Prisoner's Dilemma on Real Social Networks: Revisited
PPRISONER’S DILEMMA ON REAL SOCIAL NETWORKS: REVISITED
SHARON M. CAMERON AND A. CINTR ´ON-ARIAS
Abstract.
Prisoner’s Dilemma is a game theory model used to describe altruistic be-havior seen in various populations. This theoretical game is important in understandingwhy a seemingly selfish strategy does persist and spread throughout a population thatis mixing homogeneously at random. For a population with structure determined bysocial interactions, Prisoner’s Dilemma brings to light certain requirements for the al-truistic strategy to become established. Monte Carlo simulations of Prisoner’s Dilemmaare carried out using both simulated social networks and a dataset of a real social net-work. In both scenarios we confirm the requirements for the persistence of altruism inthe population. Introduction
One well-documented example of altruism is food sharing. Vampire bats (
Desmodusrotundus ) have nocturnal schedules, they leave their roost for several hours during thenight in search of prey they can feed from. On following nights, it is customary for them tolocate the prey they had previously fed on and continue their extraction of warm blood. Ahandful of these bats may be unsuccessful in their food supply search, however, they will notstarve as some of their peers will regurgitate a portion of the blood they acquired on thatnight, and share it with them. Wilkinson [45] discovered that sharing food by regurgitation,among wild vampire bats, is a function of reciprocation and it is independent of the degreeof relatedness. In other words, as explained by Nowak [32], if a bat has previously fedanother one, it is more likely this beneficiary re-pays the favor in the future. Food sharingis often cited as an example of direct reciprocity [45, 32], which is another way of referringto cooperation, the main topic of this study. Here we focus on mathematical modeling andsimulation of cooperation.The time evolution of cooperation is a subject of fascination for evolutionary biologists,that finds it roots in the foundations of game theory [42, 26, 3, 31]. Prisoner’s Dilemmais perhaps one of the best-studied theoretical games that describes altruistic behavior inorganisms. Typically, Prisoner’s Dilemma is formulated as a two-strategy and two-playergame, where the payoffs are determined by years served in a jail sentence. Indeed, thename of this game is coined from a scenario where two partners in crime are being held for
Date : August 27, 2012.1991
Mathematics Subject Classification.
Primary: 91A40; Secondary: 91D30.
Key words and phrases.
Game theory, social networks, small-world networks.S.M.C. received funding from NSF under grant DUE-0525447 and from ETSU Honors College under aResearch Discovery Position. A.C.-A. was funded by ETSU Presidential-Grant-in-Aid number E25150. a r X i v : . [ phy s i c s . s o c - ph ] A p r SHARON M. CAMERON AND A. CINTR ´ON-ARIAS interrogation in separate rooms at police quarters, and they weigh in their options whilethe questioning takes place.According to classical game theory, for populations of players that mixed homogeneouslyat random, cooperation is doomed to become extinct [31, 14]. On other hand, more recentdevelopments suggest that when Prisoner’s Dilemma is considered not just as a time-evolving process but rather as a spatio-temporal evolving process, there are certain con-ditions that prevent the extinction of cooperation [33]. In this study we address spacein the context of a social landscape for players of Prisoner’s Dilemma. We consider so-cial networks with both simulated datasets and a dataset sampled from a collegiate socialnetworking site. Specifically, our main contribution is to successfully validate a necessarycondition for the establishment of cooperation (see [33] and references therein) against anempirical dataset of a social network (friendship in a social networking site [40]).This paper is organized in the following way. Versions of Prisoner’s Dilemma in well-mixed populations and those with network structure are introduced in Sections 2 and3, respectively. Models for social networks with small-world properties are discussed inSection 4. In Section 5 a dataset of a real social network is introduced. A discussion of theresults is offered in the last section.2.
Prisoner’s Dilemma in Well-Mixed Populations
In a well-mixed game, everyone is assumed to interact with one another, homogeneouslyat random. Under this assumption, cooperators may receive a benefit b > c > b − c . On the other hand, defectors, whom mayonly receive a benefit b from cooperators, and whom neither pay a cost nor distribute anybenefit, end up having a payoff equal to b . These payoffs are summarized into the strategypayoff matrix A = (cid:20) b − c − cb (cid:21) . At time t we have that x ( t ) + x ( t ) = 1, where x ( t ) denotes the density (fraction orproportion) of cooperators in the well-mixed population, while x ( t ) = 1 − x ( t ) denotesthe density of defectors. The fitness vector f stores the expected fitness for each strategy(cooperation and defection), and results from the matrix-vector multiplication(1) f = Ax = (cid:20) b − c − cb (cid:21) (cid:20) x x (cid:21) = (cid:20) ( b − c ) x − cx bx (cid:21) . In other words, the fitness of the strategy cooperation is f = ( b − c ) x − cx , whilethe fitness of the strategy defection f = bx . By defining the average fitness as follows φ = x f + x f we can write the replicator equations for the Prisoner’s Dilemma game[31]: dx dt = x ( f − φ ) = − cx x (2) dx dt = x ( f − φ ) = cx x (3) RISONER’S DILEMMA ON REAL SOCIAL NETWORKS: REVISITED 3
This system supports a stable equilibrium, such that ( x , x ) → (0 , u = 1 − x . Thereduced replicator equation becomes du/dt = − cu (1 − u ) = F ( u ). Clearly, u = 0 is a stableequilibrium because F (cid:48) (0) = − c < c > naturalselection favors defectors over cooperators [31]. Figure 1.
