Cascading failures and the emergence of cooperation in evolutionary-game based models of socialand economical networks

Wen-Xu Wang,1 Ying-Cheng Lai,1, 2 and Dieter Armbruster3

1School of Electrical, Computer, and Energy Engineering,Arizona State University, Tempe, Arizona 85287, USA

2Department of Physics, Arizona State University, Tempe, Arizona 85287, USA3School of Mathematics and Statistics, Arizona State University, Tempe, Arizona 85287, USA

We study catastrophic behaviors in large networked systems in the paradigm of evolutionary games by incor-porating a realistic “death” or “bankruptcy” mechanism. We find that a cascading bankruptcy process can arisewhen defection strategies exist and individuals are vulnerable to deficit. This finding yields insights into sig-nificant social and economical phenomena such as large-scale failures of financial institutions and corporationsduring an economic recession. Strikingly, we observe that, after the catastrophic cascading process terminates,cooperators are the sole survivors, regardless of the game types and of the connection patterns among individ-uals as determined by the topology of the underlying network. It is necessary that individuals cooperate witheach other to survive the catastrophic failures. Cooperation thus becomes the optimal strategy and absolutelyoutperforms defection in the game evolution. Our results can be useful for devising strategies in policy makingto prevent large-scale crises in social and economical systems.

PACS numbers: 05.45.-a, 87.23.Ge, 87.23.Kg, 89.75.Fb

Evolutionary games have been a powerful paradigm tostudy a variety of self-organized behaviors in natural, so-cial, and economical systems. So far, the tolerance of indi-viduals to elimination or death in the paradigm has re-ceived relatively little attention. A relevant example isthe bankruptcy of agents in economical systems. For anyagent, a lowest amount of profit should be maintainedfor it to survive, which comes from the interactions withother agents in a certain time period for continuous in-vestment into the future. Another example is ecosystems,where individuals compete and cooperate for essential life-sustaining resources. If some minimal requirement for re-sources cannot be satisfied, individuals will die. In thispaper, we incorporate an elimination mechanism into thegaming rules to better mimic the evolution of cooperativebehavior in realistic systems. In particular, we assign atolerance parameter to every individual in the network,which is the lowest payoff needed for an individual to sur-vive. Taking into account the diversity in social and bi-ological systems, each individual can have its own toler-ance. For example, the number of interactions of an indi-vidual is a characteristic to distinguish it from others, soit can be used to define the tolerance. In a network, thedeath of an individual leads to the removal of its nodes to-gether with all its connections with the others. The gameand network thus co-evolve as a result of the eliminationwith respect to the survival tolerance. Our main findingis that, in the presence of defectors, a cascading processof death of individuals can occur in relatively short time,which can even spread to the whole network, leading toa complete extinction. Strikingly, we find that a pure co-operation state emerges after the cascade terminates, inwhich the exclusive survivors are cooperators. This phe-nomenon occurs regardless of the type of games and ofthe network topology. This finding strongly suggests thatdefectors, despite their temporary advantages, are vulner-able to catastrophic cascading process. Cooperation be-

comes the optimal strategy to maximize benefit and avoiddeath for an individual. This resolves the social dilemmaof profit versus cooperation in a natural manner. Our re-sults can yield insights into the mechanism of catastrophicevents in economical and ecosystems. For example, duringthe recent economic recession, the large scale bankruptcyof financial organizations is a typical cascading processwhere high-risk investments, a kind of defection behav-ior, decrease the capacity of agents to resist deficiency andtrigger the outbreak of the bankruptcy cascade. For evo-lutionary biology, our result may also provide hints to themechanism of species extinctions in relatively short timescales.

I. INTRODUCTION

A hallmark of the recent great economical recession is thecollapse and bankruptcy of a large number of financial institu-tions and corporations on a scale that has not been seen sincethe great depression. The manner by which the failures occurmay be described as a cascading process, where the initial col-lapse of one or a few institutions, for example, triggered thefailures of many others. While sophisticated economical andsocial models can conceivably be built up to describe the pro-cess of cascading collapses, from the standpoint of physics weare interested in a “minimal” model that can capture the majorgeneric ingredients of the process, which are independent ofthe system details. It is hoped that the model can then lead toinsights into the prevention of such cascades. The purposes ofthis paper are to construct such a model, to present analysesand numerical results, and to explore the implications.

For convenience, we shall refer to financial institutions,banks and corporations etc., collectively as agents. In gen-eral, agents are connected with each other through a net-worked structure that often can be complex. While com-plex networks are common in natural and man-made systems,

2

the interactions among the agents can be much more compli-cated than these seen in natural systems such as biological net-works where the couplings can usually be described by somesimple mathematical functions. In this regard, evolutionarygames are appropriate and effective to describe the interac-tions among agents. The basic feature of our model is thusevolutionary games on large networks. In the following weshall give a brief description of background on evolutionarygames and existing works, justify a new feature that we intro-duce to model failures or elimination of agents, and state ourfindings.

Evolutionary game theory has been a powerful tool to studya variety of self-organized behaviors in natural, social, andeconomical systems [1–4]. A ubiquitous behavior is cooper-ation that is necessary for generating and maintaining ordersin these systems. Understanding how cooperation emergesamong selfish individuals has been a challenging problem,especially in view of the dilemma in social science that dis-favors cooperation [5–8]. The Prisoner’s Dilemma games(PDGs) [9], the snowdrift games (SGs) [10], and the publicgoods games (PGGs) [11] have been used to model interac-tions among selfish individuals and how the social dilemmacan be resolved through self-organization. In particular, thePDG and the SG are two-player games, while PGGs are usu-ally played by groups of agents. The SG, also known as theHawk-Dove game, is more favorable for the emergence of co-operation than the PDG. Based on these games, a number ofmechanisms have been discovered that facilitate cooperation,which include reputation and punishment [12–17], networkreciprocity [18–22], and success-driven migration [23], etc..In most of these works, since the focus was on the emergenceof cooperation, it is not necessary to incorporate any failuremechanism. That is, during the process no individual agent iseliminated from the game. Apparently, this assumption is notsuitable for situations where agents can go bankrupt.

