
7 motives. But the important feature is that individuals
off. If individuals maximize their material payoff, it is
9 usually input costs to defectors at their own expenses.
well know that defection will dominate [1]. Departing
3
Thisbehavioriscalledaltruisticpunishment,becausein-
1 from these initial ideas, evolutionary game theory has
dividualspayacostto punishevenifthey nevermetthe
. emergedandstrategyevolutioninpopulations wasstud-
1 punished opponent again and because the punishment
ied. In this approachit is implicit assumed the principle
0
acts weakenthe defectors and the entire population gets
of natural selection, where the payoff is equated to fit-
0
better off [15]. Reputation, rewards or repeated inter-
1 ness and the fittest strategy survives [2]. In this context
action, as internal motives, they all interact with pun-
: it was shownthat the classicaltheory is recoveredin the
v ishment motives [5, 6]. Punishment involvessome subtle
replicatorequation,wherepopulationisconsideredtobe
i
X well-mixed, that is, a population where everybody inter- questions and gives rise to another evolutionary puzzle:
altruisticpunishment, althoughseeminglyusual, maybe
r acts with everybody [2].
a amaladaptivetraitasthepunishersgetworstpayoffs[8].
Cooperationcannotbesupportedwithoutextramech-
anisms[3]. Essentiallytwoactionstakeplacefor cooper- Recently it was introduced the possibility of an “im-
ation survival: maintenance of mutual cooperation and plicitpunishment”withoutturntoextraindividualpref-
preventionfromexploitation[4]. Cooperatorscanbebet- erencesexceptthedesiretomaximizetheowngain. This
ter off only if they meet each other so that their profits was accomplished by the adoption of different strategies
exceed defectors’ profits. If the individuals perceive that againstdifferentopponentsinthecontextofnetworkreci-
it is important what the opponents are doing in order to procitywithsynchronousupdate [18]. Insteadofplaying
attend these two essential actions, reciprocalpreferences the same strategy against all of the neighbors, individu-
cancomeup[5–8]. Reciprocitymeansthatwhatanindi- als can choose one different strategy against each oppo-
vidualdodepends onwhatothersdotohim/herdirectly nent. If eachplayerstrategicallyupdates their strategies
orindirectly. DirectreciprocitymeansthatIchoosewhat possibly imitating a successful random neighbor and re-
to do against you depending on what you do to me. In- places the interaction that gives the worst payoff by the
direct reciprocity means that my behavior toward you imitated strategy, it was shown for square lattices that
also depends on what you do to others. Another subtle cooperationwasstronglysupported,evenforhugedefec-
2
tion tendency, and was robust against misjudgments of 3. Evolutionary analysis
the worstinteraction. The possibility of opponentdiffer-
We made our simulations using networks of size N =
entiation introduces a mechanism of punishing without
40 and N = 100 and evaluate the fraction of cooper-
costsandwithoutanykindofinternalpreferencesexcept
ation (f ) adopted by all of the players in all of their
the desire of maximize own payoff. We call this punish- c
interactions. If n is the quantity of C strategies used
ment “implicit punishment”. But in that work [18], the c
in all of the interactions by all of the players, we have
possibility of adoption of different strategies was intro-
0 ≤ n ≤ N(N −1) and f = n /N(N −1). The ran-
duced in the context of network reciprocity. What hap- c c c
dom initial configuration consists of 50% of cooperation
pens if networkreciprocityis excluded? Here we analyze
and the averagesare made over 100 different initial con-
this model in well mixed populations, what means that
ditions. We use ǫ = 0.001 and we show here only the
we are excluding network reciprocity effects. The other
case b = 2, although we simulated the model also for
important feature of the model is the synchronous up-
other values of b. In fact, the b value has no effect in the
date assumption. It is well known that results may be
simulations and in the mean-field results.
striking different if asynchronousupdate is used[19, 20].