Numerical solutions to the replicator equations for the Prisoner’sDilemma. Densities of cooperators x ( t ) (solid curve) and defectors x ( t )(dashed curve) are displayed versus time t . Initial conditions and parametervalues: x (0) = 0 . x (0) = 0 . b = 1 . c = 0 . x ( t ), while the dashed curve represents the density of defectors x ( t ). For thisparticular numerical solution, we set b = 1 . c = 0 .
3. Also, we started this simulationwith 95 % of cooperators and only 5 % defectors. As can be seen, the density of cooperatorsapproaches zero as time progresses (i.e., ( x ( t ) , x ( t )) → (0 ,
1) as t → Prisoner’s Dilemma in Social Networks
In this section we consider a population of individuals who may engage in a decision-making scheme equivalent to Prisoner’s Dilemma. In fact, social connections by means ofacquaintanceship, friendship, or levels of influence that can factor in decision-making are
SHARON M. CAMERON AND A. CINTR ´ON-ARIAS modeled with an undirected graph (network) , where each vertex (node) represents anindividual and an edge (link) denotes potential social ties [17].A social network provides a landscape where each node plays one of two strategies,cooperation or defection, and at each time step nodes decide whether to switch to a newstrategy or keep playing the same. The key for these decisions is the payoff per-node,which is now a space and time dependent quantity. All nodes connected to a node say i ,form its neighborhood, say Ω i . To compute the payoff of a node one needs to account forall pair interactions (cooperator-cooperator, cooperator-defector, defector-cooperator anddefector-defector) happening in the node’s neighborhood. The strategy played by node i is denoted with a binary vector v i defined as v i = (cid:20) (cid:21) if node i is cooperator (cid:20) (cid:21) if node i is defector . The payoff of node i at time t is given by(4) P ( i, t ) = (cid:88) j ∈ Ω i v iT Av j , where A = (cid:20) b − c − cb (cid:21) denotes the strategy payoff matrix. The fitness of a node is thepayoff re-scaled by an intensity of selection parameter w , such that 0 < w <
1. When w → w → i at time t is defined as follows: F ( i, t ) = 1 − w + wP ( i, t ) , where the functional form of F is known as linear fitness (see [20, 33]).The time evolution of Prisoner’s Dilemma in a social network of players is subject to anupdating rule. In this study we considered the so-called “death-birth” updating [33]: ateach time step a node is chosen uniformly at random (unbiased) to die and its neighborscompete proportional to their fitness. Once this dying node is determined it becomestemporarily empty. This action may also be seen not necessarily as an actual death of thatmember of the social network, but rather as if that node becomes a free-agent and is opento be persuaded into playing other strategies. The neighbors of this empty node competefor it, meaning that the persuasion is proportional to their fitness. Fitness is computed foreach node in the neighborhood of the empty node (the empty node has to be excluded fromthe neighborhoods of each of node linked to it because it has no strategy for time being),then the aggregate fitness for each strategy is calculated. By aggregate fitness we mean thetotal fitness of nodes playing cooperation and that of those playing defection. The emptynode decides which strategy to play in the next time step in proportion to the aggregate The words graph and network, vertex and node, and edge and link will be used interchangeably.