In existing works on evolutionary games on networks [18–22], agents’ payoffs are determined by both their and theiropponents’ strategies according to a certain set of rules. Allagents imitate their neighbors to acquire strategies to gainmore payoffs, where the neighbors of an agent are those thatare directly connected to the agent, as determined by the net-work topology. We shall incorporate a failure or an elimina-tion mechanism into the gaming rules. In particular, we assigna tolerance parameter to every agent in the network, which isthe lowest payoff needed for the agent to survive. Due to thediversity in the properties of the agents, this parameter canbe different for different agents. To define the tolerance, wefollow the phenomenology of the crisis in financial marketsassociated with the great recession. For any profitable agent,a lowest amount of profit should be maintained, which comesfrom the interactions with other agents in a certain time pe-riod for continuous investment into the future. The minimalvalue of the profit is related to the size of the agent: largerorganizations require higher profits to maintain their normalfunctioning and to avoid bankruptcy. An appropriate defini-tion of the tolerance with respect to the diversity of agents canbe obtained by using the normal payoff of agents as a refer-ence, which is the payoff gained in a healthy market where

all agents are cooperators. Since information about the neigh-borhood has been incorporated into the normal payoff, infor-mation about the diversity of agents is naturally embedded inthe tolerance. When the lowest payoff of an agent cannot beachieved during the game, agent becomes bankrupt and is re-moved, together with all its links, from the network.

So far, the death and removal of agents in evolutionarygames has received little attention, and there has been nowork on sudden, cascading-like large-scale failures of agents.For example, in some previous studies, the death of an agentis usually accompanied by the birth of a new one at theempty site, resulting from the competition in the neighbor-hood [15, 24, 25] or from imitating strategies of those neigh-bors with higher fitness. While the assumption of simultane-ous death and birth is suitable for addressing issues such aswhether natural selection favors cooperation and what topol-ogy promotes cooperation, it is not suitable for studying cas-cading failures. Our working hypothesis is then that the deathof agents is much faster than the generation of new ones ina certain time period so that the birth process can be effec-tively neglected. That is, when an agent is removed from thegame, the site in the network that it originally occupies re-mains empty through the dynamical process. This is a keydifference between our model and previous ones, which al-lows us to investigate catastrophic behavior through a “mini-mal” model. Our model also differs much from previous onesof games on adaptive networks [26, 27] and of the evolutionof cooperation under topological attacks or errors [28].

Our main finding is that in the presence of defectors so thatagents are vulnerable to defection for instantaneous higherpayoff, a cascading process of agent death can occur in rela-tively short time, which can even spread to the whole network,leading to a complete extinction. This result captures the es-sential feature of what happened in financial markets duringthe recent economic recession, where the defection strategycan generally be represented by several types of harmful eco-nomic activities, such as sub-prime mortgage securities. Suchhigh-risk investments decrease the capacity of agents to resistdefection. Our finding implies that deleterious economic strat-egy can play the key role in the outbreak of bankruptcy cas-cades, which is consistent with intuitive understanding. Strik-ingly, we find that a complete cooperation state emerges af-ter the cascade terminates and the exclusive survivors are co-operators, which holds regardless of the type of games or ofthe network topology. This finding strongly suggests that de-fectors, although they can gain much more payoffs and pre-vail temporally, are extremely vulnerable to the occurrence ofthe catastrophic cascading behavior. For an agent, in orderto maximize its payoff while avoiding death, cooperation be-comes the optimal strategy. When most agents cooperate, thesystem can be maintained in a healthy state in that no large-scale cascading events are likely. Our finding suggests thatrational agents can certainly survive and make profit throughcooperation, which naturally resolves the social dilemma ofprofit versus cooperation.

A brief account of some of the results with a focus on regu-lar networks has appeared recently [29]. The purposes of thispaper are to provide an extensive analysis of the cascading

3

failures on both regular and complex networks and to addressnew issues such as cascades triggered by a single defector.The paper is organized as follows. In Sec. II, we detail ourmodel. Numerical results and analyses for regular and com-plex networks are presented in Secs. III and IV, respectively.In Sec. V, we treat the case where cascading failures are trig-gered by a single defector. Conclusions and discussions areoffered in Sec. VI.

II. MODEL

We shall use all three types of games (PDG, SG, and PGG)studied commonly in the literature to simulate interactionsamong agents in the network. Our goal is to identify genericdynamical features that hold regardless of the type of thegame. The main ingredients of these games are as follows. (i)In the PDG, there are two players and they can choose either tocooperate or to defect. Both players are offered a reward R formutual cooperation and a lower payoff, a punishment P , formutual defection. If one player decides to cooperate but theother defects, the defector gets the highest payoff T (temp-tation to defect), while the cooperator gets the lowest payoffS. The payoff rank for the PDG is thus T > R > P > S.(ii) In the SG, there are also two players but the payoff rank isT > R > S > P , where the positions of S and P are reversedas compared with the PDG. This means that mutual defectionis an irrational strategy in the SG. (iii) The PGG differs fromthe PDG and the SG in that it is played by a group of players.In such a game, cooperators contribute a cost c to the publicgood and defectors do nothing. The total reward is the productbetween the total contribution and an enhancement factor η,which is equally distributed among all members in the group.Thus, a defector’s payoff without cost always outperforms acooperator’s.

The particular sets of players involved in a game are deter-mined by the network topology. At each time step, the actualpayoff gained by any agent is the sum of payoffs resultingfrom all interactions with others. Initially, all nodes of a con-nected network are occupied by both cooperators and defec-tors. At each iteration (time step), there are three dynamicalprocesses.

