Cooperation in Evolutionary Games on Complex Networks

Cooperation in Evolutionary Games on Complex Networks

49th IEEE Conference on Decision and Control

December 15-17, 2010

Hilton Atlanta Hotel, Atlanta, GA, USA

ong>Cooperationong> ong>inong> ong>Evolutionaryong> ong>Gamesong> on Complex Networks

Jianlei Zhang, Chunyan Zhang, Tianguang Chu and Zhifu Chen

Abstract— We consider a population engaged ong>inong> contong>inong>uous

public goods games. In our study, the lowest contributor ong>inong>

each game will be removed from the group, meanwhile new

players will be added to the network to maong>inong>taong>inong> the constant

population size. Here, each new node would establish some

connections with the existong>inong>g ones ong>inong> the network. We conclude

that appropriate ong>inong>teraction numbers of each newcomer

support the emergence and maong>inong>tenance of cooperation for

the contong>inong>uous public goods game. Furthermore, the ong>inong>itial

regular network has been driven to a slowly varyong>inong>g nonregular

heterogeneous network which facilitates the evolution

of cooperation. By ong>inong>vestigatong>inong>g the resultong>inong>g distribution of

ong>inong>dividual contributions, we show that provision of common

goods can be fostered by permittong>inong>g players target punishment

on the lowest contributor ong>inong> each contong>inong>uous public goods

game. We hope that our proposed model presents a feasible

mechanism for promotong>inong>g the emergence and evolution of

cooperative behaviors.


Cooperative phenomena are common between unrelated

and selfish ong>inong>dividuals ong>inong> natural, social, economical and

other systems. For example, replicatong>inong>g molecules had to

cooperate to form the first cells. Song>inong>gle cells had to cooperate

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

the body cooperate and help the cells of the germ long>inong>e

to reproduce. Animals cooperate to form social structures,

groups and societies. More examples about altruistic behavior

ong>inong> natural and social systems would not be listed

here. The evidence ong>inong> biology and sociology demonstrates

that cooperation and coordong>inong>ation among selfish ong>inong>dividuals

may arise and be sustaong>inong>able. However, natural selection

ong>inong> Darwong>inong>ian theory favors egoists who pursuit maximizong>inong>g

their ong>inong>dividual benefits even at the expense of the public

ong>inong>terest. Thus social dilemmas occur and constitute situations

where the collective payoff is at odds with ong>inong>dividual success.

Durong>inong>g the last few decades much work has been carried out

ong>inong> order to get an idea of what the drivong>inong>g forces behong>inong>d

cooperative behaviors of selfish ong>inong>dividuals are. Resolvong>inong>g

the social dilemma ong>inong> which a conflict arises between the

ong>inong>dividual benefits and public ong>inong>terest is a challengong>inong>g topic

attractong>inong>g much attention from various scientific fields.

ong>Evolutionaryong> game theory has presented a competent

framework for ong>inong>vestigatong>inong>g the emergence and evolution of

cooperation among selfish agents [1]–[9]. A widely studied

model, which illustrates the conflict between cooperation

This work is supported by National Nature Science Foundation of Chong>inong>a

(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 Engong>inong>eerong>inong>g, Pekong>inong>g University, Beijong>inong>g 100871,

Chong>inong>a. (J. Zhang), (C. Zhang), (T. Chu), (Z. Chen).

and defection, is the prisoner’s dilemma game (PDG) [10]–

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

has to simultaneously and ong>inong>dependently make its decision

concernong>inong>g about the strategy adoption. In the origong>inong>al PDG,

each of two players has two optional choices, cooperation

(C) or defection (D). They will both collect the payoff R and

P for mutual cooperation and mutual defection, respectively.

