Noise induced enhancement of network reciprocity in social dilemmas
aa r X i v : . [ n li n . AO ] M a r Noise induced enhancement ofnetwork reciprocity in social dilemmas
Gui-Qing Zhang a ‡ , Qi-Bo Sun b † , Lin Wang c ∗ a Department of Physics, College of Science, Nanjing Forestry University,Nanjing 210037, China b Department of Resource and Environmental Science, School of Agriculture and Biology,Shanghai Jiao Tong University, Shanghai 200240, China c Adaptive Networks and Control Laboratory, Department of Electronic Engineering,Fudan University, Shanghai 200433, ChinaE-mail: † [email protected]; ∗ [email protected]; ‡ [email protected]; Abstract
The network reciprocity is an important dynamic rule fostering the emer-gence of cooperation among selfish individuals. This was reported firstly inthe seminal work of Nowak and May, where individuals were arranged onthe regular lattice network, and played the prisoner’s dilemma game (PDG).In the standard PDG, one often assumes that the players have perfect ra-tionality. However, in reality, we human are far from rational agents, as weoften make mistakes, and behave irrationally. Accordingly, in this work, weintroduce the element of noise into the measurement of fitness, which is de-termined by the parameter α controlling the degree of noise. The considerednoise-induced mechanism remarkably promotes the behavior of cooperation,which may be conducive to interpret the emergence of cooperation withinthe population. Keywords:
Noise, Fitness, Prisoner’s Dilemma Game, Cooperation,Imitation
1. Introduction
The emergence and maintenance of cooperation among selfish individu-als is a ubiquitous phenomenon extensively presenting in various complexsystems, from human or animal societies to self-replicating chemical or bio-logical systems [1, 2]. To explain this question, the evolutionary game theory
Preprint submitted to Chaos, Solitons & Fractals September 24, 2018 reates a universal theoretical framework that has been widely studied bydiverse disciplines over the past decades [3, 4, 5, 6]. In particular, the evo-lutionary prisoner’s dilemma game (PDG) has attracted numerous attentionfrom broad theoretical issues to specific experimental ones, as it reflects thesocial conflict among the independent and selfish people in a simple butaccurate way [7, 8, 9, 10, 11, 12, 13].In a typical PDG, two involved players simultaneously make the choice:cooperate or defect. Each of them will receive a reward R if they bothcooperate, and a punishment P if both defect. If a player defects while theopponent chooses to cooperate, the former one receives a temptation T , whilethe latter loser receives a sucker’s payoff S . The ranking of these four payoffsis T > R > P > S . This implies that players are more prone to defect ifthey both wish to maximize their own payoff, regardless of the opponent’sdecision. The resulting is a social dilemma inducing the widespread defection,which is however inconsistent with the fact that the cooperative and altruisticbehavior is widely observable in reality. To answer this puzzle, multifariousmechanisms have been proposed, such as reward and punishment [14, 15,16, 17, 18], voluntary participation [19, 20], spatially structured population[21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], heterogeneity ordiversity [37], the mobility of players [38, 39, 40, 41], age structure [43, 44, 45],to name but a few.The network reciprocity is a well-known dynamical rule that fosters theprevalence of cooperation [46]. It says that if the game players are arrangedon a network, where the individuals occupy the network nodes and the linksdetermine who interacts with whom, the cooperators can shape compactclusters to prevent the invasion of defectors [47, 48, 49, 50]. Thus the numberof cooperators will preserve at a high level. Nowak and May [51] proposedthe first model of the networking PDG, where the players were located on thesquare lattices. At each round of the dynamical process, players first gatheredtheir own payoffs via the neighboring interactions based on the regulation ofPDG. Then per player had a chance to adopt the strategy of his neighbors,if they had higher fitness. By simply introducing the interaction structureinto the consideration, the cooperative behavior can be prevalent.In the standard PDG, it is assumed that the players have perfect rational-ity. However, from the perspective of behavior economics, we human are farfrom rational calculators, as we often make mistakes, and behave irrationally[52, 53]. Therefore, at each round of the game, players may not have theability to obtain their payoff exactly, due to the presence of noise deriving2rom the fact such as the error of observation. By concerning the diversity ordiscrepancy of people [54, 55, 56, 57], the degree of noise for different playersmay have some heterogeneity. With the above arguments, we extend the tra-ditional PDG by introducing the element of noise into the definition of fitnessof individuals. We assume that the noise among different players are inde-pendent. During the strategy updating stage at each round of the game, thepresence of noise impacts the likelihood of strategy switching for each playeraccording to the imitation dynamics. To compare with the standard PDG,we still arrange the players on regular lattice networks following Ref. [51].We extensively perform the computer simulations to elucidate the impact ofdifferent levels of noise. In the remainder of this paper, we first specify ourmodified model of PDG; subsequently, we present the main results; and atlast, we will summarize the conclusions.
