Cooperation in Evolutionary Games on Complex Networks

49th IEEE C**on**ference **on** Decisi**on** and C**on**trol

December 15-17, 2010

Hilt**on** Atlanta Hotel, Atlanta, GA, USA

**on****on**ary**on** **Complex** **Networks**

Jianlei Zhang, Chunyan Zhang, Tianguang Chu and Zhifu Chen

Abstract— We c**on**sider a populati**on** engaged **on**t

public goods games. In our study, the lowest c**on**tributor

each game will be removed from the group, meanwhile new

players will be added to the network to ma**on**stant

populati**on** size. Here, each new node would establish some

c**on**necti**on**s with the exist**on**es **on**clude

that appropriate **on** numbers of each newcomer

support the emergence and ma**on** for

the c**on**t

regular network has been driven to a slowly vary**on**regular

heterogeneous network which facilitates the evoluti**on**

of cooperati**on**. By **on** of

**on**tributi**on**s, we show that provisi**on** of comm**on**

goods can be fostered by permitt

**on** the lowest c**on**tributor **on**t

game. We hope that our proposed model presents a feasible

mechanism for promot**on** of

cooperative behaviors.

I. INTRODUCTION

Cooperative phenomena are comm**on** between unrelated

and selfish **on**omical and

other systems. For example, replicat

cooperate to form the first cells. S

to form the multi-cellular organisms. The soma cells of

the body cooperate and help the cells of the germ l

to reproduce. Animals cooperate to form social structures,

groups and societies. More examples about altruistic behavior

here. The evidence **on**strates

that cooperati**on** and coord**on** am**on**g selfish

may arise and be susta**on**

their

**on**stitute situati**on**s

where the collective payoff is at odds with

Dur

cooperative behaviors of selfish

the social dilemma **on**flict arises between the

attract**on** from various scientific fields.

**on**ary

framework for **on** of

cooperati**on** am**on**g selfish agents [1]–[9]. A widely studied

model, which illustrates the c**on**flict between cooperati**on**

This work is supported by Nati**on**al Nature Science Foundati**on** of Ch

(NSFC) under Grants No. 60774089, No. 10972003, No. 10926195, No.

60674047 and No. 60974064.

The authors are with the State Key Lab of Turbulence and **Complex**

Systems, College of Eng

Ch

chutg@pku.edu.cn (T. Chu), zfchen@pku.edu.cn (Z. Chen).

and defecti**on**, is the pris**on**er’s dilemma game (PDG) [10]–

[12]. In the (two-player and **on**e-shot) PDG, each participator

has to simultaneously and **on**

c**on**cern**on**. In the orig

each of two players has two opti**on**al choices, cooperati**on**

(C) or defecti**on** (D). They will both collect the payoff R and

P for mutual cooperati**on** and mutual defecti**on**, respectively.

The mixed choice gives the cooperator the sucker’s payoff

S and the defector the temptati**on** T respectively. Moreover,

the four elements satisfy the order rank

and usually the additi**on**al c**on**stra

**on**. So mutual cooperati**on** leads to the highest ga

for both of **on** is the optimal decisi**on**

regardless of the choice of the competitor. Thus a socalled

social dilemma occurs, where mutual cooperati**on** is

beneficial **on**g perspective, but defecti**on** will br

shot-term profits.

However, many social situati**on**s are associated with collective

acti**on** based **on** jo**on**s made by a group often

when it comes to study**on** of cooperati**on**

through pairwise **on**s, the related public goods game

(PGG) accounts for group **on**s as well [13]–[21].

In the PGG, be**on** of the

two-player PDG, multiple agents (more than two players)

task which is beneficial to an entire group. And the task

requires the cooperati**on** of several

group who share the workload required to perform the

task. Players make their choices between cooperati**on** and

defecti**on**

received by all participants, while any cost is undertaken

by the cooperators **on**ly. Therefore it is well known that

c**on**tributors participat

free-rider problem

benefits of liv**on**tribut

S**on** is crucial for the prosperity

of society and also frequently encountered

situati**on**s, various mechanisms aimed at f

c**on**diti**on**s the cooperati**on** emerges **on**ary games.

Prom**on**s [22], [23],

direct reciprocity [24],

[25]–[30], spatially structured populati**on** [31]–[41], punishment

[19], [42], social diversity and the associated c**on**tributi**on**

diversity [21], and voluntary participati**on**

**on**s [15], [17].

In the simplest form of PGG, cooperati**on** or defecti**on** is

the **on**ly strategy that an **on**

978-1-4244-7744-9/10/$26.00 ©2010 IEEE 1785

means mak**on**or

and accrues a benefit to both the d**on**or and the recipient.