The mixed choice gives the cooperator the sucker’s payoff

S and the defector the temptation T respectively. Moreover,

the four elements satisfy the order rankong>inong>g T > R > P > S

and usually the additional constraong>inong>t 2R > T +S ong>inong> repeated

ong>inong>teraction. So mutual cooperation leads to the highest gaong>inong>

for both of ong>inong>dividuals and defection is the optimal decision

regardless of the choice of the competitor. Thus a socalled

social dilemma occurs, where mutual cooperation is

beneficial ong>inong> a long perspective, but defection will brong>inong>g large

shot-term profits.

However, many social situations are associated with collective

action based on joong>inong>t decisions made by a group often

ong>inong>volvong>inong>g more than two ong>inong>dividuals simultaneously and

ong>inong>dependently. So while the PDG is unrivaled ong>inong> popularity

when it comes to studyong>inong>g the evolution of cooperation

through pairwise ong>inong>teractions, the related public goods game

(PGG) accounts for group ong>inong>teractions as well [13]–[21].

In the PGG, beong>inong>g thought as a natural extension of the

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

ong>inong>teract withong>inong> their designated group and perform a given

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

requires the cooperation of several ong>inong>dividuals of the same

group who share the workload required to perform the

task. Players make their choices between cooperation and

defection ong>inong>dependently and simultaneously. Any benefit is

received by all participants, while any cost is undertaken

by the cooperators only. Therefore it is well known that

contributors participatong>inong>g a public good are subject to a

free-rider problem ong>inong> which players can always exploit the

benefits of livong>inong>g ong>inong> a group without contributong>inong>g to the task.

Song>inong>ce widespread cooperation is crucial for the prosperity

of society and also frequently encountered ong>inong> real-life

situations, various mechanisms aimed at fong>inong>dong>inong>g under what

conditions the cooperation emerges ong>inong> evolutionary games.

Promong>inong>ent examples ong>inong>clude repeated ong>inong>teractions [22], [23],

direct reciprocity [24], ong>inong>direct reciprocity or image score

[25]–[30], spatially structured population [31]–[41], punishment

[19], [42], social diversity and the associated contribution

diversity [21], and voluntary participation ong>inong> social

ong>inong>teractions [15], [17].

In the simplest form of PGG, cooperation or defection is

the only strategy that an ong>inong>dividual may adopt. ong>Cooperationong>

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

means makong>inong>g an ong>inong>vestment which ong>inong>curs a cost to the donor

and accrues a benefit to both the donor and the recipient.

However, ong>inong>dividuals often have a variety of possible choices

ong>inong> real social situations, ong>inong>stead of contributong>inong>g with all their

wealth or none. Inspired by this often encountered situation,

our study endows players with diverse choices ong>inong>stead of

bong>inong>ary option. In other words, we consider a population

whose members take the contong>inong>uously varyong>inong>g contributions,

aimong>inong>g to describe various behavioral patterns ong>inong> real life.

In this variation of the traditional PGG, each ong>inong>dividual can

choose not only full cooperation or no cooperation, but also

ong>inong>termediate cooperation levels ong>inong> the form of contributong>inong>g

arbitrary fund to a collectively advantageous group project.

This adopted model for ong>inong>vestigatong>inong>g this social problem is

referred to as contong>inong>uous public goods game (CPGG) ong>inong> our

study. We assume that the CPGG is played ong>inong> ong>inong>teraction

groups of size N, ong>inong> which each player contributes to the

public good at a cost to itself. Let c i be the contribution

of any given ong>inong>dividual i ong>inong> the population, which can

vary between 0 and 1 (as an additional simplification but

without loss of generality, hereong>inong> we set the maximum

contribution value to be 1). Each contribution ong>inong>creases the

common resource by rc, where r determong>inong>es the efficiency of

ong>inong>vestments and the attractiveness of the public goods. Then

each member ong>inong> the group receives an equally distributed

payoff of r ∑ N

i=1 c i/N from their contributions to the public

goods, hence the fong>inong>al actual payoff of player i is

P i = r


c i /N − c i


with N beong>inong>g the number of players ong>inong> the given ong>inong>teraction

group. For the Public Goods game one always assumes

r > 1. If r is smaller than N, to contribute is always

disadvantageous agaong>inong>st any combong>inong>ation of actions by other

group members. The game directly leads to the tragic outcome

ong>inong> which everybody ends up contributong>inong>g nothong>inong>g,

hence foregoong>inong>g the public goods.

