Evolution of Cooperation and Coordination in a Dyn

Networked Society

E.Pestelacci ∗M.Tomassini ∗

L.Luthi ∗

Abstract

Situations of conflict giving rise to social dilemmas are widespread in society and game theory is one major way in which they can be investigated.Starting from the observation that individuals in society interact through networks of acquaintances,we model the co-evolution of the agents’strategies and of the social network itself using two prototypical games,the Prisoner’s Dilemma and the Stag Hunt.Allowing agents to dismiss ties and establish new ones,we find that cooperation and coordination can be achieved through the self-organization of the social network,a result that is non-trivial,especially in the Prisoner’s Dilemma case.The evolution and stability of cooperation implies the condensation of agents exploiting particular game strategies into strong and stable clusters which are more densely connected,even in the more difficult case of the Prisoner’s Dilemma.1Introduction In this paper we study the behavior of a population of agents playing some simple two-person,one-shot non-cooperative game.Game theory [13]deals with social interactions where two or more individuals take decisions that will mutually influence each other.It is thus a view of collective systems in which global social outcomes emerge as a result of the interaction of the individual decisions made by each agent.Some extremely simple games lead to puzzles and dilemmas that have a deep social meaning.The most widely known among these games is the Prisoner’s Dilemma (PD),a universal metaphor for the tension that exists between social welfare and individual selfishness.It stipulates that,in situations where individuals may either cooperate or defect,they will rationally choose the latter.However,cooperation would be the preferred outcome when global welfare is considered.Other simple games that give rise to social dilemmas are the Hawk-Dove and the Stag-Hunt (SH)games.In practice,however,cooperation and coordination on common objectives is often seen in human and animal societies [3,20].Coordinated behavior,such as having both players cooperating in the SH,is a bit less problematic as this outcome,being a Nash equilibrium,is not ruled out by theory.For the PD,in which cooperation is theoretically doomed between rational agents,several mechanisms have been invoked to explain the emergence of cooperative behavior.Among them,repeated interaction,reputation,and belonging to a recognizable group have often been mentioned [3].Yet,the work of Nowak and May [16]showed that the simple fact that players are arranged according to a spatial structure and only interact with neighbors is sufficient to sustain a certain amount of cooperation even when the game is played anonymously and without repetition.Nowak and May’s study and much of the following work were based on regular structures such as two-dimensional grids (see [17]for a recent review).

Nevertheless,many actual social networks usually have a topological structure that is neither regular nor random but rather of the small-world type.Roughly speaking,small-world networks are graphs in which any node is relatively close to any other node.In this sense,they are similar to random graphs but unlike regular lattices.However,in contrast with random graphs,they also have a certain amount of local structure,as measured,for instance,by a quantity called the clustering coefficient which essentially represents the probability that two neighbors of a given node are themselves connected (an excellent review of the subject appears in [15]).Some work has been done in recent years in the direction of using ∗Information Systems Department,University of Lausanne,Switzerland

a r X i v :0805.0481v 1 [p h y s i c s .s o c -p h ] 5 M a y 2008

Most of the above mentioned studies have assumed afixed population size and structure,which amounts to dealing with a closed system and ignoring anyfluctuations in the system’s size and internal interactions.However,real social networks,such as friendship or collaboration networks,are not in an equilibrium state,but are open systems that continually evolve with new agents joining or leaving the network,and relationships(i.e.links in network terms)being made or dismissed by agents already in the network[4,10,23].Thus,the motivation of the present work is to re-introduce these coupled dynamics into our model and to investigate under which conditions,if any,cooperative and coordinated behavior may emerge and be stable.In this paper,we shall deal with networked populations in which the number of players remains constant but the interaction structure,i.e.who interacts with whom,does not stay fixed;on the contrary,it changes in time and its variation is dictated by the very games that are being played by the agents.A related goal of the present work is to study the topological structures of the emergent networks and their relationships with the strategic choices of the agents.

Some previous work has been done on evolutionary games on dynamic networks[11,19,21,28]. Skyrms and Pemantle[21]was recently brought to our attention by a reviewer.It is one of thefirst important attempts to study the kind of networks that form under a given game and,as such,is closely related to the work we describe here.The main ideas are similar to ours:agents start interacting at ran-dom according to some game’s payoff matrix and,as they evolve their game strategy according to their observed payoffs,they also have a chance of breaking ties and forming new ones,thus giving rise to a social network.The main differences with the present work is that the number of agents used is low,of the order of10instead of the103used here.This allows us to study the topological and statistical nature of the evolving networks in a way that is not possible with a few agents,while Skyrms’and Pemantle’s work is more quantitative in the study of the effects of the stochastic dynamics on the strategy and net-work evolution process.The work of Zimmermann and Egu´ıluz[28]is based on similar considerations too.There is a rather large population which has initially a random structure.Agents in the population play the one-shot two-person Prisoner’s Dilemma game against each other and change their strategy by copying the strategy of the more successful agent in their neighborhood.They also have the possibility of dismissing interactions between defectors and of rewiring them randomly in the population.The main differences with the present work are the following.Instead of just considering symmetrical undirected links,we have a concept of two directed,weighted links between pairs of agents.In our model there is afinite probability of breaking any link,not only links between defectors,although defector-defector and cooperator-defector links are much more likely to be dismissed than cooperator-cooperator links. When a link is broken it is rewired randomly in[28]while we use a link redirection process which fa-vors neighbors with respect to more relationally distant agents.In[28]only the Prisoner’s Dilemma is studied and using a reduced parameter space.We study both the Prisoner’s Dilemma and the Stag Hunt games covering a much larger parameter space.Concerning timing of events,we use an asynchronous update policy for the agents’strategies,while update is synchronous in[28].Finally,instead of a best-takes-over discrete rule,we use a smoother strategy update rule which changes an agent’s strategy with a probability proportional to the payoffs difference.Santos et al.[19]is a more recent paper also dealing with similar issues.However,they use a different algorithm for severing an undirected link between two agents which,again,does not include the concept of a link weight.Furthermore,the Stag Hunt game is only mentioned in passing,and their strategy update rule is different.In particular,they do not analyze in detail the statistical structure of the emerging networks,as we do here.Other differences with the above mentioned related works will be described in the discussion and analysis of results.Finally,ourown previous work[11]also deals with the co-evolution of strategy and structure in an initially ran-dom network.However,it is very different from the one presented here since we used a semi-rational threshold decision rule for a family of games similar,but not identical to the Prisoner’s Dilemma in[11]. Furthermore,the idea of a bidirectional weighted link between agents was absent,and link rewiring was random.