However,

**on**s, **on**tribut

wealth or n**on**e. Inspired by this often encountered situati**on**,

our study endows players with diverse choices

b**on**. In other words, we c**on**sider a populati**on**

whose members take the c**on**t**on**tributi**on**s,

aim

In this variati**on** of the traditi**on**al PGG, each

choose not **on**ly full cooperati**on** or no cooperati**on**, but also

**on** levels **on**tribut

arbitrary fund to a collectively advantageous group project.

This adopted model for

referred to as c**on**t

study. We assume that the CPGG is played **on**

groups of size N, **on**tributes to the

public good at a cost to itself. Let c i be the c**on**tributi**on**

of any given **on**, which can

vary between 0 and 1 (as an additi**on**al simplificati**on** but

without loss of generality, here

c**on**tributi**on** value to be 1). Each c**on**tributi**on**

comm**on** resource by rc, where r determ

each member

payoff of r ∑ N

i=1 c i/N from their c**on**tributi**on**s to the public

goods, hence the f

P i = r

N∑

c i /N − c i

i=1

with N be**on**

group. For the Public Goods game **on**e always assumes

r > 1. If r is smaller than N, to c**on**tribute is always

disadvantageous aga**on** of acti**on**s by other

group members. The game directly leads to the tragic outcome

**on**tribut

hence forego

After extend

cooperative games to c**on**t**on** games,

we would like to draw attenti**on** to the

of allow**on**

We

its members with public goods through a CPGG and the

group has the right to expel the worst member, c**on**vert

it **on**-member excluded from shar**on**

goods. Notably, examples **on**s

organizati**on**s, and comities from which members can be

expelled for n**on**-fulfill**on**s. More precisely,

we focus **on** permitt**on** the

lowest c**on**tributor who provides the least c**on**tributi**on** to the

comm**on** pool

c**on**tributi**on**s are higher than the lowest subject will stay

and participate **on**

has been implemented

number of newcomers will jo

c**on**stant amount of the whole populati**on** size.

The rema

next secti**on** we describe the model **on**

III features the results, whereas **on** we

summarize our f**on**clud

II. MODEL

To vividly imitate the c**on**tacts am**on**g players

world, spatial evoluti**on**ary games **on** complex network which

mechanism account**on**

discards the well-mixed assumpti**on** for the populati**on**,

where complex network provides a natural and c**on**venient

framework to describe the populati**on** structure **on** which the

evoluti**on** of cooperati**on** is studied. Generally, a typical setup

is the follow

which can be a regular lattice or have a more complex

structure. Then, agents are c**on**stra

their immediate neighbors al**on**g the edges of the underly

network. The system under our **on** is the spatial

CPGG **on**omous players. Initially, we

employ a two-dimensi**on**al regular network as the populati**on**

structure

self-**on**s are omitted. Here, the populati**on** structure

is not static, but vary**on** process. In our

study, players **on** the nodes of the underly

choose from a large set of strategies.

S

network that is c**on**structed from the aforementi**on**ed N

**on** the

two dimensi**on**al network with degree k i = 4, where k i

is its c**on**nectivity degree. Thus, all

same k nearest neighbors

lattice network. Follow

each player i acts as an organizer of the comm**on** pool i with

size k i + 1, where there occurs the CPGG

and its neighbor

by itself, player i also engages

by its nearest neighbors.

Initially each player i is designated a random c**on**tributi**on**

amount c i (0 ≤ c i ≤ 1). Then player i pays its c**on**tributi**on**

c i to all k i + 1 comm**on** pools it participates

given group with k +1

c**on**tributi**on** is multiplied by a factor r (1 < r < N). Then

the collective goods is redistributed to all k + 1 members

equally, irrespective of their actual c**on**tributi**on**s. In this

case each player i receives the freely-shared public

r ∑ k+1

i=1 c i/(k + 1), thus its f

is r ∑ k+1

i=1 c i/(k + 1) − c i . Evidently, players are faced with

the temptati**on** of be**on**tribut

(c i = 0). Thus the ga

neighborhood centered at

k

∑ j+1

P ij = η c r − c i (1)

r=1

1786

where c r is the **on**tributi**on** amount of a member

r

renormalized enhancement factor **on** the public goods. Then,

the total payoff P i of player i is the sum of ga

**on**s

P i = ∑

j∈Ω i

P ij (2)

where Ω i represents i’s nearest neighborhood plus itself.

As a disguised punishment, here we suppose that the

lowest c**on**tributor will be expelled from the populati**on** after

each game round, meanwhile, the same number of new

players will be added after remov**on**tributors.