After extendong>inong>g the scope of two-player two-strategy

cooperative games to contong>inong>uous and multi-person games,

we would like to draw attention to the ong>inong>triguong>inong>g possibilities

of allowong>inong>g network evolution ong>inong> cooperative games.

We ong>inong>troduce a feedback between network structure and

ong>inong>dividual performance. We model a group that provides

its members with public goods through a CPGG and the

group has the right to expel the worst member, convertong>inong>g

it ong>inong>to a non-member excluded from sharong>inong>g the common

goods. Notably, examples ong>inong> real situations ong>inong>clude teams,

organizations, and comities from which members can be

expelled for non-fulfillong>inong>g their obligations. More precisely,

we focus on permittong>inong>g players to target punishment on the

lowest contributor who provides the least contribution to the

common pool ong>inong> the game, and the rest ong>inong>dividuals whose

contributions are higher than the lowest subject will stay

and participate ong>inong> the next game round. Still, after expulsion

has been implemented ong>inong> the ong>inong>vestigated system, the same

number of newcomers will joong>inong> ong>inong> the group to maong>inong>taong>inong> the

constant amount of the whole population size.

The remaong>inong>der of this paper proceeds as follows. In the

next section we describe the model ong>inong> ample detail. Section

III features the results, whereas ong>inong> the last section we

summarize our fong>inong>dong>inong>gs and provide concludong>inong>g comments.


To vividly imitate the contacts among players ong>inong> real

world, spatial evolutionary games on complex network which

ong>inong>teracts the participatong>inong>g partners are meanong>inong>gful. The possible

mechanism accountong>inong>g for the establishment of cooperation

discards the well-mixed assumption for the population,

where complex network provides a natural and convenient

framework to describe the population structure on which the

evolution of cooperation is studied. Generally, a typical setup

is the followong>inong>g: agents are assigned to the nodes of a network,

which can be a regular lattice or have a more complex

structure. Then, agents are constraong>inong>ed to play games with

their immediate neighbors along the edges of the underlyong>inong>g

network. The system under our ong>inong>vestigation is the spatial

CPGG ong>inong>dulged by N autonomous players. Initially, we

employ a two-dimensional regular network as the population

structure ong>inong>corporated ong>inong>to the CPGG. Duplicate long>inong>ks and

self-ong>inong>teractions are omitted. Here, the population structure

is not static, but varyong>inong>g ong>inong> the coevolution process. In our

study, players on the nodes of the underlyong>inong>g network can

choose from a large set of strategies.

Song>inong>ce the studied type of neighborhood is a regular

network that is constructed from the aforementioned N

ong>inong>dividuals, each player i ong>inong>itially occupies a node on the

two dimensional network with degree k i = 4, where k i

is its connectivity degree. Thus, all ong>inong>dividuals have the

same k nearest neighbors ong>inong>itially due to the underlyong>inong>g

lattice network. Followong>inong>g the significant work [21], here

each player i acts as an organizer of the common pool i with

size k i + 1, where there occurs the CPGG ong>inong>volvong>inong>g i itself

and its neighborong>inong>g partners. Besides the CPGG organized

by itself, player i also engages ong>inong> other k i CPGGs organized

by its nearest neighbors.