(i) Game playing and payoffs. For the PDG, we follow pre-vious studies and use the rescaled parameters R = 1, T = b(b > 1), S = P = 0 [18]. Thus b is the only parameter. Forthe SG, we set R = 1, T = 1 + r, S = 1 − r, P = 0, so thesingle parameter is 0 < r < 1 [20]. For the PGG, in an ar-bitrary group formed by node x and its neighbors, the payoffsof a defector and a cooperator are

P (D) =cηn(C)

kx + 1and P (C) = P (D)− c, (1)

respectively, where η is the enhancement parameter, n(C) isthe number of cooperators in the group and kx is the numberof neighbors of node x. Without loss of generality, we set c tobe unity [22].

(ii) Failure and agent removal. At each iteration, the node

FIG. 1: For the PDG on a two-dimensional lattice with four neigh-bors for each site, time series of the size of death sd and the fractionsof cooperators ρc and defectors ρD in the survivors for α = 0.5 andb = 1.1. Initially, same numbers of cooperators and defectors arerandomly distributed over the lattice.

that hosts agent i and all its links will be removed if

Pi < Ti ≡ αPNi , (2)

where Ti is the tolerance to death, PNi is the normal payoff

when the system is in a healthy state in which all agents arecooperators, and 0 ≤ α ≤ 1 is a tolerance parameter. For thePDG and the SG, we have

Ti = αPNi = αki, (3)

where ki is the number of neighbors of i. For the PGG, wehave

Ti = αPNi = α(η − 1)(ki + 1). (4)

For α = 1, agents have zero tolerance to breakdown, whilefor α = 0, agents are completely tolerant.

(iii) Strategy updating. At each time step, agent i randomlychooses a survived neighbor j and imitates j’s strategy withthe probability [19, 20]

Wi→j =1

1 + exp [−(Pj − Pi)/K], (5)

where K is the level of “noise” representing the uncertaintiesin assessing the payoffs. In our simulations we set K = 0.1(quite arbitrarily).

III. CASCADING FAILURES AND SURVIVAL STRATEGYON REGULAR NETWORKS

We consider lattices with varying numbers of neighbors foreach site. Figure 1(a) shows, for a two-dimensional lattice

4

FIG. 2: For the PDG on 2D4n, evolution of spatial patterns for b = 1.1. The lattice size is 50× 50 and all sites are occupied initially. (a) Fort = 0, 10% of the occupants are defectors randomly distributed in the lattice. The color coding is: red for defectors, blue for cooperators, andwhite for empty sites. (b) For t = 5, defectors reproduce themselves and invade the domain of cooperators. Note that empty sites arise nearthe defectors. (c) For t = 9, the number of defectors has been reduced considerably, the vacant areas enlarge, and some clusters of cooperatorsbegin to form. (d) For t = 27, all defectors have failed and have been removed, marking the end of the cascading process. For t > 27, thespatial pattern is invariant and the number of survivors is a constant.

where each site has four neighbors (2D4n), a typical time se-ries of the size sd of failed (or dead) agents, defined as thenumber of removed nodes normalized by the network size.Time evolutions of the fractions of cooperators ρC and ofdefectors ρD among survivors are shown in Fig. 1(b). Wesee that a cascading process of failures occurs, where about80% of the agents eventually fail and are removed. Associ-ated with the death of agents, ρc first decreases with time andthen reaches unity after the cascading process is complete andthe system reaches a new steady state. In contrast, after asmall increment at the beginning of the cascading process, ρDdecreases continuously and tends to zero eventually. These re-sults indicate that, after a cascading process, cooperators arethe sole survivors. Similar results have been obtained for theSG and the PGG. The general observation is that, when de-fecting strategies are practiced, the occurrence of large-scalecascading processes is common and, in order to survive, anagent needs to cooperate persistently. Our computations havealso revealed that the value of sd depends on the toleranceparameter α, the temptation to defect (b or r), and the en-hancement factor η. If defections occur significantly more of-ten than cooperations, a complete breakdown of the system islikely, where no agent can survive.

To gain insights into the cascading process, we examinethe evolution of spatial patterns. Figures 2(a-d) show, for thePDG, spatial patterns at four instants of time. Initially 10% ofagents are randomly placed in the ocean of cooperators [Fig.2(a)]. At some early stage, there is an increase in the numberof defectors but empty sites begin to arise around defectors,and both cooperators and defectors begin to die, as shown inFig. 2(b). At a later time, the death rate of defectors exceedsthat of cooperators, and clusters of cooperators begin to form,as shown in Fig. 2(c). Finally, defectors become extinct andthe lattice is shared by cooperator clusters and empty sites[Fig. 2(d)]. After the extinction of defectors, the death ofcooperators stops and the patterns become time-invariant.

The evolution of spatial patterns leads to a qualitative ex-planation for the outbreak of death and the survival of coop-

erators. In particular, the spread of the defection strategy andthe loss of interactions among agents induce continuous deathand occurrence of empty sites adjacent to defectors, but theincrease of empty sites and the formation of cooperator clus-ters lead ultimately to the extinction of defectors. This canbe explained by focusing on the interaction among defectorsand their neighbors. Suppose that a defector is surroundedby cooperators. The defector’s payoff will be the highest andlarger than the tolerance value. However, two situations mayarise that can cause the death of the defector. Firstly, due tothe highest payoff gained by the defector, cooperating neigh-bors tend to imitate its strategy and betray with a high prob-ability. Secondly, the death of cooperating neighbors can oc-cur because of their insufficient cooperations. Regardless ofwhich situation actually occurs, a negative feedback mecha-nism is induced by the defector, resulting in the reduction ofits payoff. Consequently, either the defector becomes vulner-able to death or it is overwhelmed by neighboring coopera-tors. For cooperating neighbors, once they turn to be defec-tors, the defection strategy spreads and further death can fol-low until the emergence of cooperator clusters. At the bound-ary of the clusters, cooperators receive sufficient mutual co-operations to resist both invasion of defectors and insufficientpayoffs to death. Defectors adjacent to the boundary are sur-rounded by many empty sites and cannot gain enough payoffsfrom cooperators to survive. This behavior is similar to whathappened in a financial market during a crisis. For example,although high-risk investments may bring higher short-termprofits, they can trigger large-scale failures.