Before going on, let us state one fundamental feature
Here we analyze the model with both synchronous and
of the model that is independent if the update is syn-
asynchronousupdates. Weshowthatcooperationstillre-
chronousornot. Supposea focalplayerimitatesadefec-
mainsalive,althoughforasynchronousupdateitachieves
tionstrategy. Westatethatanyinteractionoftype(C,C)
its lower bound level. We analyze the model using com-
willneverbereplacedby(D,C).Ifsomeopponentadopts
puter simulations and a mean-field technique.
defection, the focal player must have at least one (C,D)
2. The model or (D,D) interaction. But these interactions give payoffs
worst than (C,C). So (C,C) will never be replaced. This
Let us state the model formally. We study the pris- proves the existence of a lower bound for the fraction of
oner dilemma in a population of size N as the scenario cooperationgiven by the initial fraction of mutual coop-
for the cooperation problem. We consider a well-mixed eration. Onthe otherhand,if the focalplayerimitates a
population, what means that each player can interact D strategy, he/she will first seek for (C,D) interactions.
with everybody. The strategy vector of an individual is Iftheyarepresent,(C,D)willbereplacedby(D,D).One
S~ = (S1,...,Sj,...,SN−1), where Sj can be C (coop- can see that mutual cooperation is never destroyed and
eration) or D (defection). So individuals are merged in every exploitation is punished.
N − 1 interactions. If in one of these interactions an For the usual game, where each player adopts a sin-
individual plays C against an opponent who is playing glestrategyagainstallofitsopponents,asingledefector
D, we denote this interaction as (C,D) (the first entry can invade a population of cooperators in infinity well-
is the strategy of the focal player and the second en- mixed population [2]. The first remarkable feature of
try is the opponent strategy). The payoff of a D strat- “implicitpunishment”isthatcooperationisevolutionary
egy against a C strategy is P(D,C) = b, where b > 1 stableinwell-mixedpopulationforbothsynchronousand
is the defection tendency. Using the same notation, we asynchronous update. If a mutant that adopts defection
have that P(D,D) = ǫ, with ǫ << 1, P(C,C) = 1 and against everybody appears in a population where every-
P(C,D) = 0. For synchronous update, each player in- body is cooperating, the mutant initially earns a huge
teracts with the other N −1 players, plays a round of payoff. But as soon as others imitate it, the exploited
onegameagainsteachopponent,andearnsacumulative (C,D)interactionswillbereplacedby(D,D),neutralizing
payoff. After that, each player chooses randomly one theexploitations. Whatissimple,butremarkable,isthat
neighbor and compare their cumulative payoffs. If the theinteractionsthatarechangingarejustthoseinwhich
opponent cumulative payoff is bigger than its own one, theexploiterisinvolvedandalloftheothermutualcoop-
it imitates the strategy the opponent is using against erationsaremaintained. The “implicitpunishment” will
him/her with probability proportional to the difference takeplace untilthe mutantcumulative payoffis equalto
of cumulative payoffs, Π=|∆P |/((N −1)b) [21]. On cooperator cumulative payoff. If we call n the quan-
cum exp
theotherhand,iftheopponentcumulativepayoffislower tity of defection adopted by the mutant exploiter, the
thanits ownone,the focalplayerremainswith the same payoff of the mutant exploiter and the payoff of the co-
strategy. If imitation takes place, the new strategy re- operators are P = n (b + ǫ) and P = N − 2,
exp exp coop
places the strategy used in the interaction that gives the respectively. By equating both expression we have the
worstpayoff. Theworstpayoffofthefocalplayerisgiven equilibrium fraction
by the interaction (C,D), followed by (D,D), (C,C) and
N −2
(D,C). For asynchronous update, a random individual is n = .
exp
b+ǫ
chosen so that it can imitate and possibly replace one
of its strategies like in the synchronous case. After this Let us now discuss the results obtained by numerical
individual update, the entire population play a round of simulationswithbothsynchronousandasynchronousup-
the game,andeachplayerearnsnew cumulative payoffs, dates. In the initial conditions, we set the players to co-
and another random individual is chosen to update its operatewithprobability0.5againsteachoneofitsoppo-
strategies. A time step consists of N of such processes. nents. This gives an initial cooperation fraction of 50%.