RISONER’S DILEMMA ON REAL SOCIAL NETWORKS: REVISITED 5 fitness of cooperation and defection. (See appendix for additional details in pseudo codeform.)The ratio of benefit to cost serves as a threshold quantity that determines persistenceof cooperation. When this ratio is compared to the average degree of the network, averagenumber of edges per node, denoted by (cid:104) k (cid:105) , one obtains that(5) bc > (cid:104) k (cid:105) is a necessary condition for selection to favor cooperation. This threshold result is derivedfrom combining pair approximations and diffusion approximations [33], where the fixationprobability of a strategy is calculated. This latter quantity represents the probability thata single player of a strategy (either cooperation or defection) which starts in a uniformlyat random position in the network (unbiased), then gives rise to a lineage of players of thesame strategy, invading the whole population (see supplemental materials of [33]).In contrast to the well-mixed case, where for any values b, c > x ( t ) → t gets large, for populations withstructure, such as those with social network ties, it is seen that cooperation is not doomed tobe outcompeted. Clusters of cooperators can persist, provided some conditions are satisfied(with death-birth update using aggregate fitness per strategy and when b/c > (cid:104) k (cid:105) ). In thisstudy we intend to illustrate this feature using both synthetic data and a dataset of a realsocial network. The former are generated using Watts-Strogatz algorithm for small-worldnetworks, to be discussed in the next section.4. Models of Social Networks: Small-World Phenomenon andWatts-Strogatz Network Model
Imagine we consider the following conditions for an experiment on a social network.Randomly selected seed individuals are asked to forward a letter with the ultimate goalof reaching a target recipient who resides in Sharon, Massachusetts. Even though seedindividuals are given the name, address, and occupation of the target person, they arerequired to only pass the letter along to someone in their circle of acquaintances that theyknow by their first-name. S. Milgram [27] was the designer of this experiment which resultedin measuring the average number of intermediaries in these forwarding-letter chains: onaverage it took six individuals from seed to target for the letter to arrive in Sharon, MA(see [27] and chapter 20 of [14]).This became known as the “small-world phenomenon” and it speaks to structural prop-erties of networks, where distance between nodes is measured in terms of edges. Moreprecisely, paths are the concatenation of edges that connect a seed node to a targetnode, the discovery of Milgram’s experiment would translate in saying that on averagethe forwarding-letter paths consisted of six edges, indeed a short path [27, 43, 44].Watts and Strogatz [43] proposed a model to construct families of networks with shortpaths, while also keeping track of an additional feature called clustering. The latter refersto the existence of close triads or triangles, which denotes the ability of neighbors ofneighbors to also be connected to each other by means of homophily (nodes connecting to
SHARON M. CAMERON AND A. CINTR ´ON-ARIAS (a)$ p = . p = . p = . Density of cooperators at stopping time (b)$ p=0.0
Density of cooperators at stopping time F r equen cy p=0.1 Density of cooperators at stopping time F r equen cy p=1.0 Density of cooperators at stopping time F r equen cy (c)$ Figure 2.
Prisoner’s Dilemma on small-world networks obtained withrewiring probability values set to p = 0 . p = 0 .
1, and p = 1 .
0. Panel(a) displays the density of cooperators versus time, for 10 realizations withstopping time equal to 7 . × . Panel (b) portrays boxplots, while Panel(c) displays frequency histograms, of the density of cooperators with stop-ping time of 1 . × , for 100 realizations. Network and game parametervalues: n = 1000, (cid:104) k (cid:105) = 4, b = 1 . c = 0 . w = 0 . RISONER’S DILEMMA ON REAL SOCIAL NETWORKS: REVISITED 7 other nodes who resemble themselves). Watts-Strogatz network model transitions betweentwo regimes: regular graphs, known to have high levels of clustering; and random graphs,known to have small characteristic path lengths. Gradual increments in the level of disorderare parametrized by a tuning quantity: the probability of rewiring existing edges in anetwork with a fixed number of nodes, denoted by p where 0 < p <
1. Commonly, Watts-Strogatz networks are referred to as small-world networks (additional details can be foundin [13, 14, 29, 43, 44] and references therein).Our interest in the small-world networks relies in using them as a theoretical controlgroup in the context of Prisoner’s Dilemma. More concretely, we are going to simulatePrisoner’s Dilemma using networks generated with the Watts-Strogatz algorithm. In thisway, we simulate social influence by means of small-world networks while the theoreticalgame evolves in time.
Parameter values.
Networks of size n = 1000 and average degree (cid:104) k (cid:105) = 4 wereemployed. The Prisoner’s Dilemma parameter values were chosen equal to those used inwell-mixed populations for illustrations purposes (see Figure 1): b = 1 . c = 0 .
3. Wedecided to set the intensity of selection to a medium level ( w = 0 .
5) between strong( w = 1) and weak ( w = 0) selection. Rewiring probabilities were set at three differentvalues: p = 0 . p = 0 . p = 1 . Initial conditions.
Simple Random Sampling (SRS) was used to determine initialconditions in the following sense. Nodes were initially set to be cooperators or defectorswithout preference (by means of SRS) due to degree, clustering, path length, or any othernetwork attribute. On the other hand, the number of nodes playing each strategy was cho-sen uniformly at random, with the only constraint that total population remains constantat n = 1000. Stopping time and realizations.
A stopping time of T = 1 × was employed. Atotal of 100 stochastic realizations of Prisoner’s Dilemma were carried out for a fixed valueof rewiring probability p . A network was drawn from Watts-Strogatz algorithm, with eachfixed value of p , which was kept static during time steps t = 1 through t = T (i.e., over thecourse of one stochastic realization of the theoretical game). Update rule.