Initially each player i is designated a random contribution

amount c i (0 ≤ c i ≤ 1). Then player i pays its contribution

c i to all k i + 1 common pools it participates ong>inong>. In a

given group with k +1 ong>inong>dividuals ong>inong>cludong>inong>g i, the collected

contribution 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 contributions. In this

case each player i receives the freely-shared public ong>inong>terest

r ∑ k+1

i=1 c i/(k + 1), thus its fong>inong>al payoff ong>inong> this game round

is r ∑ k+1

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

the temptation of beong>inong>g free-riders by contributong>inong>g nothong>inong>g

(c i = 0). Thus the gaong>inong> P ij of agent i associated with the

neighborhood centered at ong>inong>dividual j can be expressed as


∑ j+1

P ij = η c r − c i (1)



where c r is the ong>inong>dividual contribution amount of a member

r ong>inong> j’s k j + 1 nearest neighbors, and η = r/(k j + 1) is a

renormalized enhancement factor on the public goods. Then,

the total payoff P i of player i is the sum of gaong>inong>s from all

ong>inong>teractions ong>inong> which it participates.

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 contributor will be expelled from the population after

each game round, meanwhile, the same number of new

players will be added after removong>inong>g the lowest contributors.

Thus the structure of the population itself is not static, but a

consequence of coevolutionary dynamics. Hereong>inong> we assume

that each newcomer preferentially establish its m ong>inong>teractions

with the old nodes accordong>inong>g to their degrees. Here, m

is positive parameter and we can study the effects of its

value on the cooperation level. Accordong>inong>g to this preferential

attachment rule, those existong>inong>g players who own more

ong>inong>teraction neighbors show more attractiveness to newcomers

than others. Thus ong>inong> our study, an existong>inong>g node i is chosen

by a new player with the followong>inong>g probability

q i =

k i

∑ M

j=1 k j

where M (0 < M < N ) is the existong>inong>g population

size ong>inong> the network before newcomers joong>inong>, and k j is the

connectivity number of ong>inong>dividual j. Notably, to be fair,

the ong>inong>itial contribution amount of the newcomer equals the

average contribution of its m ong>inong>teractong>inong>g neighbors.

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

random strategy (namely the ong>inong>dividual contribution ong>inong> our

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

ong>inong> accordance with the Monte Carlo simulation procedure.

Existong>inong>g players on the network update their strategies (the

ong>inong>dividual contribution amount) ong>inong> accordance with followong>inong>g

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

by playong>inong>g games with its nearest neighbors. Next, its payoff

is compared to a randomly selected neighbor j with payoff

p j . Last, player i will adopt the strategy of ong>inong>dividual j with

a probability W(s i → s j ) proportional to the difference ong>inong>

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 + exp[(P i − P j )/T]

where T characterizes the magnitude of noise ong>inong>volvong>inong>g

many different effects (fluctuations ong>inong> payoffs, errors ong>inong>

decision, ong>inong>dividual trials, etc.) [43]. T = 0 and T → ∞

correspond to the completely determong>inong>istic and completely

random selection of the j’s strategy s y , respectively. Namely,

T = 0 denotes that the agent always adopts the best

strategy determong>inong>ately, and T → ∞ implies that the agent

is absolutely irrational ong>inong> makong>inong>g decisions. For any fong>inong>ite

positive values, T ong>inong>troduces some uncertaong>inong>ties or irrational



factor ong>inong> the strategy adoption, such as the case of small

possibility to select the worse one. Here, we only consider

the simple situation for ong>inong>dividuals’ selection probability and

simply set T = 50.


In the followong>inong>g, we show the simulation results carried

out for a population of N = 2500 players occupyong>inong>g the

vertices of the lattice network ong>inong>itially. The ong>inong>itial strategies

(ong>inong>dividual contribution amounts ong>inong> our study) of the population

are randomly distributed ong>inong> [0,1]. The simulation results

are obtaong>inong>ed by averagong>inong>g over the last 10 4 generations of the

entire 10 5 generations. Moreover, each data poong>inong>t averages

over 100 realizations of both the networks and the ong>inong>itial


Average contribution











0.2 0.4 0.6 0.8 1

Fig. 1. (Color onlong>inong>e) The simulation results of average contribution f c

of the whole population as a function of η with different m values. The

network size is 2500. Each data poong>inong>t is obtaong>inong>ed by averagong>inong>g over the

last 10 4 generations of the entire 10 5 generations. Here, all curves were

obtaong>inong>ed by settong>inong>g m = 4. Each value corresponds to an average over 100

runs. Long>inong>es are just guides for the eye.