This article is structured as follows.In sect.2,we give a brief description of the games used in our study.This part is intended to make the article self-contained.In sect.3,we present a detailed description of our model of co-evolving dynamical networks.In sect.4,we present and discuss the simulation results and their significance for the social networks.Finally,in sect.5,we give our conclusions and discuss possible extensions and future work.

2Social Dilemmas

The two representative games studied here are the Prisoner’s Dilemma(PD)and the Stag-Hunt(SH)of which we briefly summarize the significance and the main results.More detailed accounts can be found elsewhere,for instance in[3,20].In their simplest form,they are two-person,two-strategies,symmetric games with the following payoff bi-matrix:

C D

C(R,R)(S,T)

D(T,S)(P,P)

In this matrix,R stands for the reward the two players receive if they both cooperate(C),P is the punishment for bilateral defection(D),and T is the temptation,i.e.the payoff that a player receives if it defects,while the other cooperates.In this case,the cooperator gets the sucker’s payoff S.In both games, the condition2R>T+S is imposed so that mutual cooperation is preferred over an equal probability of unilateral cooperation and defection.For the PD,the payoff values are ordered numerically in the following way:T>R>P>S.Defection is always the best rational individual choice in the PD; (D,D)is the unique Nash equilibrium(NE)and also an evolutionarily stable strategy(ESS)[13,27]. Mutual cooperation would be preferable but it is a strongly dominated strategy.

In the SH,the ordering is R>T>P>S,which means that mutual cooperation(C,C)is the best outcome,Pareto-superior,and a Nash equilibrium.However,there is a second equilibrium in which both players defect(D,D)and which is somewhat“inferior”to the previous one,although perfectly equivalent from a NE point of view.The(D,D)equilibrium is less satisfactory yet“risk-dominant”since playing it“safe”by choosing strategy D guarantees at least a payoff of P,while playing C might expose a player to a D response by her opponent,with the ensuing minimum payoff S.Here the dilemma is represented by the fact that the socially preferable coordinated equilibrium(C,C)might be missed for “fear”that the other player will play D instead.There is a third mixed-strategy NE in the game,but it is commonly dismissed because of its inefficiency and also because it is not an ESS[27].Although the PD has received much more attention in the literature than the SH,the latter is also very useful,especially as a metaphor of coordinated social behavior for mutual benefit.These aspects are nicely explained in[20]. 3Model Description

Our model is strictly local as no player uses information other than the one concerning the player itself and the players it is directly connected to.In particular,each agent knows its own current strategy and payoff,and the current strategies and payoffs of its immediate neighbors.Moreover,as the model is an evolutionary one,no rationality,in the sense of game theory,is needed[27].Players just adapt their behavior such that they copy more successful strategies in their environment with higher probability,a process commonly called imitation in the literature[8].Furthermore,they are able to locally assess the

worth of an interaction and possibly dismiss a relationship that does not pay off enough.The model and its dynamics are described in detail in the following sections.

3.1Network and Interaction Structure

The network of agents will be represented as an undirected graph G (V,E ),where the set of vertices V represents the agents,while the set of edges (or links)E represents their symmetric interactions.The population size N is the cardinality of V .A neighbor of an agent i is any other agent j such that there is an edge {ij }∈E .The set of neighbors of i is called V i and its cardinality is the degree k i of vertex i ∈V .The average degree of the network will be called ¯k

.Although from the network structure point of view there is a single undirected link between a player i and another player j ∈

V i ,we shall maintain two links:one going from i to j and another one in the reverse direction (see fig.1).Each link has a weight or “force”f ij (respectively f ji ).This weight,say f ij ,represents in an indirect way an abstract quality that could be related to the “trust”player i attributes to player j ,it may take any value in [0,1]and its variation is dictated by the payoff earned by i in each encounter with j ,as explained below.

Figure 1:Schematic representation of mutual trust between two agents through the strengths of their links.

We point out that we do not believe that this model could represent,however roughly,a situation of genetic relatedness in a human or animal society.In this case,at the very least,one should have at the outset that link strengths between close relatives should be higher than the average forces in the whole network and such groups should form cliques of completely connected agents.In contrast,we start our simulations from random relationships and a constant average link strength (see below).Thus,our simplified model is closer to one in which relationships between agents are only of socio-economic nature.

The idea behind the introduction of the forces f ij is loosely inspired by the potentiation/depotentiation of connections between neural networks,an effect known as the Hebb rule [7].In our context,it can be seen as a kind of “memory”of previous encounters.However,it must be distinguished from the memory used in iterated games,in which players “remember”a certain amount of previous moves and can thus conform their future strategy on the analysis of those past encounters [13].Our interactions are strictly one-shot,i.e.players “forget”the results of previous rounds and cannot recognize previous partners and their possible playing patterns.However,a certain amount of past history is implicitly contained in the numbers f ij and this information may be used by an agent when it will come to decide whether or not an interaction should be dismissed (see below)1.This bilateral view of a relationship is,to our knowledge,new in evolutionary game models on graphs.