A death-birth updating rule was implemented (see Section 3), such that b/c exceeds the average degree: b/c = 1 . / . (cid:104) k (cid:105) = 4. This necessary conditionfor the establishment of cooperation is precisely what we intend to validate with the presentstudy.Figure 2(a) displays a snapshot of the density of cooperators versus time, i.e., x ( t )versus t . For the sake of resolution only 10 realizations are displayed with time between t = 1 and t = 70000, where each discrete time step t represents a round of the gamebeing played. Left-side, middle, and right-side figures in Figure 2(a) depict time seriescorresponding to p = 0 . p = 0 .
1, and p = 1 .
0, respectively.Figures 2(b)–(c) summarize results of 100 realizations where the stopping time is T =1 × . In Figure 2(b) we find boxplots of x ( T ), while Figure 2(c) displays histogramsof x ( T ) for the three values of rewiring under consideration. Across these three typesof rewiring we observe a consistent unimodal shape of the distributions of 100 samples, SHARON M. CAMERON AND A. CINTR ´ON-ARIAS
Table 1.
Mean and five-number summary of density of cooperators atstopping time, for 100 realizations of Prisoner’s Dilemma on small-worldnetworks (see caption of Figure 2).Rewiring Minimum First Quartile Median Mean Third Quartile Maximum p = 0 . p = 0 . p = 1 . x ( T ) are specified in Table 1. Foreach value of p , the mean and median are fairly close to one another, such that theymatch when round off to one decimal digit. In a community with regular graph structure, p = 0 .
0, the median fraction of cooperators at stopping time is 0.7935 with interquartilerange (IQR) of 0.0475. Simulated communities with high levels of clustering and smallaverage distance between nodes, p = 0 .
1, exhibit a median of x ( T ) equal to 0.6805 withIQR equal to 0.0513. On the other hand, in communities with simulated random graphstructure, the median and IQR of x ( T ) are equal to 0.6595 and 0.0433, respectively. Thevalues of IQR in these three cases confirm what is observed in Figure 2(b)–(c), i.e., nostrong fluctuations in variability of x ( T ) samples is noticed.We now use the median to comment on average behavior of Prisoner’s Dilemma timeevolution, among simulated social networks (communities). It is well-known that Watts-Strogatz algorithm provides families of networks at p = 0 . x ( T ) drops substantially from p = 0 . p = 0 . . × − . On the other hand, while the median of x ( T ) decreases again from p = 0 . p = 1 .
0, it is not as drastically as in the previous case.The comparison of Prisoner’s Dilemma across two extremes of small-world networks, topversus bottom bloxplot in Figure 2(b), suggests the coverage of cooperators in the simulatedcommunities drops from 80% to 66%. In other words, structure plays a role in the finalnumber of cooperators at stopping time. In a more general sense, these boxplots in Figure2(b) confirm that clusters of cooperators persist in these simulated social networks overtime. The choice of stopping time at T = 1 × guarantees a burn-in phase. Longitudinaltrends of x ( t ) with t exceeding 5 × (not displayed here) assure a steady-state-likebehavior.A closer examination of the solid curve displayed in Figure 1, along with the realizationsof Figure 2(a), leads to compare Prisoner’s Dilemma on well-mixed communities versussmall-world networks. As it was discussed in Section 3, for social networks with b/c > (cid:104) k (cid:105) cooperators are not condemned to extinction, unlike in well-mixed populations. The same RISONER’S DILEMMA ON REAL SOCIAL NETWORKS: REVISITED 9 parameter values were used in the numerical solutions of Figure 1 and the simulations ofFigure 2(a): b = 1 . c = 0 .
3. We see in Figure 1 that after 25 rounds of the game,cooperators basically disappear in a well-mixed community, while Figure 2 illustrates asustained persistence of cooperators over time.5.
Dataset of a Social Network and Simulated Prisoner’s Dilemma
The first decade of the twenty-first century has seen the rise and establishment of readilyaccessible technology to communicate with others simply by hitting a key stroke in a mobiledevice, whether it is a laptop, a smartphone, or a tablet. The World Wide Web continuesto host the so-called “social networking sites” (SNS). These are the up-to-date versionsof forums that facilitate exchanges which are remarkably casual and informal, occurringremotely in real-time.According to Boyd and Ellison [5] SNS are web-based tools that accomplish three mainobjectives: (1) easy development of a profile with the option of making it public; (2)intuitive interface for constructing lists of users to connect with; (3) access to lists of userssharing a connection.Today, one of the well established SNS is Facebook , where users easily share personalinformation by means of photos, videos, and email. Facebook also facilitates surveyingopinions on topics of specific interest and it is known to even promote organization ofevents. In the early days Facebook membership was restricted to university affiliation. Inother words, it served as collegiate social networking site requiring users to have a validemail with an edu-suffix. It first launched at Harvard University in early 2004 and itgradually expanded to other universities. The email requirement made Facebook users feelexclusive because they had membership to a private community [5]. By September 2005,Facebook moved forward to integrate professionals working within corporate networks andhigh school students. However, Facebook did not allow its users to make their profilespublic to all users right away. This was a substantial difference relative to other SNS [5],and it meant that it preserved a strong sense of local community. Figure 3.