We shall start by revealong>inong>g the properties of the aboveong>inong>troduced

model by examong>inong>ong>inong>g the effect of m on the

stationary average contibution f c ong>inong> the proposed settong>inong>g of

CPGG. Fig. 1 shows results obtaong>inong>ed by a given combong>inong>ation

of the renormalized amplification factor η and the ong>inong>teraction

number m of newcomers. Evidently, the cooperative behaviors

gets promoted by the ong>inong>creasong>inong>g η, irrespective of the

m value. For small m (e.g., m = 2), cooperation emerges

when η < 0.4. By further ong>inong>creasong>inong>g m, the cooperationfacilitative

effect of m deteriorates. Nevertheless, ong>inong>creasong>inong>g

m has significant and non-monotonic effect on the resultong>inong>g

cooperation level when η = 1. Specifically, small m (e.g.,

m = 2) leads to about f c = 0.6, ong>inong>termediate m (e.g.,

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

brong>inong>gs about f c = 0.5 (see Fig. 1). Therefore, the concludong>inong>g

observation is that the ong>inong>teraction number m of each newcomer

generates a noticeable impact on the evolution of the

strategies ong>inong> CPGG, and the appropriately adjusted values of

m warrant the best facilitation of cooperative behaviors.

Moreover, it is worthwhile to explore the topological structures

of the resultong>inong>g partner network because the structure

of the population itself is not static, but a consequence of

coevolutionary dynamics. Symbols presented ong>inong> Fig. 2 show





















0 0.5 1




0 0.5 1








0 5 10 15 20






Fig. 2. (Color onlong>inong>e) The resultong>inong>g degree distributions for various values

of η. Here, all curves were obtaong>inong>ed by settong>inong>g m = 4. Long>inong>es are just guides

for the eye.

results for the degree distribution of the resultong>inong>g network.

Clearly, it can be observed that the ong>inong>itial underlyong>inong>g regular

network with average degree k = 4 has been changed ong>inong> the

coevolution process. In our model, the lowest contributor ong>inong>

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

same number of new players will be added to the network

with preferential attachong>inong>g ong>inong>teractions. Such a mechanism

tends to ong>inong>crease the ong>inong>teraction number of players with

larger connectivity degrees, leadong>inong>g to a certaong>inong> amount of

degree heterogeneity of the network. Previous studies have

ong>inong>vestigated that network structure results ong>inong> an advantageous

environment for the emergence and promotion of cooperation.

Moreover, the substantial promotion of cooperative

behaviors is often associated with strongly heterogeneous

states, either ong>inong> the form of the host network or social

diversity [21], [41], [44]. Our coevolution rule has driven the

regular lattice to a slowly varyong>inong>g non-regular complex network.

Thus, the results summarized ong>inong> Fig. 2 ong>inong>dicate that the

resultong>inong>g heterogeneous degree distribution emergong>inong>g withong>inong>

the employed CPGG is crucial for the fortified facilitative

effect on cooperation as outlong>inong>ed ong>inong> Fig. 1. This concurs

nicely with the above conclusion.

Furthermore, to quantify the distribution of ong>inong>dividual

contribution amounts on public good provision withong>inong> the

CPGG, we calculate the distribution of ong>inong>dividual contributions

for different η values, and a fixed connectivity number

m = 4 of each newcomer. Results ong>inong> Fig. 3 visually show

that most ong>inong>dividual contributions are particularly low when

the cooperation environment is harsh (e.g., η ≤ 0.5). And

this is consistent with the results ong>inong> the same condition ong>inong>

Fig. 1. However, when the cooperation environment turns

mild from harsh (e.g., η ≥ 0.7), the contribution amounts

of ong>inong>dividuals ong>inong>dulgong>inong>g ong>inong> the presently applied game show


0 0.5 1



0 0.5 1


Fig. 3. (Color onlong>inong>e) The proportion distribution of ong>inong>dividual contributions

for different values of the multiplication factor η. The X axis

represents the value of ong>inong>dividual contribution amount. The Y axis is the

proportion of ong>inong>dividuals with such contribution to the whole population.

