連続型スノードリフトゲームにおける協力とインセンティブ行動の進化 (第12回生物数学の理論とその応用 : 遷移過程に現れるパターンの解明に向けて)

連続型スノ $-$ ドリフトゲームにおける協力とインセンティブ行動の進化 Evolution ofcooperationand costly incentives inthecontinuoussnowdrift

game

佐々木 達矢

ウィーン大学数学

TatsuyaSasaki

FacultyofMathematics,

University of Vienna

tatsuya.

_{[email protected]}

Abstract:Gametheory research

on

the snowdriftgamehas showed that gradualevolution of the

continuouslyvaryinglevel ofcooperation injoint enterprises

can

demonstrateevolutionary merging

as

well

as

evolutionarybranching.However, littleis known aboutthe

consequences

of changesin diversity$at$thecooperation level. Inthepresentstudy I consider effects of costly

rewards

on

thecontinuoussnowdriftgame. Ishowthatnotevolutionary merging but evolutionary branching

can

promote theemergenceof poolreward,which

can

thenenhance social welfare.

1 Introduction

IncontrasttothePrisoner’s Dilemma,the snowdrift

game

can

leadto

a

coexistenceof cooperators and freeriders [1], Thedivergence scenarioforthecooperation level in the continuous snowdnftgame(SDG)hasbeen termed the “tragedy ofthecommune”[2]: gradual evolution

can

favorsuch

a

stateinwhich

a sense

of fairnessmaybe minimizedrather than

a

state

inwhich alladoptthesamecooperationlevel. Viewedinthislight, of muchinterestwould beto

investigatethe question: whether

or

notthatevolutionary branching

can

makepositiveeffects

on

thepopulation’s welfare? To date,the elementary classification of adaptive dynamics has been wellprovided for models with simple payoff functions [2,3], inwhich mostly onlythenegative conclusion of decreasein

a sense

offaimess has been considered. Tomyknowledge, the question has

never

seriously beentackled,despite the fact thatcoexistenceof clearly different levels of cooperation is

common

innatureandhuman societies.

Here

we

approachthe

issue

with considering poolreward

as a

facilitatorofcooperation. It is well known that selective

incentives

such

as

rewardandpunishment

are

common

toolstocurb human behaviors [4,5]. In

a

standard pool-rewardmodelbased

on

discretepublic good

games,

eachplayeris offered the opportunities tocontribute, first,to

a

fundpool that rewards cooperativebehaviors, andsecond, to

a

pool that providespublicgoods; finally, the resulting goods

are

equallyshared$axt’1ong$allplayers,but the resultingrewards, amongonly those who

contributedtothe public goods [6,7]. Therefore,poolreward is

a

club good, not public, and its

amountshared may depend

on

the level ofaltruism.

Voluntarily rewardingincurs

some

cost

on

each contributor.Thisthus

can

give riseto

a

dilemmatic situation

among

thosewho$contrib\dagger Jte$ toboth(pro-socialists”)and these who

contributetothepublicgood but not to therewardfund(second-orderfree

riders

Recent results haveshown,nevertheless, that voluntaryreward for the poolsystem

can

evolve and improve cooperation andpayoffs in discretepublicgood

games,

despite thepresenceof second-order free riders. Therefore,this implies that similarly,in the continuous public good

game

voluntaryreward may thrive and then facilitate the socialwelfare, if

a

population

was

tobe diversified

as

inthediscrete

case.

Inthepresentstudy

we

focus

on

the SDG,

a

simple two-player public goodgame[2]. In

whatfollows

we

investigatethegradual co-evolution ofthe

investment

levelinthe SDG and the donation levelintherewardfund. Weexaminethemodel in paKiculal.for

a

range

ofparameter settings inwhich thecontinuousSDG leads initially monomorphic populations toundergo evolutionarybranching

on

theinvestmentlevel.