Figures 3(a-c) show representative final patterns for two-dimensional lattices with 4 and 8 neighbors. We denote thetwo network systems by 2D4n and 2D8n, respectively, andwill use a similar notation for other cases treated in this pa-per. When the initial fraction of defectors is small and thetolerance to death is high, a large number of cooperators cansurvive and they tend to form large areas of clusters, as shownin Fig. 3(a) for a 2D8n lattice. For high temptation to defec-tion and low tolerance, only small groups of cooperators can

5

FIG. 3: Representative spatial patterns for the PDG on two-dimensional lattices with 8 and 4 neighbors, where the initial state is that 10% ofthe agents are defectors: (a) for a lattice with 8 neighbors (2D8n), a small cascade of death with a large fraction of cooperators that survivefinally for b = 1.01 and α = 0.1; (b) for the same lattice but for b = 1.05 and α = 0.4, a number of small clusters of cooperators that remaineven when death of agents is severe; and (c) for a 2D4n lattice for b = 1.05 and α = 0.4, small surviving clusters of cooperators. In (b) and(c), two typical cooperator clusters are marked, which are the smallest clusters surviving through the cascading process.

survive in the sea of vacant sites, as shown in Fig. 3(b) for a2D8n lattice and Fig. 3(c) for a 2D4n lattice. We see that, evenwhen failures are massive, there can still be small clusters ofcooperators that survive the catastrophe. As we will discussbelow, for a regular lattice, a sudden transition from survivalstate to extinction can occur, and the transition point is deter-mined by nothing but the stabilities of these small clusters.

To obtain a better understanding of the cascading dynamics,we investigate the dependence of the fraction of failed agentssd on the tolerance parameter α for three games on four typesof lattices with two, four, six, and eight neighbors for eachsite (1D2n, 2D4n, 2D6n, and 2D8n, respectively). The ba-sic cells of these lattices are shown schematically at the topof Fig. 4, and the computational results in Fig. 4 are withrespect to the PDG. We observe the appearance of step struc-tures for all lattices, where for such a step, sd changes discon-tinuously from one constant value to another. For differentlattices, the numbers of steps are different. Since all survivorsare cooperators, Nc (the number of cooperators) as a functionof α displays step structures as well because of the simple re-lation Nc = N(1 − sd). A striking phenomenon is that thetransition from a survival to an extinction state occurs at thecritical value αc = 0.5, regardless of the lattice type and ofother parameters such as the temptation to defection and theinitial fraction of defectors. Similar results have been foundfor the SG with the same value of αc. For the PGG, becauseof the intrinsic group interactions, the behavior of sd versus αis somewhat different from those with the PDG and the SG.However, the phenomenon of transition to extinction persists,as shown in Fig. 5. We observe that, except for the 1D2n lat-tice, there are no clear step structures and the transition pointsdiffer for different lattices.

To explain the transition to extinction, we focus on the sta-bilities of various surviving clusters of cooperators. For in-stance, we can study such clusters for parameter α slightly

below the critical value 0.5. The structures of the “minimal”clusters, one for each lattice type, are shown schematically inFig. 6. Their stabilities can be assessed by calculating thepayoffs of agents in the respective clusters. For example, forthe 1D2n lattice, the two cooperators’ payoff is Pi = 1 andtheir tolerance is Ti = αki = 2α. For α < 0.5, we havePi > Ti so that both cooperators will survive and the clus-ter is stable. Similarly, all clusters in Fig. 6 are stable forα < 0.5. For α > 0.5, these clusters become unstable andthere are no longer survivable structures.

The transition associated with the PGG can be understoodsimilarly. For an arbitrary surviving node i, its payoff satisfiesPi > Ti. Combining Eqs. (2) and (4), we have

Pi = (η − 1)(k′i + 1) > α(η − 1)(ki + 1) = Ti, (6)

where ki is the original degree of i and k′i is the remainingdegree in the aftermath of the cascading event. The criticalvalue αc for the transition point is then given by

αc =k′i + 1

ki + 1, (7)

which is independent of the enhancement parameter η. Sincefor the PGG, the smallest stable cluster has the same structureas that for the PDG, the transition point αc is also determinedby the cluster structure in Fig. 6. A common property amongthese minimal cluster structures is that the remaining node de-gree k′i for any node is large than or equal to kL/2, where kL isthe node degree of the original lattice. Since nodes with moreremaining connections are more stable for identical originaldegrees, the extinction-transition point αc is determined bythe nodes at boundary. We then have

αc =kL + 2

2(kL + 1). (8)

6

FIG. 4: For the PDG on four types of regular networks (1D2n, 2D4n,2D6n, 2D8n, as schematically illustrated at the top), the fraction offailed (dead) agents sd as a function of the tolerance parameter α.The dashed vertical lines are theoretical predictions for various tran-sitions between distinct states, including the extinction transition.The network size is 100 × 100 and all data points are obtained aftera steady-state is reached for which sd remains to be a constant.

The values of αc for the four types of lattices can then becalculated as α1D2n

c = 2/3, α2D4nc = 3/5, α2D6n

c = 4/7,α2D8nc = 5/9. These predictions are verified by simulation

results for the PGG, as shown in Fig. 5.We can consequently explain the presence of step structures

in Fig. 4 associated with the transitions from one survivingstate to another by examining the condition for survival:

Pi = k′i > αki = Ti, (9)

or

αc =k′iki. (10)

Because the remaining degree k′i satisfies k′i ≤ ki, its possi-ble values are 1, 2, · · · , ki. However, since no stable clusterexists for α > 0.5, there is an additional constraint for k′i:k′i ≤ ki/2. All possible values of k′i determine the num-bers of steps in Fig. 4. For the 1D2n lattice, k′i = 1 is theonly choice so that αc = 1/2, which separates two steps insd. For the 2D4n lattice, k′i can be 1, 2, which results inαc = 1/4, 1/2, corresponding to 3 steps. Similarly, for the2D6n lattice, we have αc = 1/6, 1/3, 1/2, which separate thewhole α-interval into 4 steps. For the 2D8n lattice, we haveαc = 1/8, 1/4, 3/8, 1/2, so there are 5 steps. These predic-tions are in good agreement with numerical computations, asshown in Fig. 4.