We also define a quantity s i called satisfaction of an agent i which is the sum of all the weights of the links between i and its neighbors V i divided by the total number of links k i :

s i = j ∈V i f ij k i

.We clearly have 0≤s i ≤1.

1A further refinement of the concept could take obsolescence phenomena into account.For instance,in the same way that pheromone trails laid down by ants evaporate with time,we could introduce a progressive loss of strength of the links proportional to the time during which there is no interaction between the concerned agents.For the sake of simplicity,we prefer to stick with the basic model in this work

The constant size of the network during the simulations is N=1000.The initial graph is generated randomly with a mean degree comprised between¯k=5and¯k=20.These values of¯k are of the order of those actually found in many social networks(see,for instance,[4,10,14,25]).Players are distributed uniformly at random over the graph vertices with50%cooperators.Forces between any pair of neighboring players are initialized at0.5.With¯k>1a random graphfinds itself past the percolation phase transition[5]and thus it has a giant connected component of size O(N)while all the other components are of size O(log(N)).We do not assure that the whole graph is connected,as isolated nodes will draw a random link during the dynamics(see below).

Before starting the simulations,there is another parameter q that has to be set.This is akin to a “temperature”or noise level;q is a real number in[0,1]and it represents the frequency with which an agent wishes to dismiss a link with one of its neighbors.The higher q,the faster the link reorganization in the network.This parameter has a role analogous to the“plasticity”of[28]and it controls the speed at which topological changes occur in the network.As social networks may structurally evolve at widely different speeds,depending on the kind of interaction between agents,this factor might play a role in the model.For example,e-mail networks change their structure at a faster pace than,say, scientific collaboration networks[10,23].A similar coupling of time scales between strategy update and topological update also occurs in[21,19].

3.3Timing of Events

Usually,agents systems such as the present one,are updated synchronously,especially in evolutionary game theory simulations[12,16,18,28].However,there are doubts about the physical signification of simultaneous update[9].For one thing,it is strictly speaking physically unfeasible as it would require a global clock,while real extended systems in biology and society in general have to take into accountfinite signal propagation speed.Furthermore,simultaneity may cause some artificial effects in the dynamics which are not observed in real systems[9,11].Fully asynchronous update,i.e.updating a randomly chosen agent at a time with or without replacement also seems a rather arbitrary extreme case that is not likely to represent reality very accurately.In view of these considerations,we have chosen to update our population in a partially synchronous manner.In practice,we define a fraction f=n/N(with N=an,a∈N)and,at each simulated discrete time step,we update only n≤N agents randomly chosen with replacement.This is called a microstep.After N/n microsteps,called a macrostep,N agents will have been updated,i.e.the whole population will have been updated in the average.With n=N we recover the fully synchronous update,while n=1gives the extreme case of the fully asynchronous update.Varying f thus allows one to investigate the role of the update policy on the dynamics.We study several different values of f,but we mainly focus on f=0.01.

3.4Strategy and Link Dynamics

Here we describe in detail how individual strategies,links,and link weights are updated.Once a given node i is chosen to be activated,i.e.it belongs to the fraction f of nodes that are to be updated in a given microstep,i goes through the following steps:

•if the degree of agent i,k i=0then player i is an isolated node.In this case a link with strength

0.5is created from i to a player j chosen uniformly at random among the other N−1players in

the network.

•otherwise,

–either agent i updates its strategy according to a local replicator dynamics rule with prob-

ability1−q or,with probability q,agent i may delete a link with a given neighbor j and

creates a new0.5force link with another node k;–the forces between i and its neighbors V i are updated

Let us now describe each step in more detail.

Strategy Evolution.We use a local version of replicator dynamics(RD)as described in[6]and further modified in[12]to take into account the fact that the number of neighbors in a degree-inhomogeneous network can be different for different agents.The local dynamics of a player i only depends on its own strategy and on the strategies of the k i players in its neighborhood V i.Let us callπij the payoff player i receives when interacting with neighbor j.This payoff is defined as

πij=σi(t)MσT j(t),

where M is the payoff matrix of the game(see sect.2)andσi(t)andσj(t)are the strategies played by i and j at time t.The quantity

Πi(t)=

πij(t)

j∈V i

is the accumulated payoff collected by player i at time step t.The rule according to which agents update their strategies is the conventional RD in which strategies that do better than the average increase their share in the population,while those that fare worse than average decrease.To update the strategy of player i,another player j is drawn at random from the neighborhood V i.It is assumed that the probability of switching strategy is a functionφof the payoff difference,whereφis a monotonically increasing function[8].Strategyσi is replaced byσj with probability

p i=φ( Πj− Πi).

The major differences with standard RD is that two-person encounters between players are only possible among neighbors,instead of being drawn from the whole population,and the latter isfinite in our case. Other commonly used strategy update rules include imitating the best in the neighborhood[16,28], or replicating in proportion to the payoff[6,24].Although,these rules are acceptable alternatives, they do not lead to replicator dynamics and will not be dealt with here.We note also that the straight accumulated payoff Πi has a technical problem when used on degree-inhomogeneous systems such as those studied here,where agents(i.e.nodes)in the network may have different numbers of neighbors. In fact,in this case Πi does not induce invariance of the RD with respect to affine transformations of the game’s payoff matrix as it should[27],and makes the results depend on the particular payoff values. Thus,we shall use a modified accumulated payoffΠinstead as defined in[12].This payoff,which is the standard accumulated payoff corrected with a factor that takes into account the variable number of neighbors an agent may have,does not suffer from the standard accumulated payoff limitations.