Network visualization of the largest connected component in asocial network dataset sampled at the California Institute of Technology[40]. A Facebook friendship between two users means there is a link connecting their profiles.Moreover, for these links to be established Facebook requires confirmation of a “friendshiprequest”. In this sense, Facebook friendships determine a network of users, in so manywords: a graph of undirected edges, where each node represents a Facebook user. Forexamples of social network analyses using this type of datasets see [22, 25, 41].The dataset employed here is a subset of those used by Traud, et al. [40], it consists ofa complete set of users and all the links between them occurring on September 2005 at theCalifornia Institute of Technology. Figure 3 displays a network visualization of the Caltechdataset, where nodes and links denote Facebook members and friendships, respectively.In their comprehensive analysis, Traud, et al. [40] quantify some of the basic networkcharacteristics of the Caltech dataset. For example, the network size is n = 1099 with only762 nodes belonging to the largest connected component. Moreover, there are 16651 edgeswithin the largest connected component. The average degree is (cid:104) k (cid:105) = 43 .
70, while the meanclustering coefficient is 0.41. Traud, et al. [40] point out that when comparing clustering,by two different measures, against the datasets of another four universities, the Caltechdataset has the largest clustering. In their study, Traud, et al. [40], one of their maingoals is to detect significant clusters of nodes (community structure), by using unbiasedalgorithms. They find the Caltech dataset has 12 communities. Using the Caltech dataset,we carried out Monte Carlo simulations of Prisoner’s Dilemma and below we give detailsof the implementation.
Parameter values.
Because the necessary condition b/c > (cid:104) k (cid:105) is at the central stageof this study, we decided to explore the ratio b/c as a linear function of the average degree (cid:104) k (cid:105) . In other words, for diagnostic tests we supposed that b/c = β (cid:104) k (cid:105) for some β ≥ β = 1 denotes a borderline case scenario. Values of β were considered in [1 . , . , t = 1, and stopping time T = 1 × , led to results displayed in Figure 4(a)for 10 realizations. It is seen in Figure 4(a) that the mean of x ( T ) is an increasingfunction of β , where β ∈ [1 , β = 1 it is seen that x ( T ) is above 0.1 (at least10% of the network remains playing cooperation), while for β ≥ x ( T ) is no lessthan 0.7 (more than 70% of cooperators remain in the network). Based on this diagnosticwe opted to set b/c = 3 . (cid:104) k (cid:105) (a value of β between 3 and 4): more specifically, we set b/c = (3 . . ≈ b = 150, c = 1. The value of intensity of selection was set at w = 0 .
5, halfway through weak and strong selection.
Initial conditions.
The effect of two types of initial conditions was also vetted. Weconsider the borderline case b/c = (cid:104) k (cid:105) and set the stopping time as T = 1 × . Figure 4(b)depicts results of 10 realizations. The top boxplot of the samples of x ( T ), corresponds tofixed initial conditions, i.e., where 50% of the nodes were initially set to be cooperators. Onthe other hand, the bottom boxplot corresponds to initial conditions determined by SimpleRandom Sampling (SRS), where the initial number of cooperators was chosen uniformlyat random between 1 and n = 1099. The choice of which nodes were initially set toplay cooperation was made independently of any network attributes. Comparison of themedian in these boxplots displayed in Figure 4(b) suggests cooperators reach very low levelsat t = T (but yet they are not extinct, at least on average), something that is expected in RISONER’S DILEMMA ON REAL SOCIAL NETWORKS: REVISITED 11 M ean den s i t y o f c oope r a t o r s Multiplier (a)$ S R S F i x ed Density of cooperators at stopping time (b)$
Figure 4.
Diagnostic tests of Prisoner’s Dilemma simulations on a datasetof a real social network [40]. Panel (a) displays mean density of cooperatorsat stopping time versus a multiplier β , where it is assumed b/c = β (cid:104) k (cid:105) for β ∈ { . , . , . , . , . , . , . , . , . , . , . } . Dashedcurve denotes variability: the mean plus minus one standard error. Theseaverages were computed out of 10 realizations with stopping time T =1 × . In Panel (b) boxplots of the density of cooperators at stoppingtime are depicted. The samples in these boxplots were obtained from 10realizations with stopping time T = 1 × and b/c = (cid:104) k (cid:105) . Two types ofinitial conditions were tested: Fixed initial conditions (top boxplot), wherehalf of the population were initialized as cooperator while the other halfwere set as defectors; Initial conditions by simple random sampling (bottomboxplot), where the initial number of cooperators was chosen uniformly atrandom between 1 and n .the borderline case b/c = (cid:104) k (cid:105) . Even though fixed initial conditions exhibit an outlier for x ( T ) samples, and some skewness, the variability remains substantially narrower in fixedversus SRS initial conditions. We opted for setting initial conditions by SRS to allow morevariability in the simulations outcome. Stopping time, realizations and updating rule.