Thus the structure of the populati**on** itself is not static, but a

c**on**sequence of coevoluti**on**ary dynamics. Here

that each newcomer preferentially establish its m **on**s

with the old nodes accord

is positive parameter and we can study the effects of its

value **on** the cooperati**on** level. Accord

attachment rule, those exist

**on** neighbors show more attractiveness to newcomers

than others. Thus

by a new player with the follow

q i =

k i

∑ M

j=1 k j

where M (0 < M < N ) is the exist**on**

size

c**on**nectivity number of

the **on**tributi**on** amount of the newcomer equals the

average c**on**tributi**on** of its m

Initially, a given player **on** the site i is designated a

random strategy (namely the **on**tributi**on**

paper) with equal probability, and then the game is iterated

**on**te Carlo simulati**on** procedure.

Exist**on** the network update their strategies (the

**on**tributi**on** amount)

rule. First, a randomly selected player i acquires its payoff P i

by play

is compared to a randomly selected neighbor j with payoff

p j . Last, player i will adopt the strategy of

a probability W(s i → s j ) proporti**on**al to the difference

performance P i − P j . Otherwise, the focal node i will stick

to its current strategy. The probability W(s i → s j ) is given

as follows

W(s i → s j ) =

1

1 + exp[(P i − P j )/T]

where T characterizes the magnitude of noise

many different effects (fluctuati**on**s

decisi**on**,

corresp**on**d to the completely determ

random selecti**on** of the j’s strategy s y , respectively. Namely,

T = 0 denotes that the agent always adopts the best

strategy determ

is absolutely irrati**on**al **on**s. For any f

positive values, T **on**al

(3)

(4)

factor **on**, such as the case of small

possibility to select the worse **on**e. Here, we **on**ly c**on**sider

the simple situati**on** for **on** probability and

simply set T = 50.

III. SIMULATION RESULTS AND ANALYSIS

In the follow**on** results carried

out for a populati**on** of N = 2500 players occupy

vertices of the lattice network

(**on**tributi**on** amounts **on**

are randomly distributed **on** results

are obta**on**s of the

entire 10 5 generati**on**s. Moreover, each data po

over 100 realizati**on**s of both the networks and the

c**on**diti**on**s.

Average c**on**tributi**on**

1

0.8

0.6

0.4

0.2

0

m=2

m=4

m=6

m=8

0.2 0.4 0.6 0.8 1

Fig. 1. (Color **on**l**on** results of average c**on**tributi**on** f c

of the whole populati**on** as a functi**on** of η with different m values. The

network size is 2500. Each data po

last 10 4 generati**on**s of the entire 10 5 generati**on**s. Here, all curves were

obta**on**ds to an average over 100

runs. L

We shall start by reveal

model by exam**on** the

stati**on**ary average c**on**tibuti**on** f c

CPGG. Fig. 1 shows results obta**on**

of the renormalized amplificati**on** factor η and the **on**

number m of newcomers. Evidently, the cooperative behaviors

gets promoted by the

m value. For small m (e.g., m = 2), cooperati**on** emerges

when η < 0.4. By further **on**facilitative

effect of m deteriorates. Nevertheless,

m has significant and n**on**-m**on**ot**on**ic effect **on** the result

cooperati**on** level when η = 1. Specifically, small m (e.g.,

m = 2) leads to about f c = 0.6,

m = 4) results f c > 0.8, whereas larger m (e.g., m = 8)

br**on**clud

observati**on** is that the **on** number m of each newcomer

generates a noticeable impact **on** the evoluti**on** of the

strategies

m warrant the best facilitati**on** of cooperative behaviors.

Moreover, it is worthwhile to explore the topological structures

of the result

of the populati**on** itself is not static, but a c**on**sequence of

coevoluti**on**ary dynamics. Symbols presented

η

1787

0.4

0.3

η=0.3

η=0.5

η=0.7

η=0.9

1

0.8

0.6

0.4

1

0.8

0.6

0.4

P(k)

0.2

0.2

0

0 0.5 1

η=0.3

0.2

0

0 0.5 1

η=0.5

0.1

0.2

0.2

0.15

0.15

0

0 5 10 15 20

k

0.1

0.05

0.1

0.05

Fig. 2. (Color **on**l**on**s for various values

of η. Here, all curves were obta

for the eye.

results for the degree distributi**on** of the result

Clearly, it can be observed that the

network with average degree k = 4 has been changed

coevoluti**on** process. In our model, the lowest c**on**tributor

each game will be removed from the system. Meanwhile, the

same number of new players will be added to the network

with preferential attach**on**s. Such a mechanism

tends to **on** number of players with

larger c**on**nectivity degrees, lead

degree heterogeneity of the network. Previous studies have

envir**on**ment for the emergence and promoti**on** of cooperati**on**.