Link Evolution.The active agent i,which has k i=0neighbors will,with probability q,attempt to dismiss an interaction with one of its neighbors.This is done in the following way.Player i will look at its satisfaction s i.The higher s i,the more satisfied the player,since a high satisfaction is a consequence of successful strategic interactions with the neighbors.Thus,there should be a natural tendency to try to dismiss a link when s i is low.This is simulated by drawing a uniform pseudo-random number r∈[0,1]and breaking a link when r≥s i.Assuming that the decision is taken to cut a link,which one,among the possible k i,should be chosen?Our solution again relies on the strength of the relevant links.First a neighbor j is chosen with probability proportional to1−f ij,i.e.the stronger the link, the less likely it will be chosen.This intuitively corresponds to i’s observation that it is preferable to dismiss an interaction with a neighbor j that has contributed little to i’s payoff over several rounds of play.However,in our system dismissing a link is not free:j may“object”to the decision.The intuitive idea is that,in real social situations,it is seldom possible to take unilateral decisions:often there is a cost associated,and we represent this hidden cost by a probability1−(f ij+f ji)/2with which j may refuse to be cut away.In other words,the link is less likely to be deleted if j appreciates i,i.e.when

f ji is high.A simpler solution would be to try to cut the weakest link,which is what happens most of the time anyway.However,with afinite probability of cuttin

g any link,our model introduces a small

amount of noise in the process which can be considered like“trembles”or errors in game theory

[13]

and which roughly reproduces decisions under uncertainty in the real world.

Assuming that the{ij}link isfinally cut,how is a new link to be formed?The solution adopted here is inspired by the observation that,in social networks,links are usually created more easily between people who have a mutual acquaintance than those who do not.First,a neighbor k is chosen in V i\\{j} with probability proportional to f ik,thus favoring neighbors i trusts.Next,k in turn chooses player l in his neighborhood V k using the same principle,i.e.with probability proportional to f kl.If i and l are not connected,a link{il}is created,otherwise the process is repeated in V l.Again,if the selected node, say m,is not connected to i,a new link{im}is established.If this also fails,a new link between i and a randomly chosen node is created.In all cases the new link is initialized with a strength of0.5in both directions.This rewiring process is schematically depicted infig.2for the case in which a link can be successfully established between players i and l thanks to their mutual acquaintance k.

Figure2:Illustration of the rewiring of link{ij}to{il}.Agent k is chosen to introduce player l to i (see text).

At this point,we would like to stress several important differences with previous work in which links can be dismissed in evolutionary games on networks[11,19,28].In[28],only links between defectors are allowed to be cut unilaterally and the study is restricted to the PD.Instead,in our case,any link has afinite probability to be abandoned,even a profitable link between cooperators if it is recent,although links that are more stable,i.e.have high strengths,are less likely to be rewired.This smoother situation is made possible thanks to our bilateral view of a link which is completely different from the undirected choice made in[28].

In[19],links can be cut by an unsatisfied player,where the concept of satisfaction is different from ours,and simply means that a cooperator or a defector will wish to break a link with a defector.The cut will be done with a certain probability that depends on the strategies of the two agents involved and their respective payoffs.Once a link between i and j is actually cut and,among the two players,i is the one selected to maintain the link,the link is rewired to a random neighbor of j.If both i and j wish to cease their interaction,the link is attributed to i or j probabilistically,as a function of the respective payoffs of i and j,and rewiring takes place from there.Thus,although both i’s and j’s payoffs are taken into consideration in the latter case,there is no analogous of our“negotiation”process as the concept of link strength is absent.In[11]links are cut according to a threshold decision rule and are rewired randomly anywhere in the network.

Afinal observation concerns the evolution of¯k in the network.While in[19,28]the initial mean degree is strictly maintained during network evolution through the rewiring process,here it may increase slightly owing to the existence of isolated agents which,when chosen to be updated,will create a new link with another random agent.While this effect is of minor importance and only causes smallfluctua-tions of¯k,we point out that in real evolving networks the mean connectivityfluctuates too[4,10,23].Updating the Link Strengths.Once the chosen agents have gone through their strategy or link update steps,the strengths of the links are updated accordingly in the following way:

f ij(t+1)=f ij(t)+

πij−¯πij

k i(πmax−πmin)

,

whereπij is the payoff of i when interacting with j,¯πij is the payoff earned by i playing with j,if j were to play his other strategy,andπmax(πmin)is the maximal(minimal)possible payoff obtainable in a single interaction.This update is performed in both directions,i.e.both f ij and f ji are updated ∀j∈V i because both i and j get a payoff out of their encounter.

4Simulation Results

4.1Simulation Parameters

We simulate on our networks the two games previously mentioned in sect.2.For each game,we can explore the entire game space by limiting our study to the variation of only two parameters per game. This is possible without loss of generality owing to the invariance of Nash equilibria and replicator dynamics under positive affine transformations of the payoff matrix using our payoff scheme[27].In the case of the PD,we set R=1and S=0,and vary1≤T≤2and0≤P≤1.For the SH,we decided tofix R=1and S=0and vary0≤T≤1and0≤P≤T.The reason we choose to set T and S in both the PD and the SH is to simply provide natural bounds on the values to explore of the remaining two parameters.In the PD case,P is limited between R=1and S=0in order to respect the ordering of the payoffs(T>R>P>S)and T’s upper bound is equal to2due to the2R>T+S constraint.Had wefixed R=1and P=0instead,T could be as big as desired,provided S≤0is small enough.In the SH,setting R=1and S=0determines the range of T and P(since this time R>T>P>S).Note however,that for this game the only valid value pairs of(T,P)are those that satisfy the T>P constraint.

As stated in sect.3.2,we used networks of size N=1000,randomly generated with an average degree¯k∈{5,10,20}and randomly initialized with50%cooperators and50%defectors.In all cases, the parameters are varied between their two bounds in steps of0.1.For each set of values,we carry out 50runs of at most20000macrosteps each,using a fresh graph realization in each run.A run is stopped when all agents are using the same strategy,in order to be able to measure statistics for the population and for the structural parameters of the graphs.The system is considered to have reached a pseudo-equilibrium strategy state when the strategy of the agents(C or D)does not change over150further macrosteps,which means15×104individual updates.We speak of pseudo-equilibria or steady states and not of true evolutionary equilibria because,as we shall see below,the system never quite reaches a totally stable state in the dynamical systems sense in our simulations but only transient states that persist for a long time.