2 Model and methods

We consider

an

infinitely large,well-mixed population. Eachindividual $i$inthepopulationhas

two continuouslyva1yingtraits,$(c_{i}, e_{j})$with$c_{j},$$e_{i}\geq 0$,in which$c_{i}$represents thedegreeof

investmenttotheSDG,and$e_{i}$,the degree of donation inthe reward fund. Fromtimeto time, $a$

couple of players

are

randomly sampled fromthe population.Each player$i(=1,2)$_firstinvests$e_{i}$

$C(c_{j}\rangle.$ Finally, each

receives

(i)the benefits from the SDG

and

(ii) the

rewards

from the

reward

fund. (i)Theindividualbenefitfrom the SDG,$B(c_{1}+c_{2})$,whichdepends

on

the investment

sum

of thetwoplayers, is equal

among

them, irrespective of their

investment

levels in the SDG. (ii) The

group

reward fromthefilnd,$R(e_{1}+e_{2})$,which depends

on

the donation

sum

ofthe two

players,is sharedbetweenthem, buteach sharemaybe unequaland based

on

merit. Fund-sharingmaybemodeled invarious ways.Inthe model

we

examine

a

simple andintuitive

case

thatthe individualshare of rewards is proportionalto thedegreeof investment inthe SDG,$c_{i}.$

We study thegradual evolutionofpopulations

on

thetwo-dimensional trait

space

$(c, e)$_,

usingthetechniques ofadaptive dynamics [8,9]. Wefirstdescribe the fitness of

a

rare

mutant withstrategy$y=$$(c_{\nu},, e_{y})$ amongtheresidents with strategy$x=(c_{X}. e_{x})$. Thisis called invasion

fitness$F(x, y)$,defined by$F(x, v)=P(x,y)-P(x, x)$,where$P(x,y)$denotes theexpected payoff

for the mutant with$y$. The expectedpayoffis given by

$P(x,y)=B(c_{X}+c$ $+ \frac{c_{/}}{c_{\lambda}.+c_{1’}}R(e_{X}+e$ $C(c_{1}.)-S(e_{\}}.)$, (1)

where the firsttermdenotes the benefit fromthe SDG, and the secondterm,theindividual reward which isproportional totheinvestmentlevel,$c.$

Forsimplicity,

we assume

both the

upper

limits of$c\backslash$and

$e_{j}$

as

1. We

assume

quadratic

payofffunctions for the SDGand linear functions for the rewardfund,

as

follows: $B(x)=\ x^{2}+$

$\beta_{1}x,$ $C(x)=\gamma_{2}x^{2}+\gamma_{1}x,$_$R(x)=rx$,and_$S(x)=sx$.Inthe

case

of

no

rewarding,themonomorphic

adaptive dynamics for quadratic cost and benefit functions have been fully analyzed. According

to[2], the corresponding adaptive dynamics for monomorphic populations

can

have theunique equilibrium$(”$singularstrategy $Q=(c^{*}, 0)$with$c^{*}=(\gamma_{1}-\beta_{1})/(4\beta_{2}-2\gamma_{2})$

.

Singular strategy$Q$

enters the$c$’s interval [0,1] and alsois$a$(globally)convergence-stable, ifand onlyif

$\partial Fl\ _{y}\lfloor_{v=x=(0.0)}=\beta_{1}-\gamma_{1}>0$and$\partial Fl\ _{v}|_{v=x=(1,0)}=(4\ -2\gamma_{2})+(\beta_{1}-\gamma,)<0$

.

Furthemore, $Q$is

an

evolutionarybranching pointatwhich

a

monomorphic population will divergetotwo

subpopulations

across

thepoint, if$\ovalbox{\tt\small REJECT} Fl\ _{y}^{2}|_{v=x=Q}=b-\gamma_{2}>0$;

or

otherwise,

an

evolutionary

mergingpoint

across

which

a

dimorphic population

can converge

to

a

uniform populationatthe point [2,3].

3

Results and discussion

We shall

demonstrate

that evolutionarybranching

can

profoundly affect theevolutionary fate of populationsbeen stuck at

a

state

in

whichall

never

reward: $e=0$

.

Itis easyto

see

thatthe homogeneousstateofnon-rewarding$(e=0\rangle$isglobally convergence-stable, if $r-s<0$

otherwise,thatof$hl1$-rewarding_$(e=1)$is globally convergence-stable. Ofspecial interestis the

former case,

in

whichindividuals tendto

decrease in

thedonation level

in

monomorphic populations. In the

case

singular strategy$Q=(c^{*},$$0\rangle$

can

be globally convergence-stable for

gradualco-evolution ofinvestment and donation$(c, e)$

.

Accordingtonumerical simulations(Fig.

$1\rangle$,indeed the monomorphic populations, starting with$e=1$,evolvestowards singularstrategy$Q$

(Fig. la).Voluntarilyrewardingvanishes forthe adaptive dynamics of monomorphic populations.

