The outbreak of cooperation among success-driven individuals under noisy conditions

Abstract

According to Thomas Hobbes' Leviathan [1651; 2008 (Touchstone, New York), English Ed], "the life of man [is] solitary, poor, nasty, brutish, and short," and it would need powerful social institutions to establish social order. In reality, however, social cooperation can also arise spontaneously, based on local interactions rather than centralized control. The self-organization of cooperative behavior is particularly puzzling for social dilemmas related to sharing natural resources or creating common goods. Such situations are often described by the prisoner's dilemma. Here, we report the sudden outbreak of predominant cooperation in a noisy world dominated by selfishness and defection, when individuals imitate superior strategies and show success-driven migration. In our model, individuals are unrelated, and do not inherit behavioral traits. They defect or cooperate selfishly when the opportunity arises, and they do not know how often they will interact or have interacted with someone else. Moreover, our individuals have no reputation mechanism to form friendship networks, nor do they have the option of voluntary interaction or costly punishment. Therefore, the outbreak of prevailing cooperation, when directed motion is integrated in a game-theoretical model, is remarkable, particularly when random strategy mutations and random relocations challenge the formation and survival of cooperative clusters. Our results suggest that mobility is significant for the evolution of social order, and essential for its stabilization and maintenance.

Fig. 1.

Representative simulation results for the spatial prisoner's dilemma with payoffs T = 1.3, R = 1, P = 0.1, and S = 0 after t = 200 iterations. The simulations are for 49 × 49 grids with 50% empty sites. At time t = 0 we assumed that 50% of the individuals were cooperators and 50% were defectors. Both strategies were homogeneously distributed over the whole grid. For reasons of comparison, all simulations were performed with identical initial conditions and random numbers (red, defector; blue, cooperator; white, empty site; green, defector who became a cooperator in the last iteration; yellow, cooperator who turned into a defector). Compared with simulations without noise (Upper), the strategy mutations of noise 1 with r = q = 0.05 not only reduce the resulting level of cooperation, but also the outcome and pattern formation dynamics, even if the payoff values, initial conditions, and update rules are the same (Lower). In the imitation-only case with M = 0 that is displayed on the left, the initial fraction of 50% cooperators is quickly reduced because of imitation of more successful defectors. The result is a “frozen” configuration without any further strategy changes. (A) In the noiseless case, a certain number of cooperators can survive in small cooperative clusters. (D) When noise 1 is present, random strategy mutations destroy the level of cooperation almost completely, and the resulting level of defection reaches values close to 100%. The illustrations in the center show the migration-only case with mobility range M = 5. (B) When no noise is considered, small cooperative clusters are formed, and defectors are primarily located at their boundaries. (E) In the presence of noise 1, large clusters of defectors are formed instead, given P > 0. The illustrations on the right show the case where imitation is combined with success-driven migration (here, M = 5). (C) In the noiseless case, cooperative clusters grow and eventually freeze (i.e., strategy changes or relocations do not occur any longer). (F) Under noisy conditions, in contrast, the cooperative clusters continue to adapt and reconfigure themselves, as the existence of yellow and green sites indicates.

Fig. 2.

Representative simulation results after t = 200 iterations in the “defector‘s paradise” scenario, starting with a single defector in the center of a cooperative cluster at t = 0. The simulations are performed on 49 × 49 grids with n = 481 individuals, corresponding to a circle of diameter 25. They are based on the spatial prisoner's dilemma with payoffs T = 1.3, R = 1, P = 0.1, and S = 0 and noise parameters r = q = 0.05 (red, defector; blue, cooperator; white, empty site; green, defector who became a cooperator; yellow, cooperator who turned into a defector in the last iteration). For reasons of comparison, all simulations were carried out with identical initial conditions and random numbers. (A–D) In the noisy imitation-only case with M = 0, defection (red) eventually spreads all over the cluster. The few remaining cooperators (blue) are due to strategy mutations. (E–H) When we add success-driven motion, the result is very different. Migration allows cooperators to evade defectors. That triggers a splitting of the cluster, and defectors end up on the boundaries of the resulting smaller clusters, where most of them can be turned into cooperators. This mechanism is crucial for the unexpected survival and spreading of cooperation.

Fig. 3.