4.2Cooperation and Stability

Cooperation results for the PD in contour plot form are shown infig.3.We remark that,as observed in other structured populations,cooperation may thrive in a small but non-negligible part of the parameter space.Thus,the added degree of freedom represented by the possibility of refusing a partner and choosing a new one does indeed help tofind player’s arrangements that help cooperation.Thisfinding is in line with the results of[19,28].Furthermore,the fact that our artificial society model differs from the latter two in several important ways also shows that the result is a rather robust one.When considering the dependence on thefluidity parameter q,one sees infig.3that the higher q,the higher the cooperation level.This was expected since being able to break ties more often clearly gives cooperators more possibilities forfinding and keeping fellow cooperators to interact with.This effect has been previously observed also in the works of[19,28]and,as such,seems to be a robustfinding,relatively

T

T

T

0.1

0.20.30.40.50.60.70.80.910

0.1

0.20.30.40.50.60.70.80.910

0.1

0.20.30.40.50.60.70.80.910.5

0.5

0.5

Figure 3:Cooperation level for the PD in the game’s configuration space.Darker gray means more defection.

independent of the other details of the models.The third parameter considered in fig.3is the mean

degree ¯k

.For a given value of q ,cooperation becomes weaker as ¯k increases.We believe that,as far as ¯k

is concerned,a realistic average characterization of actual social networks is represented by ¯k =10(middle row in fig.3)as seen,for instance,in [4,10,14,25].Higher average degrees do exist,but they are found either in web-based pseudo-social networks or in fairly special collaboration networks like the particle physics community,where it is customary to include as coauthors tens or even hundreds of authors [14].Clearly,there is a limit to the number of real acquaintances a given agent may manage with.

We have also performed many simulations starting from different proportions of randomly distributed cooperators and defectors to investigate the effect of this parameter on the evolution of cooperation.In Fig.4we show five different cases,the central image corresponding to the 50%situation.The images

correspond to the lower left quarter of the right image in the middle row of Fig.3with ¯k

=10,q =0.8,1Compared with the level of cooperation observed in simulations in static networks,we can say that

results are consistently better for co-evolving networks.For example,the typical cases with ¯k

=10and q =0.5,0.8show significantly more cooperation than what was found in model and real social

0.5

0.50

0.1

0.20.30.40.50.60.70.80.91Figure 4:Cooperation level for the PD starting with different fractions of cooperators increasing from

20%to 80%from left to right.Only the lower left quarter of the parameter space is shown.Results are the average of 50independent runs.

networks in previous work

[12].Even when there is a much lower rewiring frequency,i.e.with q =0.2,the cooperation levels are approximately as good as those observed in our previous study in which exactly the same replicator dynamics scheme was used to update the agents’strategies and the networks were of comparable size.The reason for this behavior is to be found in the added constraints imposed by the invariant network structure.The seemingly contradictory fact that an even higher cooperation level may be reached in static scale-free networks [18],is theoretically interesting but easily dismissed as those graphs are unlikely models for social networks,which often show fat-tailed degree distribution functions but not pure power-laws (see,for instance,[2,14]).As a further indication of the latter,we shall see in sect.4.3that,indeed,emerging networks do not have a power-law degree distribution.

From the point of view of the evolutionary dynamics,it is interesting to point out that any given simulation run either ends up in full cooperation or full defection.When the full cooperation state of the population is attained,there is no way to switch back to defection by the intrinsic agent dynamics.In fact,all players are satisfied and have strong links with their cooperating neighbors.Even though a small amount of noise may still be present when deciding whether or not to rewire a link,since there are only cooperators around to only very little link rewiring.

On the other hand,well before many defectors around,the system may experience some The converse may also happen,but when the full different.In this case

agents are unsatisfied,they all the other players around being also defectors,there will Thus the system will find itself in a fluctuating state,properties of the population

and of the network.To be some very long runs with all-defect end states.degree tends to increase slightly with time and the degree distribution function continues to evolve (see sect.4.3).

P 0.5

P P T

P T

00.1

0.20.30.40.50.60.70.80.91

Cooperation percentages as a function of the payoff matrix parameters for the SH game are shown

in fig.5for ¯k

=10and q =0.2,0.5,and 0.8.Note that in this case only the upper left triangle of the configuration space is meaningful (see sect.4.1).The SH is different from the PD since there are two evolutionarily stable strategies which are therefore also NEs:one population state in which everybody defects and the opposite one in which everybody cooperates (see sect.2).Therefore,it is expected,and absolutely normal,that some runs will end up with all defect,while others will witness the emergence of full cooperation.In contrast,in the PD the only theoretically stable state is all-defect and cooperating states may emerge and be stable only by exploiting the graph structure and creating more favorable neighborhoods by breaking and forming ties.The value of the SH is in making manifest the tension that exists between the socially desirable state of full cooperation and the socially inferior but less risky state of defection [20].The final outcome of a given simulation run depends on the size of the basin of attraction of either state,which is in turn a function of the relative values of the payoff matrix entries.To appreciate the usefulness of making and breaking ties in this game we can compare our results with what is prescribed by the standard RD solution.Referring to the payoff table of sect.2,let’s assume that the column player plays C with probability αand D with probability 1−α.In this case,the expected payoffs of the row player are:

E r [C ]=αR +(1−α)S and

E r [D ]=αT +(1−α)P

The row player is indifferent to the choice of αwhen E r [C ]=E r [D ].Solving for αgives:

α=

P −S

R −S −T +P

.