FIG. 5: For the PGG, dependence of sd on the tolerance parameter αfor the four types of networks as in Fig. 4. The dashed vertical linesare theoretical predictions for the extinction-transition points for thefour types of networks as given by Eq. (8).

FIG. 6: Four smallest surviving clusters in four types of lattices forα slightly below the critical value αc for all three games. Theseclusters determine the transition point to extinction. The stabilitiesof nodes inside the clusters can be determined by comparing theirremaining payoffs with their tolerance payoffs. The nodes inside thecluster are more stable than those at boundaries. For α > αc, nodesat boundaries die out and the clusters disappear.

IV. CASCADING FAILURES ON COMPLEX NETWORKS

A. Scale-free networks

Extensive research in the past decade has revealed that alarge number of real-world networks possess the small-worldand/or the scale-free topology [30–32]. The small-worldtopology is especially relevant to social and economical net-works where node-to-node interactions are best described byevolutionary games. It is thus important to test whether ourfindings with respect to regular networks in Sec III, namely,that defection strategy can lead to large-scale cascading fail-ures and cooperators are the sole survivors, apply to complexnetworks.

For evolutionary games on complex networks, it has beenestablished that cooperation can be supported by several nat-

7

b

α

1 1.5 2

0.2

0.4

0.6

0.8

1

r

0 0.5 1

0.2

0.4

0.6

0.8

1

η

2 4 6 8 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.2

0.4

0.6

0.8

C survival

Extinction

FIG. 7: For the PDG, the SG, and the PGG on scale-free networks, dependence of the death size sd on the tolerance parameter α and gameparameters b, r, and η, respectively. The initial fraction of defectors is 0.1 for the PDG and the SG, and 0.5 for the PGG. There are two distinctasymptotic phases: extinction and survival of cooperators. The boundaries between the two phases are marked by the white curves. In theentire parameter space, defectors cannot survive. For the three games, larger values of α lead to larger values of sd because agents are morevulnerable to payoff decrease from losing connections with or being betrayed by neighbors. Failures, however, can be made less severe byreducing the temptation to defection. Decreasing b (r) or increasing η can enhance the formation of cooperation clusters and their abilities toresist both death and the invasion of defectors. Ensemble average is based on 10 network realizations and 10 independent gaming processesfor each network realization.

ural mechanisms, for example, repeated interactions [5], pun-ishment [12–17], and migration [23]. It turns out, counter-intuitively, that heterogeneity in node degrees can be benefi-cial to the emergence and persistence of cooperation both fortwo-player games (the PDG and the SG) [21] and for gamesinvolving groups of players (the PGG) [22]. To make an un-biased comparison with our results from regular networks, weshall simply implement our evolutionary-game model in Sec.II on both scale-free and small-world networks. For scale-freenetworks, we use the standard Barabasi-Albert model [33],while for small-world networks, we use the Newman-Wattsmodel [34]. We focus on how the size of death and the sur-viving strategies depend on two key game parameters, α andb (r) or η, as defined in Sec. II.

Figures 7(a-c) show the contour plots of sd in the two-dimensional parameter plane for the PDG, the SG, and thePGG on scale-free networks, respectively. Analogous to whathas been observed on regular lattices, there exist two exclu-sive asymptotic phases: extinction without any survivors forlarge values of α and b (r, η) and a survival phase in whichonly cooperators can resist death and survive, regardless ofthe values of α and b (r, η). Cooperators and defectors can-not coexist for any parameter combinations. Similar to whatwe have observed for regular lattices, the defectors are des-tined to be eliminated despite their temporal high advantagewhen encountering cooperators. Thus, the finding that defec-tion can only bring short-term benefit to agents with respectto cascading failures and cooperation is the essential strategyfor survival, holds true also for complex networks. A differ-ence from the case of regular networks is that, for complex

networks, sd is more sensitive to the variation of α and thesd-versus-α curve tends to be continuous. This is understand-able considering the underlying complex topology that pro-vides a richer spectrum of agent tolerances due to the diversityof node degrees. This difference is mostly at a detailed leveland it does not affect our general conclusion on cascading fail-ures and surviving strategy, which apparently holds regardlessof the game type and of the network topology.

The boundary between extinction and cooperator survivalfor scale-free networks can be estimated, as follows. Becauseof the highly heterogeneous degree distribution and the asso-ciated complex connecting pattern, stable clusters resistant toextinction can no longer be expected. The boundary in factdepends on parameters (e.g., b and r for the PDG and the SG,and η for the PGG). However, in certain regimes the parame-ter dependence can be weak. For example, we find that for thePGG, the boundary hardly depends on η if it is large. In thiscase, the boundary is solely determined by the network struc-ture, which can then be treated by a stability analysis as wehave done for regular networks. This is particularly the casewhen cooperation is facilitated by large values of η.