A death-birth updating rule wasemployed (Section 3), while the stopping time was set as T = 1 × and 100 realizationsof Prisoner’s Dilemma were carried out using the Caltech dataset. Table 2.
Mean and five-number summary of density of cooperators atstopping time, obtained from Caltech dataset.Minimum First Quartile Median Mean Third Quartile Maximum0.2185 0.7055 0.8381 0.7881 0.9038 1.0000Panel (a) of Figure 5 displays only 10 (out of the 100 realization) curves of cooperatorsdensity, for the sake of enhanced resolution. There is clear evidence of patterns supportingpersistence of cooperation, as it is revealed in this subset of the 100 realizations.Another observed feature in Panel (a) is downward-spike temporal patterns, for a handfulof realizations. In other words, the density of cooperators in these cases drops remarkably,but it seems to return back to sustained levels. Similar patterns of drops in cooperationdensity have been reported before by Egu´ıluz, et al. [15] (see Figure 5), and by Kim, et al.[21] (see Figure 3(b)), albeit with different versions of Prisoner’s Dilemma.The histogram of samples of cooperators density at stopping time T = 1 × is displayedin Figure 5(b). Considerable skewness is observed, in comparison to small-world networks(see Figure 2(c)). Moreover, skewness is also confirmed by the boxplot in Figure 5(c),where a handful of outliers appear. The latter suggest low levels of sustained cooperation,but no necessarily extinction.The five-number summary and mean of x ( T ) are given in Table 2. As expected becauseof skewness the mean (0.7881) and median (0.8381) are distant from one another, relativeto the simulations on small-world networks (see Table 1). Also, the IQR of the samples is0.1983, implying IQR of the simulations with the Caltech dataset is one order of magnitudelarger than the IQR’s obtained with small-world networks.Since the median of x ( T ) is 0.8381 one concludes that, on average, clusters of coop-erators in the network make up at least 80% of the population, over the long run. Thisis considered a validation of b/c > (cid:104) k (cid:105) , as a necessary condition for the establishment ofcooperation in a social network [33]. Such validation against empirical data [40] is the maincontribution of this study. 6. Discussion
Some of the very first formulations of the theory of games surfaced during the first halfof the twentieth century, when von Neumann and Morgenstern [42], followed by Nash [28],seeded foundations for a new field of study.Prisoner’s dilemma was invented by Merrill Flood and Melvin Dresher at the RANDcorporation in 1950 [32]. Although its original formulation came from the point of viewof classical game theory, that is, with well-mixed populations. Consideration of popula-tion structure in Prisoner’s Dilemma was first conveyed with lattices or regular networks.For example, Nowak and May [30] proposed a purely deterministic version of Prisoner’sDilemma on a two dimensional lattice. This led to a system that was extremely sensitiveto initial conditions giving rise to fluctuations in the densities of cooperators and defectors
RISONER’S DILEMMA ON REAL SOCIAL NETWORKS: REVISITED 13 (a)$
Density of cooperators at stopping time F r equen cy (b)$ Density of cooperators at stopping time (c)$
Figure 5.
Prisoner’s Dilemma on a real social network, dataset sampledat California Institute of Technology [40]. Panel (a) displays 10 realizationsof the density of cooperators versus time with stopping time T = 1 × .Panel (b) depicts the histrogram while Panel (c) displays the boxplot ofsamples of cooperators density at stopping time, for 100 realizations ofPrisoner’s Dilemma. SRS initial conditions were used. Game parametervalues: b = 150, c = 1 and w = 0 .