2. Model
Let us consider an evolutionary prisoner’s dilemma game, where the play-ers occupy the nodes of a regular L × L square lattice network with theperiodic boundary condition. Each player (each node), e.g. x , is initiallyassigned as either a cooperator ( s x = C ) or a defector ( s x = D ) with theequal probability. In the standard PDG governed by the imitation dynamics,a player updates his strategy according to the following rules: At each round,the player x performs the PDG with each of his four nearest neighbors, bywhich he gathers the resulting gains as his total payoff or fitness. Then herandomly selects one neighboring player y , and measures the difference oftheir fitness at this round, by which he decides whether changes his ownstrategy with a probability based the on the Fermi function W ( s y → s x ) = 11 + exp[( p x − p y ) /K ] , (1)where K is the intensity of selection [58], and p x , p y are the fitness of players x, y , respectively. As mentioned above, in practice, players may not measuretheir fitness exactly, due to the presence of noise deriving from many reasonssuch as the error of observation. Here we use the parameter µ to reflect theinfluence of noise, with α controlling the degree of noise. We define µ = αχ, (2)3here χ is a uniformly distributed random number in the interval [-1,1].Following the notation suggested in Refs. [49, 59], we utilize the rescaledpayoff matrix: the temptation to defect T = b (the highest payoff received bya defector playing with a cooperator), reward for mutual cooperation R = 1,and both the punishment for mutual defection P and the sucker’s payoff S (the lowest payoff required by a cooperator encountering a defector) is equalto 0, P = S = 0. The condition 1 ≤ b ≤ x acquires his total payoff p x via interacting with his nearestneighbors. With the influence of noise, he evaluates his fitness accordingto the relation F x = (1 + µ ) p x . To some extent, this setup is similar toa previously studied model [45]. Additionally, compared with the effects ofnoise considered in works [60, 61, 62, 63], we mainly consider this factor fromthe viewpoint of whole population. Next, by randomly choosing one neighbor y (his fitness F y is measured in the same way as x ), the focal player x adopts y ’s strategy with the probability W ( s y → s x ) = 11 + exp[( F x − F y ) /K ] . (3)During one full Monte Carlo step (MCS), each player has a chance to up-date his strategy according to the above procedure. Starting from a randominitial state, this evolutionary process proceeds until the system reaches thestationary state, at which we record the key statistics: the average densityof cooperator ρ C in dependence on parameters K , b , and the connectivitystructure.The simulation results are mainly obtained by implementing the evolu-tionary dynamics on the square 100 ×
100 lattice network. The density ofcooperators ρ C is measured by averaging the last 10 full steps of the overall2 × MCS. To overcome the impact of randomness, final results have beenaveraged over 40 independent runs for each set of parameters.
3. Simulation Results
We start by visualizing the spatial distribution of cooperators and defec-tors at the equilibrium with four typical values of the parameter α . Figure1 illustrates the results acquired with b = 1 . K = 0 .
1. As shown in theupper left panel, the cooperators vanish finally when α = 0, which conforms4o what is expected in the standard model [51]. When α >
0, the cooperatorsbegin mushrooming. Strikingly, when α is large enough, the cooperators willprevail in the system with a negligible number of defectors (see the bottomright panel). This implies that the cooperation behavior can be remarkablypromoted by increasing the value of α . a bc d Figure 1: (
Color Online ) Snapshots of the spatial distribution of cooperators and defectorsat the stable states with four typical values of α . Each site corresponds to a player. Thecooperators C are represented by the red color, and the defectors D are denoted by green.From panel (a) to (d), α = 0 , . , . , and 0 .
75, respectively. We fix b = 1 .