To proceed, we note that, for a scale-free network, a vastmajority of the nodes have in fact the smallest degrees. Thestabilities of these nodes can play a key role in the extinction.In reality, the number of connections of an agent possessesdetermines its robustness. We can thus assume that all agentswith the smallest degree (kmin) have the identical critical tol-erance to death, αc, where for α > αc, the death of theseagents can possibly trigger the extinction due to their largenumbers. The critical tolerance of agents with kmin can thus

8

FIG. 8: For the standard scale-free network, extinction boundary αc

as a function of the average degree ⟨k⟩ for the three types of games,where the initial fraction of defectors is 0.85 (relatively high) andthe temptation-to-defection parameter is b = 1.01 and r = 1.01(for the PDG and the SG, respectively) or the enhancement param-eter is η = 10 (for the PGG). The network size is 1000. Each datapoint is obtained by averaging over 10 network realizations and 10independent gaming processes for each network realization. The stargraph is a typical survivable cooperator cluster when α is close to theboundary αc.

be used to estimate the extinction boundary. This approachalso makes use of the fact that hubs are usually more stablethan small-degree agents in that the latter usually fail moreeasily than the former. From Eq. (7), the stability conditionfor the smallest-degree agents in the PGG can be written as

αc =k′min + 1

kmin + 1, (11)

where k′min is the minimal degree of the network after cas-cading. The range of possible values of k′min is from 1 tokmin, which depends on both the enhancement parameter ηand the initial fraction of defectors. However, when both ηand the initial fraction of defectors are large, only the moststable agents can survive. An agent can survive when thereis at least a single interaction. We thus have k′min = 1. Ifα is reduced such that the single interaction cannot provideenough payoff for the agent to sustain, extinction will arise.For the standard scale-free network [33], the average degreeis ⟨k⟩ = 2kmin. The extinction threshold thus depends on ⟨k⟩and can be written as

αc =4

2 + ⟨k⟩. (12)

Similarly, for the PDG and the SG, we have

αc =2

⟨k⟩, (13)

which is valid in the regime of small temptation to defec-tion and large initial fraction of defectors. Figure 8 shows

these theoretical estimates together with results from directnumerical simulations, where a good agreement is observed.Our computations also reveal that, for scale-free networks, thestable cluster possesses a star-like structure, as indicated inFig. 8, where all connections are originated from a hub node.For α > αc, the star-like structure becomes unstable and noagents can survive, signifying the onset of extinction.

B. Small-world networks

The contour plots for the PD, the SG and the PGG onNewman-Watts small-world networks are shown in Fig. 9, re-spectively. Similar to the observations on lattices and scale-free networks, there are two phases, the extinction phase andthe survival phase that are separated by the white curves. Inthe survival phase, cooperators are the only survivors. Thesurvival region is smaller compared to that on scale-free net-works, due to the fact that the scale-free structure promotescooperation.

V. CASCADING FAILURES TRIGGERED BY A SINGLEDEFECTOR

A. Regular lattices

Here we discuss the scenario in which the cascading pro-cess of death is triggered by a single defector on regular andscale-free networks. In Fig. 10, we show some typical spatialpatterns on regular lattices starting from a central defector forthe PDG as an example. Figures 10(a), 10(b) and 10(c) arefor a 2D4n lattice. In Fig. 10(a), for small b and large α, e.g.,b = 1.1 and α = 0.8, the single defector results in continu-ous death starting from the central site and all agents becomeextinct eventually. In this case, the death process is inducedexclusively by the loss of the interactions from the removalof neighbors (not by the diffusion of defection strategy). InFig. 10(b), for large b and small α, during the cascading pro-cess, most agents die except a few small clusters that are stableenough to survive. In contrast, for small b and small α values,as shown in Fig. 10(c), more cooperator clusters of larger sizessurvive as compared to (b). In Fig. 10(b) and (c), the large-scale death is triggered by both the diffusion of the defectionstrategy and the loss of interaction with cooperators amongagents. Figure 10(d) exhibits the spatial pattern on a 2D8nlattice for b = 1.3 and α = 0.4. The survival clusters ap-pear different from that for the 2D4n lattice and the pattern ofdefectors invading the cooperator clusters is different as well.

The contour plot for the PDG on the 2D4n lattice with a sin-gle defector is shown in Fig. 11. An interesting phenomenonis that the cascading process of death is prohibited in the mid-dle range of the threshold parameter α. This means that therobustness of the network system is a non-monotonic functionof the agents’ tolerances and, as a result, strong tolerance canlead to extinction more easily. To explain the counterintuitivebehavior, we investigate the evolution of a sample lattice, asshown in Fig. 12. For simplicity, we consider large values of b

9

b

α

1 1.5 2

0.2

0.4

0.6

0.8

1

r0 0.5 1

0.2

0.4

0.6

0.8

1

η2 4 6 8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.2

0.4

0.6

0.8

FIG. 9: (Color online) Dependence of the size of death sd on the tolerance parameter α and the game parameter b, r and η for the PD, the SGand the PGG respectively on NW small-world networks. For the PD and the SG, the initial fraction of defectors is 0.1 and for the PGG is 0.5.There are exclusive two phases: extinction and cooperator survival. In each case, the boundary between them is marked by the white curve.

FIG. 10: (Color online) Representative spatial patterns for the PDGwith a single defector at the center on regular lattices: (a) b = 1.1,α = 0.8, t = 35; (b) b = 1.8, α = 0.25, t = 93; (c) b = 1.1,α = 0.1, t = 128 and (d) b = 1.3, α = 0.4, t = 119, where (a), (b)and (c) are for 2D4n, and (d) is for 2D8n. The blue regions representcooperators, the red regions are for defectors and the white regionsdenote empty sites.

for explanation. In Fig. 11 for large b, when 0.5 < α < 0.75,nearly all agents survive, whereas extinction occurs in otherregions of α. We thus consider the role of α in three re-gions separately: (i) α < 0.5, (ii) 0.5 ≤ α ≤ 0.75, and (iii)α > 0.75. For α > 0.75, according to the stability condition,

b

Cap

acity

1 1.2 1.4 1.6 1.8 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

FIG. 11: (Color online) For a 2D4n lattice, the dependence of the sizeof death sd on the tolerance parameter α and the game parameter b.

a cooperator will die if it is in the vicinity of a defector orloses interaction with a cooperator neighbor. Thus the coop-erators adjacent to the defector will die first. The defector can-not survive either and all agents will die eventually (Fig. 12).For α < 0.5, at the first step, no agent dies. Instead, dueto the advantages of defectors in gaining payoffs, the centraldefector will pass its strategy to its neighboring agents. Af-ter that the central defector cannot survive and the defectionstrategy spreads, together with the death of defectors exceptthose at the boundary of the large defector cluster. Finally, allagents are eliminated. For the robust region 0.5 < α < 0.75,as shown in Fig. 12, we can get from calculation that, at thefirst step, no agent dies and the central defector will pass itsstrategy to its neighbors. After that, since α > 0.5, cooper-