(1)

P

T

0.10.20.30.40.50.60.70.80.91Figure 6:Probabilities of cooperation for the mixed strategy NE as a function of the game’s parameters for the Stag Hunt.

Since the game is symmetric,the result for the column player is the same and (αC,(1−α)D )is a NE in mixed strategies.We have numerically solved the equation for all the sampled points in the game’s parameter space,which gives the results shown in fig.6.Let us now use the following payoff values in order to bring them within the explored game space (remember that NEs are invariant w.r.t.such a transformation [27]):

C D C (1,1)(0,2/3)D

(2/3,0)

(1/3,1/3)

Substituting in(1)gives α=1/2,i.e.the (unstable)polymorphic population should be composed by about half cooperators and half defectors.Now,if one looks at fig.5at the points where P =1/3and T =2/3,one can see that this is approximately the case for the first image,within the limits

of the approximations caused by the finite population size,the symmetry-breaking caused by the non-homogeneous graph structure,and the local nature of the RD.On the other hand,in the middle image and,to a greater extent,in the rightmost image,this point in the game space corresponds to pure coop-eration.In other words,the non-homogeneity of the network and an increased level of tie rewiring has allowed the cooperation basin to be enhanced with respect to the theoretical predictions of standard RD.Skyrms and Pemantle found the same qualitative result for very small populations of agents when both topology and strategy updates are allowed

[21].It is reassuring that coordination on the payoff-dominant equilibrium can still be achieved in large populations as seen here.

4.3Structure of the Emerging Networks

In this section we present a statistical analysis of the global and local properties of the networks that emerge when the pseudo-equilibrium states of the dynamics are attained.Let us start by considering the

evolution of the average degree ¯k

.Although there is nothing in our model to prevent a change in the initial mean degree,the steady-state average connectivity tends to increase only slightly.For example,in

the PD with q =0.8and ¯k

init =5and ¯k init =10,the average steady-state (ss)values are ¯k ss 7and ¯k

ss 10.5respectively.Thus we see that,without imposing a constant ¯k as in [19,28],¯k nonetheless tends to increase only slightly,which nicely agrees with observations of real social networks [4,10,23].There is a special case when the steady-state is all-defect and the simulation is allowed to run for a very long time (2×104macrosteps);in this case the link structure never really settles down,since players

are unsatisfied,and ¯k

may reach a value of about 12when starting with ¯k =10and q =0.8.0.050.10.150.20.25

0.3T

1.1

1.21.31.41.51.61.71.81.90.4

0.6Figure 7:Clustering coefficient level for the PD game.Lighter gray means more clustering.Another important global network statistics is the average clustering coefficient C .The clustering

coefficient C i of a node i is defined as C i =2E i /k i (k i −1),where E i is the number of edges in the neighborhood of i .Thus C i measures the amount of “cliquishness”of the neighborhood of node i and it characterizes the extent to which nodes adjacent to node i are connected to each other.The clustering

coefficient of the graph is simply the average over all nodes:C =1N N

i =1C i [15].Random graphs are locally homogeneous and for them C is simply equal to the probability of having an edge between any pair of nodes independently.In contrast,real networks have local structures and thus higher values of C .Fig.7gives the average clustering coefficient ¯C =150

50i =1C for each sampled point in the PD configuration space,where 50is the number of network realizations used for each simulation.It is apparent that the networks self-organize and acquire local structure in the interesting,cooperative parts of the parameter’s space,since the clustering coefficients there are higher than that of the random graph

with the same number of edges and nodes,which is ¯k/N =10/1000=0.01.Conversely,where

defection predominates C is smaller,witnessing of a lower amount of graph local restructuring.These impressions are confirmed by the study of the degree distribution functions (see below).The correlation between clustering and cooperation also holds through increasing values of q :C tends to increase from left to right in fig.7,a trend similar to that observed in the middle row of fig.3for cooperation.This

correlation is maintained also for ¯k

=5and ¯k =20(not shown).

0.050.10.150.20.250.3

T 0.10.20.30.40.50.60.70.80.9P 0.4Figure 8:Clustering coefficient level for the SH game.

As far as the clustering coefficient is concerned,the same qualitative phenomenon is observed for the SH namely,the graph develops local structures and the more so the higher the value of q for a given ¯k

(see fig.8).Thus,it seems that evolution towards cooperation and coordination passes through a rear-rangement of the neighborhood of the agents with respect to the homogeneous random initial situation,something that is made possible through the higher probability given to neighbors when rewiring a link,a stylized manifestation of the commonly occurring social choice of partners.

(a)(b)

Figure 9:Cumulative degree distributions.Average values over 50runs.(a):PD,(b):SH.q =0.8,¯k

=10.Linear-log scales.Figure 10:Cumulative degree distributions for the PD in case of defection before (dotted line)and after (thick line)reaching a steady-state.Linear-log scales.

The degree distribution function (DDF)p (k )of a of a graph represents the probability that a ran-domly chosen node has degree k [15].Random graphs are characterized by DDF of Poissonian form,while social and technological real networks often show long tails to the right,i.e.there are nodes that have an unusually large number of neighbors [15].In some extreme cases the DDF has a power-law

form p (k )∝k −γ;the tail is particularly extended and there is no characteristic degree.The cumulative degree distribution function (CDDF)is just the probability that the degree is greater than or equal to k and has the advantage of being less noisy for high degrees.