3
For asynchronous update, cooperation cannot dominate a
o
populationbutitcancoexistwithdefectionandassumes
2(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:14)(cid:20)(cid:21)
valuesnearthelower-bound(25%fortheinitialcondition 2(cid:22)(cid:8)(cid:18)(cid:21)u(cid:24)(cid:14)(cid:8)(cid:17)(cid:25)
assumed here). Fig. 1 shows simulation and mean-field tm(cid:6)
attfihiperelepdrspoyaoxnppicpmuhrlaroaotxtiniioomonnua.rsteiFuosuinpgldt.rsae2tsfeous,rlhtcostohwofeospraetsrhtyahenteiscoishmnyrnoudncloahomtruiooisnnnuaotpauendssdauttphem.deeaFaetonner--. (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)a(cid:7)(cid:1)a(cid:4)(cid:7)(cid:7)(cid:10)(cid:11)(cid:2)(cid:3)(cid:5)(cid:6)(cid:7)(cid:8)tmd a
One can see in Fig. 3 that, for short times, the syn- tm-
chronous update behaves like the asynchronous update.
At the beginning, all of the exploitations are neutralized
tmn
and only the initial mutual cooperation survives. After t o(cid:8)it(cid:6) n(cid:8)it(cid:6) M(cid:8)it(cid:6) -(cid:8)it(cid:6) f(cid:8)it(cid:6)
(cid:5)(cid:6)(cid:12)(cid:11)
that, cooperation starts to increase very slowly until it
dominatesthe population. So wecandefine ashort-time
FIG.2: Fraction of cooperation forsynchronousupdatewith
regime and a long-time regime for synchronous dynam- N =40 and b=2.0.
ics. The same qualitative result holds for large popula-
tions, but simulation time gets extremely huge for syn-
a
chronous update. Fig. 4 shows a finite size analysis for ai--
the time spent to reach the minimum value of the co- ai- 2(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:15)(cid:21)(cid:22)
2(cid:23)(cid:11)(cid:19)(cid:22)m(cid:25)(cid:15)(cid:11)(cid:18)(cid:26)
operation fraction before cooperation dominates in the
sr1lrayee/rggnTgiicmmehgereboo,uefnwotsorhufitiNosnchiutzep→ei,srdotac∞,hhteeai..mrlaoNpOcnlotnygeti-renttiizhgtmehedteahotbrtaeyahtgsedirtmaholhmeeoarnieinngsdaiat,sthsiioenfNnowaoseilyfnomnhccrgoape-voattepoismeteNrisec-, (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)a(cid:7)(cid:10)a(cid:4)(cid:7)(cid:7)(cid:11)(cid:12)(cid:2)(cid:3)(cid:5)(cid:6)(cid:7)(cid:8)aaaaiinMiinM-- a
ation.
aio-
aio
a ouaaa Muaaa fuaaa (cid:9)uaaa t(cid:11)SaM
(cid:5)(cid:6)(cid:13)(cid:12)
FIG.3: Short-timeregimeofthecooperationfractionforsyn-
chronousupdate with N =40 and b=2.0.
a proximation gives not only the stationary solutions, but
tmff fits reasonably the in silico time evolution, although for
tmf 2(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:14)(cid:20)(cid:21) thesynchronousupdateitfitswellonlyintheshort-time
2(cid:22)(cid:7)(cid:18)(cid:21)u(cid:24)(cid:14)(cid:7)(cid:17)(cid:25)
regime. Let us now derive the mean-field solution. We
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)a(cid:7)(cid:10)a(cid:4)(cid:7)(cid:7)(cid:11)(cid:12)(cid:2)(cid:3)(cid:5)(cid:6)(cid:7)(cid:8)ttmmttM-mmffM- a fisyrnstcharnoanloyuzseotnhee. asynchronouTs update followed by the
(cid:2)(cid:3)(cid:2)(cid:2)(cid:7)(cid:4)
tmnf
(cid:2)(cid:3)(cid:2)(cid:2)(cid:7)
tmn
t n(cid:7)it- -(cid:7)it- d(cid:7)it- (cid:10)(cid:7)it- o(cid:7)itf (cid:2)(cid:3)(cid:2)(cid:2)(cid:6)(cid:4)
(cid:5)(cid:6)(cid:13)(cid:12)
(cid:2)(cid:3)(cid:2)(cid:2)(cid:6)
FIG.1: Fractionofcooperationforasynchronousupdatewith (cid:1)(cid:2)(cid:3) (cid:2)(cid:3)(cid:2)(cid:2)(cid:5)(cid:4) T
N =100 and b=2.0.