5. This dataset has size n = 1099 andaverage degree (cid:104) k (cid:105) = 44 .
0. Game parameter values were chosen to ensure b/c > (cid:104) k (cid:105) [33]. on the lattice. In other words, their system supports spatial arrays that vary chaotically,having cooperation and defection shift in their sustained patterns [30].Regular lattices are often a good first approach while extending a dynamical modelto incorporate space. However, when the structure in the population is determined bysocial interactions, such as those maintained by players of an evolutionary game, theseregular graphs are limited descriptions. The role of social structure is better addressedby employing small-world networks [1, 2, 6, 7, 8, 10, 11, 12, 15, 16, 18, 19, 21, 24, 34, 36,37, 38, 39, 46, 47, 48, 49], heterogeneous networks [23, 33], and datasets of real networks[16, 40, 41].There is a continued interest in exploring Prisoner’s Dilemma on social networks withsmall-world properties. In their pioneer introduction to small-world networks, Watts &Strogatz [43] argued that as the fraction of rewired edges is increased, then it is less likelyfor cooperation to emerge (with a Tit-for-Tat updating rule). Moreover, Watts [44] ex-plains that networks with very shy levels of clustering tend to not enhance cooperation.Because the establishment of cooperation requires a critical mass of cooperators orches-trating against defectors, so that they optimize their fitness or payoff by cooperating witheach other. According to Watts [44], network shortcuts can enable a few defectors to break-through the seed of cooperators, leading to the eventual halt of the once sustained clusterof cooperation.On the other hand, small-world networks tend to favor cooperation under a regimeknown as strategy dynamics. Strategy dynamics is an approach in which an initial set ofupdating rules are assigned in the first round, and for the following rounds players maychoose to switch between, say for example, Generalized Tit-for-Tat and Copycat [44].For over a decade, efforts in exploring Prisoner’s Dilemma on small-world networksfootprints a growing literature. Here we comment on what we consider key citations, butwe invite the reader to consult an extended list of references [1, 2, 6, 7, 8, 10, 11, 12, 15,16, 18, 19, 21, 24, 34, 36, 37, 38, 39, 46, 47, 48, 49] and references therein.Even though several variations of Prisoner’s Dilemma (a common approach is to re-parametrize the payoff matrix, resulting in a matrix with only one parameter called thetemptation to defect) and its updating rule are considered, a distinct consistent messageis prevalent: cooperation can persist in small-world networks.For example, Abramson and Kuperman [1] argue that in small-world networks with anaverage degree of four, compact groups of cooperators are seen to persist. Moreover, longrange edges, by means of moderate values of the rewiring probability, favor cooperatorsas they start to reconnect, thus outcompeting defectors [1]. Tomochi [38] discusses howrandom connections (rewiring) enable breakthroughs of cooperation among clusters of de-fectors, leading to an unexpected scenario, where niches of defectors form and do not haveincentives to switch their strategy, thus imposing over cooperators. Hauert and Szab´o [19]use the ratio of cost to net benefit of cooperation as a parameter while exploring phasetransitions, between cooperation and defection, in models with network structure. Fur-thermore, clusters of cooperators persist with diffusion, that relocates these cooperatorsto other sites in a square lattice. Hauert and Szab´o [19] also note regular small-worldnetworks are even more favorable to sustained cooperation than square lattices. Perc [34] RISONER’S DILEMMA ON REAL SOCIAL NETWORKS: REVISITED 15 addresses the effects of extrinsic stochastic payoff functions, considered as spatio-temporalrandom variations in Prisoner’s Dilemma. Additionally, Perc [34] finds an optimal fractionof rewired edges supports noise-induced cooperation with resonance. Xia, et al. [48] em-ploy co-evolutionary small-world networks in a Prisoner’s Dilemma game and they find thatsocial structure collapses with avalanches, by attacking the best cooperator hubs. Theyargue that mutation of the wealthiest (as determined by payoff) cooperators may promotesustained cooperation on a large scale [48].Prisoner’s dilemma and social networks have been studied using samples of real data.Fu, et al. [16], analyze a dataset sampled from a Chinese social networking site, which it isdubbed the Xiaonei dataset. First, they compute the clustering coefficient and character-istic path length, and conclude this dataset has small-world properties. Second, Fu, et al.[16], explain that the evolution of cooperation in a Xiaonei dataset, is influenced by severalsocial network attributes, including: average connectivity, small-world effect, and degree-degree correlations. Their numerical simulations of Prisoner’s Dilemma on the Xiaoneidataset suggest cooperation is substantially promoted, whenever the temptation-to-defectparameter remains bounded, between 1.00 and 1.35.The contribution by Fu, et al. [16], shares similarities with this study. Because herewe also employ a dataset sampled from a social networking site along with simulations ofPrisoner’s Dilemma.This study was inspired mainly by the contributions of Ohtsuki, et al. [33], and Fu, et al.[16]. The former conveys the cooperation probability of fixation. That is, the probabilitythat a single cooperator, located in a random node of the network, in fact, converts the en-tire population from defectors into cooperators. A network of size n , according to [33], hasdefectors with a fixation probability below 1 /n and it has cooperators with a fixation prob-ability above 1 /n , provided that ratio of benefit to cost exceeds the average connectivity . Insymbols, we write b/c > (cid:104) k (cid:105) and note this condition is necessary for cooperators to be fa-vored by selection (this inequality is derived by applying pair and diffusion approximationsunder the assumption that n is considerably larger than (cid:104) k (cid:105) ). Another interpretation ofthe discovery found by Ohtsuki, et al. [33], is that natural selection promotes cooperation,with higher likelihood, when there are fewer connections.On the other hand, Fu, et al. [16], analyzed a dataset of a real social network. They em-ployed a sample of a friendship network, from a social networking site in China. Accordingto their simulations of Prisoner’s Dilemma, cooperation can reach as much as 80% of thenetwork, for a range of values of the temptation to defect parameter. Moreover, Fu, et al.[16], argued that degree heterogeneity is fundamental for the establishment of cooperationin friendship networks.Here we have confirmed that cooperation may persist among social networks, providedsome conditions are guaranteed. First, to draw a comparison, we simulate Prisoner’sDilemma on well-mixed populations and confirm that cooperation goes extinct regardlessof any values of benefit b and cost c . Then, to contrast the well-mixed scenario, we examinethe persistence of cooperation with simulated social networks and with a dataset of a realsocial network. Prisoner’s dilemma was studied in simulated networks between the twoextremes of small-world structures, that is, between regular graphs and random graphs, i.e., with rewiring p = 0 and p = 1, respectively. Cooperation keeps sustained levels in bothtypes of simulated social structures, with median levels of 80% in regular graphs and 66%in random graphs. The skewness evidenced in the boxplots of the samples of cooperatordensity, suggests that despite the fourteen percent drop in the median levels of sustainedcooperation, extinction is not a common occurrence. We must note that the simulations onwell-mixed and small-world populations were carried out using the same game parametervalues: b = 1 . c = 0 . w = 0 .