10 and K = 0 . To quantify the impact of each parameter on the behavior of cooperationin detail, we first measure the density of cooperators ρ C in dependence on thetemptation b for different values of α . Figure 2(a) clearly elucidates the factthat increasing α promotes the emergence of cooperation. It is worth men-tioning that the critical temptation value b = b c , which pinpoints the extinc-tion of cooperators if b > b c , increases with the growth of α . The monotonousincrease of b c with α is shown in Figure 2(b). This noise induced enhance-5ent of the survivability of cooperation on square lattice network raises aquestion about the universality of this mechanism. Fortunately, the quali-tatively consistent phenomenon is maintained on other types of interactionnetworks. For instance, in Figure 3, we report the results on a triangle latticegraph with several typical values of α . It is evident that the positive valuesof α enhance the emergence of cooperation. The only difference lies in thefact that the specific values of b c on the triangle lattice graph are a littlesmaller than what we can expect on the square lattice network. This impliesthe potential of the noise-induced mechanism on promoting the evolution ofcooperation. ( a ) ( b ) Figure 2: (
Color Online ) (a) The density of cooperators ρ C in dependence on the param-eters α and b on the regular square lattice network. (b) Phase diagram of the thresholdvalues of the temptation b = b c , marking the extinction of cooperators if b > b c , as theparameter α gradually increases. We fix K = 0 . As the form of the Fermi function also allows the players to make irra-tional decision (e.g., the focal player can adopt the strategy of his neighborseven if F x > F y ), we also study the impact of parameter K , which char-acterizes the intensity of selection at the strategy imitation stage. Figure4(a)(b) show the density of cooperators ρ C in dependence on the parameters α and K on the regular lattices for b = 1 .
06 (a) and 1 .
10 (b), respectively.In general, the density of cooperators ρ C decreases with the augment of K (except for the scenario with α = 0 , b = 1 .
06 and
K < .
3, also see [64, 65]for comparison). As the increase of K leads to that exp[( F x − F y ) /K ] →
0, itis more probable for players to make irrational option, inducing the decreaseof ρ C .To explain the considered mechanism, we report the time courses of ρ C for6 .0 1.1 1.2 1.30.00.20.40.60.81.0 C b =0 =0.25 =0.50 =0.75 =1.0 Figure 3: (
Color Online ) The density of cooperators ρ C in dependence on the parameter α and b on the triangle lattice network. We fix K = 0 . ( a ) ( b ) Figure 4: (
Color Online ) (a) The density of cooperators ρ C in dependence on the param-eters α and K on the square lattice network for b = 1 .
06. (b) The density of cooperators ρ C in dependence on the parameters α and K on the square lattice network for b = 1 . several typical values of α on the square lattice network. Figure 5 illustratesthe results obtained with b = 1 .
10 and K = 0 .
1. The cooperators die outwhen α = 0. By gradually increasing the values of α , the stationary stateis a mixed C + D phase, where the defectors still occupy a large portion of7he network when α < .
5. The cooperators begin prevailing after α > . α ≥ .
75. Interestingly, as timeevolves, there is always an early stage that defectors exploit the ground ofcooperators. The smaller the value of α , the more serious the defectors invadeat this stage. However, the cooperators will be aroused after this stage, andengage in beating the opponents. C t =0 =0.25 =0.5 =0.75 =1.0 Figure 5: (
Color Online ) The time courses specifying the evolution of cooperation on thesquare lattice network with several typical values of α . We fix b = 1 .
10 and K = 0 . At last, we measure the average payoff of cooperators and defectors onthe square lattice network. The total payoff of any player, e.g., i , is the sumafter he interacts with all the four neighbors, which is written as p i = X j ∈ Ω i φ Ti ψφ j . (4)where Ω i denotes the set of neighbors of i , and ψ is the payoff matrix. φ refers to the specific strategy adopted by each player, i.e., φ = (1 , T forcooperators and φ = (0 , T for defectors. < P C > is defined as the averagepayoff of all cooperators, while < P D > is the average payoff of all defectors.As seen in Figure 3, one can find that < P C > and < P D > both increasewith α when 0 . < α < .
64. Before α > . < P C > = 0, while after α > . < P D > = 0. Thus it is clear why the considered mechanism canpromote the emergence of cooperation, although players do encounter thesocial dilemmas. 8 .0 0.2 0.4 0.6 0.8 1.001234 a v e r a g e p a yo ff < P C > < P D > Figure 6: (
Color Online ) The average payoffs under different values of α . < P C > isdenoted by red, < P D > is denoted by green. We fix b = 1 .
10 and K = 0 .
4. Conclusion
In sum, by introducing the element of noise into the measurement offitness in the prisoner’s dilemma game, our modified model remarkably en-hances the emergence of cooperation. We find that the introduction of noisenot only serves as a promising mechanism for the evolutionary dynamics onthe square lattice network but also for that of the triangle lattice graph.Although the defectors can obtain more payoff at the initial stage, the fastexploitation and thus the shortage of cooperators weakens the advantage ofdefectors gradually. The remaining cooperators will form compact clusterspreventing the invasion of defectors, and also engage in beating the oppo-nents.
Acknowledgements
Gui-Qing Zhang acknowledges partial support from the National NaturalScience Foundation of China (Grant No.11247217). Qi-Bo Sun was sup-ported by the SJTU Student Innovation and Practice Project. Lin Wangalso acknowledges the partial support by Fudan University Excellent Doc-toral Research Program (985 Project).9 eferenceseferences