10

FIG. 12: (Color online) Illustration of the evolution from single de-fector in the ocean of cooperators on a 2D4n lattice. Blue representscooperators, red signifies defectors and gray denotes dead agents.The examples are for large values of temptation to defect, b. Thecounterintuitive results in Fig. 11 can be explained by the evolutionon the sample lattice.

ators who have less than two cooperator neighbors will die.The surviving defectors’ payoffs are less than those of theircooperator neighbors, preventing the spread of the defectionstrategy. At the next step, from the survival condition, thedefectors will die but their cooperator neighbors can survive.As a result, the death process ends and only 9 agents aroundthe initial defector are eliminated and the size of death as nor-malized by the total number of agents becomes negligible. Wenote that, for small values of b, for α < 0.5, defection strategymay not spread easily, leading to the shrink of the extinctionregion in Fig. 11 as compared to the case of large b.

An interesting issue then concerns the onset of cascadingprocess triggered by a single defector on different regular lat-tices for the PGG. We find two similar scenarios: (I) death ofthe defector followed by the death of neighboring cooperators;(II) diffusion of the defection strategy. These considerationslead to analytical insights into the onset of death for differenttypes of lattices, which we now elaborate.

For the 2D6n, the payoff of the defector as shown in Fig. 13is

PD = 4η, (14)

and the payoff of the cooperators in the vicinity of the defectoris

PC = 3(η − 1) + 2(4

5η − 1). (15)

For case (I), if PC > PD, the defector cannot pass its strategyto neighboring cooperators, so only its death can trigger the acascading process. Using the condition PC > PD , we have

η >25

3. (16)

FIG. 13: (Color online) Three lattice configurations with single de-fector at center. The agents inside the circle determine the onset ofthe cascading process.

In this case, the death condition for the defector is

TD = 5α(η − 1) > 4η = PD, (17)

which yields

α >4η

5(η − 1). (18)

On the other hand, the death of the defector should lead tothe death of neighboring cooperators for the occurrence of thecascading process. The payoff of the neighboring cooperatorsafter the removal of the defector is

p′C = 4(η − 1), (19)

and the death condition TC > P ′

C yields α > 4/5. CombiningEq. (19) with Eq. (18), we can get the critical value αc for caseI.

For case II (η < 253 ), the diffusion of the defection strategy

can lead to the death and the removal of nodes. So the onsetof the cascading process is determined by the survival proba-bility of the cooperator after losing one connection. We thenhave

αc =

! 4η5(η−1) , if η > 25

3 ;45 , otherwise.

(20)

For the 2D6n lattice with a central defector, there are alsotwo different situations, determined by PC > PD and PC <PD, respectively, where

PD = 6η, and PC = 3(η − 1) + 4(6η

7− 1). (21)

The condition PC > PD then leads to η > 49/3. Specificallyfor case I, we have

!

T = 7α(η − 1) > 6η = PD,T = 7α(η − 1) > 6(η − 1) = P ′

C ,(22)

which yields

α >6η

7(η − 1). (23)

For case II, the critical value αc can be obtained as

αc =

! 6η7(η−1) , if η > 49

3 ,57 , otherwise.

(24)

11

FIG. 14: (Color online) Critical value αc of the onset of the cas-cading process as a function of game parameter η for the PGG forthree types of lattices. The curves are theoretical predictions anddata points are simulation results. All the results are obtained fromsingle defector initially. The network size is 2500. Each data point isobtained by averaging over 20 game and 10 networks realizations.

For the 2D8n lattice, we have

PD = 8η and PC = 3(η − 1) + 6(8η

9− 1). (25)

The inequality PC > PD leads to η > 21. The critical valueαc for case (I) can then be obtained through

!

T = 9α(η − 1) > 8η = PD,T = 9α(η − 1) > 8(η − 1) = P ′

C ,(26)

which yields α > 8η9(η−1) . Finally, we have

αc =

! 8η9(η−1) , if η > 21,69 , otherwise.

(27)

Both the analytical predictions and numerical simulationsare displayed in Fig. 14 for the three types of lattices. Weobserve an excellent agreement.

B. scale-free networks with single defector on the biggest hub

For scale-free networks, when the largest hub becomes adefector, the survival probabilities of a large number of small-degree nodes connected to the hub determine the occurrenceof the cascading process and possible extinction. Since mostneighbors of the largest hub are the agents with the smallestdegree, their death induced by the defection of the hub canplay the dominant role for the cascading process of elimina-tion. The problem of predicting the critical value ofα is equiv-alent to analyzing the stability of those smallest-degree nodes

FIG. 15: (Color online) Dependence of the size of death sd on thetolerance parameter α for different values of game parameter η inthe PGG with a single defector initially. The left panel is for theaverage degree ⟨k⟩ = 3 and the right panel is for ⟨k⟩ = 4. The reddashed lines are analytical predictions. Each data point is obtainedby averaging over 500 realizations for 10 networks. The network sizeis 1000.

that are the direct neighbors of the hub. The critical value ofα is then given by Eq. (11). We find that the smallest-degreeagents cannot survive when they lose one interaction, a cas-cading process of death or extinction can arise. This gives thecondition k′min = kmin − 1. We then have, from Eq. (11),

αc =kmin

kmin + 1=

⟨k⟩

⟨k⟩+ 2. (28)

Simulation results are shown in Fig. 15 for two different val-ues of the average degree ⟨k⟩. Again there is a good agreementbetween the predicted and numerical values of αc.

For the PDG and the SG, αc can be calculated as

αc =kmin − 1

kmin=

⟨k⟩ − 2

⟨k⟩, (29)

which has also been verified numerically.