Moreover, the substantial promoti**on** of cooperative

behaviors is often associated with str**on**gly heterogeneous

states, either

diversity [21], [41], [44]. Our coevoluti**on** rule has driven the

regular lattice to a slowly vary**on**-regular complex network.

Thus, the results summarized

result**on** emerg

the employed CPGG is crucial for the fortified facilitative

effect **on** cooperati**on** as outl**on**curs

nicely with the above c**on**clusi**on**.

Furthermore, to quantify the distributi**on** of

c**on**tributi**on** amounts **on** public good provisi**on** with

CPGG, we calculate the distributi**on** of **on**tributi**on**s

for different η values, and a fixed c**on**nectivity number

m = 4 of each newcomer. Results

that most **on**tributi**on**s are particularly low when

the cooperati**on** envir**on**ment is harsh (e.g., η ≤ 0.5). And

this is c**on**sistent with the results **on**diti**on**

Fig. 1. However, when the cooperati**on** envir**on**ment turns

mild from harsh (e.g., η ≥ 0.7), the c**on**tributi**on** amounts

of

0

0 0.5 1

η=0.7

0

0 0.5 1

η=0.9

Fig. 3. (Color **on**l**on** distributi**on** of **on**tributi**on**s

for different values of the multiplicati**on** factor η. The X axis

represents the value of **on**tributi**on** amount. The Y axis is the

proporti**on** of **on**tributi**on** to the whole populati**on**.

Here, all results were obta

diversity. Moreover, the differences between the majority of

the **on**tributi**on**s and the average c**on**tributi**on**s

are not significant. We c**on**jecture that the observed results

rely **on** the proposed coevoluti**on** rule which implements

punishment **on** the lowest c**on**tributor

group. In our assumpti**on**, each

**on** about the c**on**tributi**on** amounts of its partners.

Thus, **on**tributor will

punishment of other members, by be

current group. This can suggest a c**on**v**on** for

the phenomen**on** shown

Variance of cooperati**on** degree

0.1

0.08

0.06

0.04

0.02

0

m=2

m=4

m=6

m=8

0.2 0.4 0.6 0.8 1

Fig. 4. (Color **on**l**on** results of variance of

c**on**tributi**on**s as a functi**on** of η with different m values. The network size

is 2500. Each value corresp**on**ds to an average over 100 runs. L

just guides for the eye.

η

To further support the above results, the variances of

1788

**on**tributi**on**s are presented

variances of **on**tributi**on**s take low values when

the system evolves to a steady state, thus clearly verify

results shown

majority of the **on**tributi**on**s and the average

c**on**tributi**on**s are little. It is evidenced from Fig. 3 that the

variances of **on**tributi**on**s equal 0 for lower values

of η, and then

adopted **on** number m. Moreover, further

η would result **on**tributi**on**s,

but shows n**on**-m**on**ot**on**ous trend

η. Small and positive variances of **on**tributi**on**s

**on**tributi**on**s fluctuate around

the average c**on**tributi**on** of the whole populati**on**, as could

be **on**jecture

c**on**cern**on**ally

strengthened by results

of maximiz

tolerate exploitati**on** by others, and the lowest c**on**tributor

will get punishment undoubtedly. Therefore, the differences

between the majority of the **on**tributi**on**s and

the average c**on**tributi**on**s take small values.

IV. CONCLUSIONS

In summary, we **on** mechanism

am**on**g members of a group affects the c**on**tributi**on** amount

by **on**t**on**. In

particular, we have c**on**structed and studied an evoluti**on**ary

c**on**t**on** **on**s,

**on**tributi**on**

(not just all or n**on**e) to the comm**on** pools. In our model,

the lowest c**on**tributor

by be

members will stay and participate

round. Meanwhile, new nodes will be added to ma

fixed populati**on** size. For the newcomers, the establishment

of their **on**s are based **on** the preferential attachment.

Simulati**on** results show that there exist optimal **on**

numbers of each newcomer for which cooperators thrive best.

Our results c**on**firm that the presence of evolv

heterogeneous network is advantageous for cooperati**on**. We

have also studied the distributi**on** and the variance of the

**on**tributi**on**s respectively and found that most

**on**tributi**on**s fluctuate around the average c**on**tributi**on**

of the whole populati**on**. We hope the study is

of a better understand**on**

envir**on**ments.

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