Fig.9(a)shows the CDDFs for the PD for three cases of which two are in the cooperative region and the third falls in the defecting region (see fig.3).The dotted curve refers to a region of the configuration space in which there is cooperation in the average but it is more difficult to reach,as the temptation parameter is high (T=1.8,P=0.1).The curve has a rather long tail and is thus broad-scale in the sense that there is no typical degree for the agents.Therefore,in the corresponding network there are cooperators that are linked to many other cooperators.On the other hand,if one considers the dotted-dashed curve,which corresponds to a defecting region (T=1.1,P=0.4),it is clear that the distribution is much closer to normal,with a well-defined typical value of the degree.Finally,the third thick curve,which corresponds to a region where cooperation is more easily attained (T=1.1,P=0.1),also shows a rather faster decay of the tail than the dotted line and a nar-rower scale for the degree.Nevertheless,it is right-skewed,indicating that the network is no longer a pure random graph.Since we use linear-log scales,the dotted curve has an approximately exponential or slower decay,given that a pure exponential would appear as a straight line in the plot.The tail of the thick curve decays faster than an exponential,while the dashed-dotted curve decays even faster.Almost the same observations also apply to the SH case,shown in fig.9(b).These are quite typical behaviors and we can conclude that,when cooperation is more difficult to reach,agents must better exploit the link-redirection degree of freedom in order for cooperators to stick together in sufficient quantities and protect themselves from exploiting defectors during the co-evolution.When the situation is either more favorable for cooperation,or defection easily prevails,network rearrangement is less radical.In the limit of long simulation times,the defection case leads to networks that have degree distribution close to Poissonian and are thus almost random.Fig.10shows such a case for the PD.The dashed curve is the CDDF at some intermediate time,when full defection has just been reached but the network is still strongly reorganizing itself.Clearly,the distribution has a long tail.However,if the simulation is continued until the topology is quite stable at the mesoscopic level,the distribution becomes close to normal (thick curve).

Figure 11:Cumulative degree distribution functions for three values of q ,for the same point in the PD configuration space in the cooperating region.

Finally,it is interesting to observe the influence of the q parameter on the shape of the degree distri-bution functions for cooperating networks.Fig.11reports average curves for three values of q .For high q ,the cooperating steady-state is reached faster,which gives the network less time to rearrange its links.For lower values of q the distributions become broader,despite the fact that rewiring occurs less often,because cooperation in this region is harder to attain and more simulation time is needed.

Influence of Timing.Fig.12depicts a particular cut in the configuration space as a function of the synchronicity parameter f .The main remark is that asynchronous updates give similar results,in spite of the difference in the number of agents that are activated in a single microstep.In contrast,fully synchronous update (f =1)appears to lead to a slightly less favorable situation for cooperation.Since fully synchronous update is physically unrealistic and can give spurious results due to symmetry,we

c o o p e r a t i o n T

f Figure 12:Cooperation levels in the PD for P =0.1and 1≤T ≤2as a function of the synchronicity parameter f .

suggest using fully or partially asynchronous update for agent’s simulation of artificial societies.

4.4Clusters

We have seen in the previous section that,when cooperation is attained in both games as a quasi-equilibrium state,the system remains stable through the formation of clusters of players using the same strategy.In

fig.13one such typical cluster corresponding to a situation in which global cooperation has been reached in the PD is shown.Although all links towards the “exterior”have been suppressed for clarity,one can clearly see that the central cooperator is a highly connected node and there are many links also between the other neighbors.Such a tightly packed structure has emerged to protect coop-erators from defectors that,at earlier times,were trying to link to cooperators to exploit them.These observations explain why the degree distributions are long-tailed (see previous section),and also the higher values of the clustering coefficient in this case (see sect.4.3).

Figure 13:Example of a tightly packed cluster of cooperators for PD networks.T =1.8,P =0.1and q =0.8.

When the history of the stochastic process is such that defection prevails in the end,the situation is totally different.Fig.14(a)and (b)show two typical examples of cluster structures found during a simulation.Fig.14(a)refers to a stage in which the society is composed solely by defectors.However,the forces of the links between them are low,and so many defectors try to dismiss some of their links.This situation lasts for a long simulated time (actually,the system is never at rest,as far as the links are concerned)but the dense clusters tend to dissolve,giving rise to structures such as the one shown in fig.14(b).If one looks at the degree distribution at this stage (fig.10)it is easy to see that the whole population graph tends to become random.

(a)(b)

Figure14:Example of defector clusters for PD networks,for T=1.8,P=0.3and q=0.8.Clusters like(a)exists only just after the all-defect state is reached.When a steady-state is reached only clusters like(b)are present in a network of defectors.

The SH case is very similar,which is a relatively surprising result.In fact,when cooperationfinally takes over in regions of the configuration space where defection would have been an almost equally, likelyfinal state,players are highly clustered and there are many highly connected individuals,while in less conflicting situations the clusters are less dense and the degree distribution shows a faster decay of the tail.On the other hand,when defection is thefinal quasi-stable state,the population graphs looses a large part of its structure.Thus,the same topological mechanisms seem to be responsible for the emergence of cooperation in the PD and in the SH.The only previous study that investigates the structure of the resulting networks in a dynamical setting is,to our knowledge,reference[28],where only the PD is studied.It is difficult to meaningfully compare our results with theirs as the model of Zimmermann et al.differs from ours in many ways.They use a deterministic hard-limit rule for strategy update which is less smooth than our stochastic local replicator dynamics.Moreover,they study the PD in a reduced configuration space,only links between defectors can be broken,and links are rewired at random.They concentrate on the study of the stability of the cooperating steady-states against perturbations,but do not describe the topological structures of the pseudo-equilibrium states in detail.Nevertheless,it is worthy of note that the degree distribution functions for cooperators and defectors follow qualitatively the same trend,i.e.cooperators networks have distributions with fatter tails to the right than defector networks. 5Conclusions and Future Work