(cid:2)(cid:3)(cid:2)(cid:2)(cid:5)
(cid:2)(cid:3)(cid:2)(cid:2)(cid:2)(cid:4)
mean-field solution provides a good equilibrium anal-
(cid:2)
ysis in well-mixed population if the usual game is con-
sidered [22]. But in structured population, it is not a (cid:1)(cid:2)(cid:3)(cid:2)(cid:2)(cid:2)(cid:4)
(cid:2) (cid:2)(cid:3)(cid:2)(cid:6) (cid:2)(cid:3)(cid:2)(cid:8) (cid:2)(cid:3)(cid:2)(cid:9) (cid:2)(cid:3)(cid:2)(cid:10) (cid:2)(cid:3)(cid:5) (cid:2)(cid:3)(cid:5)(cid:6)
good approximation [23]. Although in the present work (cid:1)(cid:2)(cid:5)
we deal with well-mixed population, the nature of the
“implicit punishment” model is not so simple. It is not FIG. 4: The plot shows the relation 1/T ×1/N, where N is
obvious that a mean-field approach would work. So it the population size and T is the time to reach the minimum
is a remarkable result the fact that our mean-field ap- valueof cooperation for synchronous update.
4
mean-field approximation for the asynchronous We first consider a typical player, that we call focal
update player. We are going to study the dynamics of the local
concentrationof(C,D),(D,D),(C,C),and(D,C)interac-
Let us firstdefine a localand a globalinteractioncon- tions around the focal player. Let k1, k2, k3, and k4 be
centrations for a population of size N +1. If in an in- thequantityofsuchinteractionsaroundthefocalplayer.
teraction a player i adopts strategy A and its opponent Note that k1 +k2 +k3 +k4 = N. Let us assume that
adopts B, where A,B ∈{C,D}, we say that playeri has the probability of having k1, k2, k3, and k4 interactions
a (A,B) interaction. Player i can have NAB(i) (A,B) in- are given by the respective global concentrations. The
teractions. We define the localconcentrationof(A,B) as probability of a focal player configuration (k1,k2,k3) is
thefractionof(A,B)interactionsaroundplayeri,namely given by
x (i) = N (i)/N. For the global concentration of
AB AB
(A,B) we define x = N (i)/((N +1)N).
AB Pi AB
N!
P = xk1 xk2 xk3 xN−k1−k2−k3
k1,k2,k3 k1!k2!k3!(N −k1−k2−k3)! CD DD CC DC
We considerthe other nodes as mean-fieldnodes. So the the focal player can imitate the D strategy from (C,D)
thelocalconcentrationof(C,D),(D,D),and(C,C)inter- or (D,D) interactions. Let us focus on the first alterna-
actions around those nodes is given by the configuration tive,thathappenswithprobabilityk1/N. Theopponent
vector N(x ,x ,x ). Now we are going to derive associated with the (C,D) interaction has a payoff of
CD DD CC
the rate of variation of the local concentration around
the focal player. PF =b+(N −1)(x b+x +x ǫ).
CD DC CC DD
Let us derive the rate of increasing of the (D,D) lo-
cal concentration. Suppose that the focal player is on
The probability of imitating the D strategy from (C,D)
a (k1,k2,k3) configuration. Note that the payoff of this is given by
configuration of the focal player is
[PF −PF]Θ(PF −PF )
PF =(N −k1−k2−k3)b+k3+k2ǫ. CD CD k ,
bN
There is just one transition that increases this quantity:
(C,D) to (D,D). In this replacement, the focal player where Θ(x) = 1 if x > 0, and Θ(x) = 0 otherwise. The
adoptsCandimitatesanopponentthatadoptsaD.Soat mean rate of increasing of (D,D) by one unit due to the
leastone interactionof type (C,D) must be present. But imitation from (C,D) is
WC+D = XN NX−k1N−Xk1−k2 kN1[PFCD−PF]bΘN(PFCD−PF)Pk1,k2,k3
k1=1 k2=0 k3=0
Following the same lines we obtain the possibility of imitating from (D,D), that is given by
WD+D = XN NX−k1N−Xk1−k2 kN2[PFDD−PF]bΘN(PFDD−PF)Pk1,k2,k3.