Here, all results were obtaong>inong>ed by settong>inong>g m = 4.

diversity. Moreover, the differences between the majority of

the ong>inong>dividual’s contributions and the average contributions

are not significant. We conjecture that the observed results

rely on the proposed coevolution rule which implements

punishment on the lowest contributor ong>inong> the participatong>inong>g

group. In our assumption, each ong>inong>dividual fully aware of the

ong>inong>formation about the contribution amounts of its partners.

Thus, ong>inong> a given group, the lowest contributor will ong>inong>cur the

punishment of other members, by beong>inong>g expelled from the

current group. This can suggest a convong>inong>cong>inong>g explanation for

the phenomenon shown ong>inong> Fig. 3.

Variance of cooperation degree











0.2 0.4 0.6 0.8 1

Fig. 4. (Color onlong>inong>e) The simulation results of variance of ong>inong>dividual

contributions as a function of η with different m values. The network size

is 2500. Each value corresponds to an average over 100 runs. Long>inong>es are

just guides for the eye.


To further support the above results, the variances of


ong>inong>dividual contributions are presented ong>inong> Fig. 4. Note that the

variances of ong>inong>dividual contributions take low values when

the system evolves to a steady state, thus clearly verifyong>inong>g

results shown ong>inong> Fig. 3 that the differences between the

majority of the ong>inong>dividual’s contributions and the average

contributions are little. It is evidenced from Fig. 3 that the

variances of ong>inong>dividual contributions equal 0 for lower values

of η, and then ong>inong>crease with the risong>inong>g of η, irrespective of the

adopted ong>inong>teraction number m. Moreover, further ong>inong>creasong>inong>g

η would result ong>inong> positive variances of ong>inong>dividual contributions,

but shows non-monotonous trend ong>inong> the whole span of

η. Small and positive variances of ong>inong>dividual contributions

ong>inong>dicate that most ong>inong>dividual contributions fluctuate around

the average contribution of the whole population, as could

be ong>inong>ferred from the results ong>inong> Fig. 3. Thus the conjecture

concernong>inong>g about results ong>inong> Fig. 3 can be additionally

strengthened by results ong>inong> Fig. 4. Motivated by the pursuit

of maximizong>inong>g their ong>inong>dividual ong>inong>terests, ong>inong>dividuals can not

tolerate exploitation by others, and the lowest contributor

will get punishment undoubtedly. Therefore, the differences

between the majority of the ong>inong>dividual’s contributions and

the average contributions take small values.


In summary, we ong>inong>vestigate how the expulsion mechanism

among members of a group affects the contribution amount

by ong>inong>dividuals ong>inong> contong>inong>uous public goods game situation. In

particular, we have constructed and studied an evolutionary

contong>inong>uous public goods game with multi-person ong>inong>teractions,

ong>inong> which participators can make arbitrary contribution

(not just all or none) to the common pools. In our model,

the lowest contributor ong>inong> each game would get punishment

by beong>inong>g expelled from the participatong>inong>g group, while other

members will stay and participate ong>inong> games ong>inong> next game

round. Meanwhile, new nodes will be added to maong>inong>taong>inong> the

fixed population size. For the newcomers, the establishment

of their ong>inong>teractions are based on the preferential attachment.

Simulation results show that there exist optimal ong>inong>teraction

numbers of each newcomer for which cooperators thrive best.

Our results confirm that the presence of evolvong>inong>g underlyong>inong>g

heterogeneous network is advantageous for cooperation. We

have also studied the distribution and the variance of the

ong>inong>dividual contributions respectively and found that most

ong>inong>dividual contributions fluctuate around the average contribution

of the whole population. We hope the study is ong>inong> help

of a better understandong>inong>g of behaviors ong>inong> collective action



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