Using two well known games that represent conflicting decision situations commonly found in animal and human societies,we have studied by computer simulation the role of the dynamically networked society’s structure in the establishment of global cooperative and coordinated behaviors,which are de-sirable outcomes for the society’s welfare.Starting from randomly connected players which only inter-act locally in a restricted neighborhood,and allowing agents to probabilistically and bilaterally dismiss unprofitable relations and create new ones,the stochastic dynamics lead to pseudo-equilibria of either cooperating or defecting agents.With respect to standard replicator dynamics results for mixing pop-ulations,wefind that there is a sizable configuration space region in which cooperation may emerge and be stable for the PD,whereas the classical result predicts total defection.For the SH,where both all-cooperate and all-defect steady-states are theoretically possible,we show that the basin of attrac-tion for cooperation is enhanced.Thus,the possibility of dismissing a relationship and creating a new one does indeed increase the potential for cooperation and coordination in our artificial society.The self-organizing mechanism consists in both games in forming dense clusters of cooperators which are more difficult to dissolve by exploiting defectors.While the beneficial effect of relational or geograph-ical static population structures on cooperation was already known from previous studies,here we haveshown that more realistic dynamic social networks may also allow cooperation to thrive.Future work will deal with the stability of the cooperating states against stronger perturbations than merely the im-plicit noise of the stochastic dynamics.We also intend to study more fully the structure of the emerging clusters and their relationships,and we plan to extend the model to other important paradigmatic games such as Hawks-Doves and coordination games.

Acknowledgments. E.Pestelacci and M.Tomassini are grateful to the Swiss National Science Foun-dation forfinancial support under contract number200021-111816/1.We thank the anonymous review-ers for useful remarks and suggestions.

References

[1]G.Abramson and M.Kuperman.Social games in a social network.Phys.Rev.E,63:030901,2001.

[2]L.A.N.Amaral,A.Scala,M.Barth´e lemy,and H.E.Stanley.Classes of small-world networks.

Proceedings of the National Academy of Sciences USA,97(21):11149–11152,2000.

[3]R.Axelrod.The Evolution of Cooperation.Basic Books,Inc.,New-York,1984.

[4]A.-L.Barab´a si,H.Jeong,Z.N´e da,E.Ravasz,A.Schubert,and T.Vicsek.Evolution of the social

network of scientific collaborations.Physica A,311:590–614,2002.

[5]B.Bollob´a s.Modern Graph Theory.Springer,Berlin,Heidelberg,New York,1998.

[6]C.Hauert and M.Doebeli.Spatial structure often inhibits the evolution of cooperation in the

snowdrift game.Nature,428:3–6,April2004.

[7]D.O.Hebb.The Organization of Behavior.Wiley,New York,1949.

[8]J.Hofbauer and K.Sigmund.Evolutionary Games and Population Dynamics.Cambridge Univer-

sity Press,Cambridge,UK,1998.

[9]B.A.Huberman and N.S.Glance.Evolutionary games and computer simulations.Proceedings of

the National Academy of Sciences USA,90:7716–7718,August1993.

[10]G.Kossinets and D.J.Watts.Empirical analysis of an evolving social network.Science,311:88–

90,2006.

[11]L.Luthi,M.Giacobini,and M.Tomassini.A minimal information prisoner’s dilemma on evolving

networks.In L.M.Rocha,editor,Artificial Life X,pages438–444,Cambridge,Massachusetts, 2006.The MIT Press.

[12]L.Luthi,E.Pestelacci,and M.Tomassini.Cooperation and community structure in social net-

works.Physica A,387:955–966,February2008.

[13]R.B.Myerson.Game Theory:Analysis of Conflict.Harvard University Press,Cambridge,MA,

1991.

[14]M.E.J.Newman.Scientific collaboration networks.I.network construction and fundamental

results.Phys.Rev E,:016131,2001.

[15]M.E.J.Newman.The structure and function of complex networks.SIAM Review,45:167–256,

2003.

[16]M.A.Nowak and R.M.May.Evolutionary games and spatial chaos.Nature,359:826–829,

October1992.[17]M.A.Nowak and K.Sigmund.Games on grids.In U.Dieckmann,R.Law,and J.A.J.Metz,

editors,The Geometry of Ecological Interactions:Simplifying Spatial Complexity,pages135–150.

Cambridge University Press,Cambridge,UK,2000.

[18]F.C.Santos and J.M.Pacheco.Scale-free networks provide a unifying framework for the emer-

gence of cooperation.Phys.Rev.Lett.,95:098104,2005.

[19]F.C.Santos,J.M.Pacheco,and T.Lenaerts.Cooperation prevails when individuals adjust their

social ties.PLoS Comp.Biol.,2:1284–1291,2006.

[20]B.Skyrms.The Stag Hunt and the Evolution of Social Structure.Cambridge University Press,

Cambridge,UK,2004.

[21]B.Skyrms and R.Pemantle.A dynamic model for social network formation.Proc.Natl.Acad.

Sci.USA,97:9340–9346,2000.

[22]G.Szab´o and G.F´a th.Evolutionary games on graphs.Physics Reports,446:97–216,2007.

[23]M.Tomassini and L.Luthi.Empirical analysis of the evolution of a scientific collaboration net-

work.Physica A,385:750–7,2007.

[24]M.Tomassini,L.Luthi,and M.Giacobini.Hawks and doves on small-world networks.Phys.Rev.

E,73:016132,2006.

[25]M.Tomassini,L Luthi,M.Giacobini,and W.B.Langdon.The structure of the genetic program-

ming collaboration network.Genetic Programming and Evolvable Machines,8(1):97–103,2007.

[26]S.Wang,M.S.Szalay,C.Zhang,and P.Csermely.Learning and innovative elements of strategy

adoption rules expand cooperative network topologies.PLoS ONE,3(4):e1917,2008.

[27]J.W.Weibull.Evolutionary Game Theory.MIT Press,Boston,MA,1995.

[28]M.G.Zimmermann and V.M.Egu´ıluz.Cooperation,social networks,and the emergence of

leadership in a prisoner’s dilemma with adaptive local interactions.Phys.Rev.E,72:056118,2005.