k1=1 k2=0 k3=0
The same analysis can be done to calculate the rate of change the quantity of (D,D). Following the previous
decreasing of (D,D). There is just one transition in- steps, we can define
volved,namely,(D,D)to(C,D).Atleastone(D,D)must
bepresent. TheCstrategycanbeimitatedfrom(C,C)or P = N! xk2 xk3 xN−k2−k3,
(D,C).Butnotethatthefocalplayercannotbecurrently k2,k3 k2!k3!(N −k2−k3)! DD CC DC
adopting a (C,D), because in that case (C,D) wouldgive
the worst payoff. So imitating a C strategy would not and we obtain that
5
WC−C = XN NX−k1 kN3[PFCC −PF]bΘN(PFCC −PF)Pk2,k3(1−δk3,0δk2,N), and
k2=1 k3=0
WD−C = XN NX−k1 N −kN2−k3[PFDC −PF]bΘN(PFDC −PF)Pk2,k3(1−δk3,0δk2,N).
k2=1 k3=0
Asallofthenodeshavethesametypicalbehavior,be- x =x0 ,
CC CC
cause the population is well-mixed, we can approximate
the global concentrations by the local ones. The above where the index 0 refers to the initial conditions. If we
expressionsdetermine the rate of(D,D) variationby one set ǫ = 0, just for simplicity, the solution reaches the
unit. If we want the time derivative of x , we need do fixed point
DD
divide the expressions by N and multiply by a factor of
x∞ =x0 +2x0 , x∞ =x0 ,
two, because there is the contribution of the opponents DD DD CD CC CC
update . So we have
and x∞ =x∞ =0.
dxDD =2 1 (W+ +W+ −W− −W− ). CD DC
dt N CD DD CC DC
Onecanseethatonly the initialmutualcooperationcan
Followingthesamereasoning,onecanseethatx does bemaintainedandalloftheotherinteractionsaremutual
CC
not changein time. Finally, as allof the mean-fieldvari- defections. Notethatallofthe exploitationsareneutral-
ables are normalized to one, we obtain that ized. Note that this approximation gives good results if
compared to simulation data.
dx 1 dx dx
CD DD CC
=− ( + ).
dt 2 dt dt mean-field approximation for the synchronous
update
We cansimplify furthertheseexpressionsifwereplace
k1, k2 and k3 inside the payoff expressions of the focal Let us treat the synchronous case. Now (C,C) can in-
player by the expected value of such quantities given crease, because it is possible to have a (D,D) to (C,C)
by the configuration probability: NxCD, NxDD, and transition whenever two players make a (D,D) to (C,D)
NxCC, respectively. With this extra approximation the transition in their shared (D,D) interaction. This is an
Θ function can be easily evaluated in the limit of large essential feature of the synchronous model. This kind
N and we have the following equation: oftransitiondoes nottake place inasynchronousupdate
andthatisthereasonwhycooperationassumesthelower
dx 1 ǫ
DD = 2 {x +x [1−(1−x )N−1] bound value in the asynchronous case. We can approxi-
dt N2 CD DDb CD
mate the rate of this transition by
x
− CC[(1−x )N−1−(x +x )N−1]}
CD CC DC
b dx 1
CC = (W− +W− )2.
dt Nx CC DC
This equation can be solved numerically. Fig.1 shows DD
the numerical solution and the simulation results. Note
Let us explain the term in the denominator. If the focal
thatthisapproximationfurnishesgoodresultswhencom-
player makes a (D,D) to (C,D) transition on an specific
pared with in silico evolution. For the initial condition
interaction, the mean-field player associated to this spe-
used here, we have that the terms inside the parentheses
cific interaction should choose exactly this interaction,
are 0.75 powered to N −1 and 0.5 powered to N −1.