5. The average degree in the simulated networks wasset to (cid:104) k (cid:105) = 4, which means that b/c = 6 > (cid:104) k (cid:105) .Furthermore, cooperation persists among a real social network. The latter determined bya snapshot sample of a friendship network, in a collegiate social networking site, during itsearly days when there were domain restrictions for members [40]. Simulations evidencingcooperation persistence were carried out with parameter values that satisfied the condition b/c > (cid:104) k (cid:105) . This serves as a validation of the main result by Ohtsuki, et al. [33], againsta dataset of a real social network. In fact, the median of sustained cooperation reaches84% of the social network. Albeit some variability, it is clear that cooperation among thefacebook friendship network explored here draws a substantial contrast with a well-mixedpopulation.We end with a note on further potential future directions of social network analysisand game theory. More and more the field of mathematical epidemiology is integratingtechniques from evolutionary game theory, in the context of vaccination and behavioralchanges [4, 9, 35]. For example, those vaccinating on-time can be considered cooperators,while those who do not vaccinate can obtain the benefit of heard immunity, and may beconsidered defectors. Studies involving datasets of real social networks can shed some newlight, when considering a game theoretic approach to control epidemics. Acknowledgements
S. Cameron was funded by Talent Expansion in Quantitative Biology program (NationalScience Foundation grant DUE-0525447) to attend a two-day undergraduate workshop heldat the Statistical and Applied Mathematical Sciences Institute (SAMSI), October 29–30,2010. S.M Cameron also received funding through a Research Discovery position givenby ETSU Honors College, Summer 2011. Contributions to this work were made while A.Cintron-Arias was visiting SAMSI, these visits were sponsored by East Tennessee StateUniversity Presidential-Grant-in-Aid E25150, and by SAMSI Working Group DynamicsOn Networks.
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The initial conditions are the following. Suppose a network with n nodes is used tosimulate the Prisoner’s Dilemma. An integer number m , such that 1 ≤ m ≤ n , is sampleduniformly at random from (1 , n ). Thus, m nodes are selected uniformly at random in thenetwork and are set with strategy D , while all the other ones are set with strategy C .(1) Choose one dying node uniformly at random, say it is node i .(2) Compute the neighborhood of the dying node, say Ω i .(3) Compute the payoff and fitness of every node j ∈ Ω i .(4) Compute the aggregate fitness in Ω i for each strategy:(a) aggregate fitness of all C -players in Ω i , say F C .(b) aggregate fitness of all D -players in Ω i , say F D . RISONER’S DILEMMA ON REAL SOCIAL NETWORKS: REVISITED 19 (5) Let the empty site (dying node) adopt a strategy proportional to aggregate fitness.Suppose α = min( F C , F D ) and β = max( F C , F D ). Consider the following cases.(a) Case 1: α, β >
0. Sample y ∼ Uniform(0 , α + β ). If y ≤ α then the emptysite adopts the strategy associated with α , i.e., it adopts C if α = F C or D if α = F D . Otherwise the dying node adopts the strategy associated with β .(b) Case 2: α, β <
0. Sample y ∼ Uniform( α + β, β ≤ y < β . Otherwise it adopts the strategyassociated with α .(c) Case 3: α < β >
0. Sample y ∼ Uniform( α, β ). If α ≤ y <
0, then thedying node adopts the strategy associated with α . Otherwise it adopts thestrategy associated with β . E-mail address : [email protected] E-mail address : [email protected]@etsu.edu