VI. CONCLUSIONS AND DISCUSSIONS

We have constructed a “minimal” physical model to investi-gate catastrophic behavior in networked systems governed byevolutionary games and have found two generic phenomenathat do not depend on the system details such as the networktopology and game types: (1) defection strategies for tempo-ral high payoff can result in large-scale cascading failures oreven the collapse of the entire system, and (2) the optimalstrategy for surviving catastrophic failures is cooperation. Inparticular, we find the emergence of clusters of cooperators af-ter a large-scale cascading failure. Defection strategies, whilebeing capable of generating high payoff in short time scales,can trigger a negative feedback mechanism that weakens theviability of defectors and leads to their death ultimately. Incontrast, cooperation, while often temporally outperformed

12

by defection, can survive eventually in the form of clustersthat ensure enough profits for their members to resist deficitas well as the invasion of defectors in the long run. Theseresults suggest that selfish and greedy strategies can be quiteharmful for the health of the underlying networked system,be social or economical. In order to sustain the normal func-tioning of the system and to maximize individual agents’ gainin the long run, cooperation is the optimal strategy. Theseprovide insights into, for example, the phenomenon of large-scale bankruptcy of financial institutions and corporations wit-nessed during the recent global economical recession.

Our model in fact describes the co-evolution of game dy-namics and network topology. In particular, the dynamicalevolution is triggered by selfish strategies that result in thedeath of both defectors and cooperators, thereby altering thenetwork connecting structure from time to time. In contrast,in the absence of defectors, no agent dies, regardless of thetolerance and of the interaction patterns among agents. Thisbehavior is consistent with the real-world situation that, for

example, in a financial market, order is broken by the emer-gence and diffusion of deleterious economical activities. Howto prevent the occurrence of the catastrophic behavior and toreduce damages is a goal of practical importance. Our modelmay be helpful to address this problem by incorporating othermechanisms, such as punishment or volunteering [19]. Ex-ploring control strategies to suppress cascading processes insocial and economical systems remains an open problem. Thepresent model may shed new light and stimulate further ef-forts.

Acknowledgement

We thank R. Yang for discussions. This work was sup-ported by AFOSR under Grant No. FA9550-10-1-0083 andby NSF under Grants No. BECS-1023101 and No. CDI-1026710.

[1] K. G. Binmore, Game Theory and the Social Contract. Vol. 1:Playing Fair (MIT Press, Cambridge, MA, 1994).

[2] A. M. Colman, Game Theory and its Applications in the So-cial and Biological Sciences (Butterworth-Heinemann, Oxford,1995).

[3] J. Hofbauer and K. Sigmund, Evolutionary Games and Popula-tion Dynamics (Cambridge University Press, Cambridge, UK,1998).

[4] G. Szabo and G. Fath, Phys. Rep. 446, 97 (2007).[5] R. Axelrod, The Evolution of Cooperation (Basic Books, New

York, 1984).[6] K. Sigmund, Games of Life (Oxford Univ. Press, Oxford, UK,

1993).[7] P. Kollock, Ann. Rev. Sociol. 24, 193 (1998).[8] P. Hammerstein, Genetic and Cultural Evolution of Coopera-

tion (MIT Press, Cambridge, MA, 2003).[9] R. Axelrod and W. D. Hamilton, Science 211, 1390 (1981).

[10] R. Sugden, The Economics of Rights, Co-operation and Welfare(Blackwell, Oxford, UK, 1986).

[11] G. Hardin, Science 162, 1243 (1968).[12] E. Fehr and S. Gachter, Nature 415, 137 (2002).[13] R. Boyd, H. Gintis, S. Bowles, and P. J. Richerson, Proc. Natl.

Acad. Sci. USA 100, 3531 (2003).[14] M. A. Nowak and K. Sigmund, Nature 437, 1291 (2005).[15] A. Traulsen and M. A. Nowak, Proc. Natl. Acad. Sci. USA 103,

10952 (2006).[16] C. Hauert, A. Traulsen, H. Brandt, M. A. Nowak, and K. Sig-

mund, Science 316, 1905 (2007).

[17] H. Ohtsuki, Y. Iwasa, and M. A. Nowak, Nature 457, 79 (2009).[18] M. A. Nowak and R. M. May, Nature 359, 826 (1992).[19] G. Szabo and C. Hauert, Phys. Rev. Lett. 89, 118101 (2002).[20] C. Hauert and M. Doebeli, Nature 428, 643 (2004).[21] F. C. Santos, J. M. Pacheco, and T. Lenaerts, Proc. Natl. Acad.

Sci. USA 103, 3490 (2006).[22] F. C. Santos, M. D. Santos, and J. M. Pacheco, Nature 454, 213

(2008).[23] D. Helbing, W. Yu, Proc. Natl. Acad. Sci. USA 106, 3680

(2009).[24] E. Lieberman, C. Hauert, and M. A. Nowak, Nature 433, 312

(2005).[25] H. Ohtsuki, C. Hauert, E. Lieberman, and M. A. Nowak Nature

441, 502 (2006).[26] M. G. Zimmermann, V. M. Eguıluz VM, and M. S. Miguel,

Phys. Rev. E 69, 065102 (2004).[27] J. M. Pacheco, A. Traulsen, and M. A.Nowak Phys. Rev. Lett.

97, 258103 (2006).[28] M. Perc, New J. Phys. 11033027 (2009).[29] W.-X. Wang, R. Yang, and Y.-C. Lai, Phys. Rev. E 81,

035102(R) (2010).[30] R. Albert and A. L. Barabasi, Rev. Mod. Phys. 74, 47 (2002).[31] S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. 51, 1079

(2002).[32] M. E. J. Newman, SIAM Rev. 45, 167 (2003).[33] A. L. Barabasi and R. Albert R, Science 286, 509 (1999).[34] M. E. J. Newman and D. J. Watts, Phys. Rev. E 60, 7332 (1999).