what happens with probability 1/Nx . If we perform
DD
If N is large, the terms that are powered to N are very
the same simplifications that was already done for the
smallandthey canbe neglected,atleastforshorttimes.
asynchronous case, we have
This gives the following simplified equations,
dx 1 x
dx 1 ǫ dx CC =2 { CC[(1−x )N−1−(x +x )N−1]}2,
DD =2 (x + x ), and CC =0. dt N4 b CD CC DC
dt N2 CD b DD dt
The solution of these equation are straightforward,
dx 1 ǫ
DD = 2 {x +x [1−(1−x )N−1]
ǫx0 +x0 2 1 ǫ dt N2 CD DDb CD
xDD =x0DD+b D1D+ ǫ CD[1−exp(−N2(2+b)t)], and − xCC[(1−xCD)N−1−(xCC +xDC)N−1]}
2 b b
6
Fig. 2 andFig. 3show the numericalsolutionofthese Conclusion
equations. One can see from the above equations that
x increasesmuchslowerthanx . Fortheinitialcon-
CC DD
dition assumed here, x time derivative at the begin- Here we analyzed the model that allows the individ-
CC
ningisalmostzero,becausethevaluesinsidethebrackets uals to choose different strategies against different op-
areequalto0.5poweredtoN. Butwhenevolutionstarts, ponents in well-mixed populations for both synchronous
greatpartof(C,D)interactionsarechangedto(D,D)and and asynchronous update. First we showed that coop-
x is reduced to some value near to zero making x eration is evolutionary stable for both synchronous and
CD CC
to increase faster. So we have two regimes: short-time asynchronous update, what means that a defector mu-
regime,whenx iskeptalmostconstantarounditsini- tant cannot invade a population of cooperators. We also
CC
tial values, and long-time regime, when x starts to showed,forainitialconditionof50%ofcooperation,that
CC
increase faster. Fig. 2 and Fig. 3 show both coopera- for synchronous update cooperation always dominates
tionevolutionregimes. Forshorttimes,ifwediscardthe while for asynchronous update the cooperation fraction
termsthatarepoweredtoN−1,wehavethe samesolu- assumes the lower bound given by the initial mutual co-
tion as the asynchronous case. This means that cooper- operation. For the synchronous update, population dy-
ation assumes a value near its lower bound value, given namics exhibits a short-time behavior that is similar to
by the initial mutual cooperations. But for long-time theasynchronousupdate,butforsuficientlongtimes,co-
regime, x is near zero and x˙ cannot be neglected. operationspreads for largeN but finite. The crucialdif-
CD CC
So x starts to increase until it become equal to one. ferencebetweensynchronousandasynchronousisthatin
CC
So for sufficient long times, the stationary solution is synchronousupdate it is possible to havea simultaneous
updatethatallowsa(D,D)to(C,C)transition. Thisdoes
x∞ =1 and x∞ =x∞ =x∞ =0. not happen in asynchronous update. In a preview work,
CC CD DC DD
the samemodelwasanalyseinasquarelattice withsyn-
Note that for the short-time regime, shown in Fig. 3, chronous update. Here we showed that the synchronous
themean-fieldapproximationfitswellwhencomparedto update is crucial for cooperation dominance while net-
in silico evolution. For the long-time regime, the time work reciprocity effects are not so important. Although
evolution of the mean-field solution does not fit well, al- the asynchronous update does not provide cooperation
though it gives the right stationary solution. dominace, it allows cooperation to survive at its lower
From the above expressions and simulation data, we bound value. Note that this result in asynchronous up-
see that the lower value of cooperation is reached very date is still better than those of the usual game.
fast in the synchronous update. But as long as popu-
lation size gets bigger, this value is reached very slowly
(see Fig. 4). Besides that, if N is large, x increases
CC
very slowly. By these reasons, for large N, in the syn-
Acknowledgments
chronous update the population seems to be wrapped in
the lower value of cooperation, although what is actu-
ally happening is that cooperation is slowly increasing, TheauthorsthanktoCNPqandFAPEMIG,Brazilian
spreading until dominates the entire population. agencies.
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