Subsequently, evolutionarybranching

on

the investmenttrait,$c$,takesplace. Acooperative

branch( $C$-branch”)whoseinvestmentlevelisgreaterthan $c^{*}$,onthe

one

hand,moyestowards_$c$ $=1$

.

Forthespecific parameters,

on

the

way

$C$-branch alsostartsevolving withrespect to the

donationtrait$e$(Fig. Id). Atthe

same

time, the

averages

ofinvestmentandpayoff abruptly

increase(Fig. le, f).Onthe otherhand,defectivebranch( $O$-branch”) first evolves to smaller

levels of$c$. When the evolutionof$C$-branchisonly

on

trait$e$,theevolution of$D$-branchturns

intoofthe oppositedirection, i.e.,increasein (Fig. Ic).Thepopulation intheendconsistsof C-branch with full

investment

anddonationand$D$-branchwith intermediate

investment

level and

no

donation. In particular, voluntanly rewarding

emerges

fortheadaptivedynamics of dimorphic populations.

Theresultsshowthatevolutionary branching

can

significantly affectthe

average

levels of payoff,investmentin the SDG, and donationintherewardfund,by facilitating the

emergence

of pool reward. Evolutionary branching at$Q$consequently

leads

to

discontinuous

increases all in

these three indexes. Wenotethatwithoutpoolreward, gradual evolution of theinvestmentlevel,

$c$,

can cause

dimorphic diversification ofthe populationintothe extreme levels: $c=0$ and$c=1.$

Inthe

case

it isknown that evolutionary branching

can

only lead to negligibleincrease in the

Sofar,mostofstudies

on

the evolution of selective

incentives

have made

much

effort

on

punishment [10].Discrete-strategy models have confirmed that costlypunishment isnotlikelyto

emerge

when rare,withoutany supportivemechanism,undertheSDG-likeconditionsthat cooperators and defectors

can

coexist[11,12]. Howevolutionary branching affects the

gradual

evolution ofpunishmentwouldbe anotherfascinatingquestionthat deserves further works.

References

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Welfare.

Oxford and New York:

Blackwcll.

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snowdriftgame.Biosystems 131, 51-59.

4. Sigmund$K$,Hauert$C$_,NowakMA. 2001 Rewardand punishment. Proc.Nall. Acad. Sci. 98,

10757-i0762.

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\’E,

Mesz\’ena$G$,MetzJAJ. 1998Evolutionarilysingularstrategies and the adaptive

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Figure

1

Investment level, $c$ $b$

$e$

Figure legend

Figure 1.Convergence,evolutionary branching,and

emergence

ofvoluntary$1^{\cdot}$eward.The

population $stal2s$its gradual evolution atthe

corner

with$(c, e)=(1,1)$indicatedby

a

star. According to the selectiongradient(depictedby stream plotin a),thepopulation firsttravels along

a

dashed

curve

and

converges

toboundary

point

$Q(0.5,0)$

indicated

by

an

empty circle.

Subsequently, evolutionary branching

on

trait$c$

occurs across

$Q$ aroundat$t=50(c)$

.

$A$

cooperative branch(darkblue;$C$-branch),

on

the

one

hand,

moves

to$c=1$ andaroundat$t=100,$

also startsincreasing intrait$e(d, e)$. Atthe

same

time,the

average

cooperation (fiandpayoff(e) jump

up.

On theotherhand,defective branch(lightyellow;$D$-branch)first diverges to smaller

levels of$c(c)$_.When$C$-branchevolvesalongtheline$c=1$,theevolution of$D$-branchturns

into

of theoppositedirection,i.e., increase in trait$c$

.

Eventually, thepopulationbecomesdimorphism

consisting of$C$-branch with$fUl1$ investment and donationat$(1, 1)$ and$D$-branchwith

intermediate investment and

no

donationat$(0.29, 0)$

.

Toobtain thetrajectory,I numerically solve the correspondingcanonical

equations

for the adaptive dynamics[8]by using Mathematica. To allowtransitionsbetween monomorphismand dimorphism, Iconsider

a

populationthat initiallyconsistsoftwopatches whose locations

are

infinitesimallyclosetoeach other and fractions

are

equally 0.5. Otherparameters: $B(x)=-1.4x^{2}$