Representative simulation results for the invasion scenario with a defector in the center of a cooperative cluster (“defector‘s paradise”). The chosen payoffs T = 1.3, R = 1, P = 0.1, and S = 0 correspond to a prisoner's dilemma. The simulations are for 49 × 49 grids with n = 481 individuals, corresponding to a circle of diameter 25 (red, defector; blue, cooperator; white, empty site; green, defector who became a cooperator; yellow, cooperator who turned into a defector in the last iteration). (Upper) Typical numerical results for the imitation-only case (M = 0) after t = 200 iterations: (A) for noise 1 (strategy mutations) with mutation rate r = 0.05 and creation of cooperators with probability q = 0.05; (B) for noise 2 (random relocations) with relocation rate r = 0.05; and (C) for noise 3 (a combination of random relocations and strategy mutations) with r = q = 0.05. Because cooperators imitate defectors with a higher overall payoff, defection spreads easily. The different kinds of noise influence the dynamics and resulting patterns considerably. Although strategy mutations in A and C strongly reduce the level of cooperation, random relocations in B and C break up spatial clusters, leading to a dispersion of individuals in space. Their combination in C essentially destroys both, clusters and cooperation. (Lower) Same for the case of imitation and success-driven migration with mobility range M = 5: (D) for noise 1 with r = q = 0.05; (E) for noise 2 with r = 0.05, and (F) for noise 3 with r = q = 0.05. Note that noise 1 just mutates strategies and does not support a spatial spreading, whereas noise 2 causes random relocations, but does not mutate strategies. This explains why the clusters in Fig. 3D do not spread out over the whole space and why no new defectors are created in Fig. 3E. However, the creation of small cooperative clusters is found in all 3 scenarios. Therefore, it is robust with respect to various kinds of noise, in contrast to the imitation-only case.

Fig. 4.

Spontaneous outbreak of prevalent cooperation in the spatial prisoner's dilemma with payoffs T = 1.3, R = 1, P = 0.1, S = 0 in the presence of noise 3 (random relocations and strategy mutations) with r = q = 0.05. The simulations are for 49 × 49 grids (red, defector; blue, cooperator; white, empty site; green, defector who became a cooperator; yellow, cooperator who turned into a defector in the last iteration). (A) Initial cluster of defectors, which corresponds to the final stage of the imitation-only case with strategy mutations according to noise 1 (see Fig. 2D). (B) Dispersal of defectors by noise 3, which involves random relocations. A few cooperators are created randomly by strategy mutations with the very small probability rq = 0.0025 (0.25%). (C) Occurrence of a supercritical cluster of cooperators after a very long time (see blue circle). This cooperative “nucleus” originates by random coincidence of favorable strategy mutations in neighboring sites. (D) Spreading of cooperative clusters in the whole system. This spreading, despite the destructive effects of noise, requires an effective mechanism to form growing cooperative clusters (such as success-driven migration) and cannot be explained by random coincidence. See the Movie S1 for an animation of the outbreak of cooperation for a different initial condition.

Fig. 5.

Representative example for the outbreak of predominant cooperation in the prisoner's dilemma with payoffs T = 1.3, R = 1, P = 0.1, S = 0, in the presence of noise 3 with r = q = 0.05. The simulations are for 49 × 49 grids with a circular cluster of defectors and no cooperators in the beginning (see Fig. 4A). (A) After defection prevails for a very long time (here, for almost 20,000 iterations), a sudden transition to a large majority of cooperators is observed. (Inset) The overall distance moved by all individuals during one iteration has a peak at the time when the outbreak of cooperation is observed. Before, the rate of success-driven migration is very low, while it stabilizes at an intermediate level afterward. This reflects a continuous evasion of cooperators from defectors and, at the same time, the continuous effort to form and maintain cooperative clusters. The graph displays the amount of success-driven migration only, whereas the effect of random relocations is not shown. (B) Evaluating 50 simulation runs, the error bars (representing 3 standard deviations) show a large variation of the time points when prevalent cooperation breaks out. Because this time point depends on the coincidence of random cooperation in neighboring sites, the large error bars have their natural reason in the stochasticity of this process. After a potentially very long time period, however, all systems end up with a high level of cooperation. The level of cooperation decreases with the noise strength r, as expected, but moderate values of r can even accelerate the transition to predominant cooperation. By using the parameter values r = 0.1 and q = 0.2, the outbreak of prevalent cooperation often takes <200 iterations.

Fig. 6.

Dependence of the fraction of cooperators for given payoff parameters T = 1.3 and R = 1 on the parameters P and S. The area above the solid diagonal line corresponds to the snowdrift game, the area below to the prisoner's dilemma. Our simulations were performed for grids with L × L = 99 × 99 sites and n = L²/2 individuals, corresponding to a density ρ = N/L² = 0.5. At time t = 0 we assumed 50% of the individuals to be cooperators and 50% defectors. Both strategies were homogeneously distributed over the whole grid. The finally resulting fraction of cooperators was averaged at time t = 200 over 50 simulation runs with different random realizations. The simulations were performed with noise 3 (random relocations with strategy mutations) and r = P = 0.05. An enhancement in the level of cooperation (often by >100%) is observed mainly in the area with P − 0.4 < S < P + 0.4 and P < 0.7. Results for the noiseless case with r = 0 are shown in Fig. S2a. The fraction of cooperators is represented by color codes (see the bar to the right of the figure, where dark orange, for example, corresponds to 80% cooperators). It can be seen that the fraction of cooperators is approximately constant in areas limited by straight lines (mostly triangular and rectangular ones). These lines correspond to Eq. 1 for different specifications of n₁, n₂, n₃, and n₄ (see main text for details). (B) The light-blue area reflects the parameters for which cooperators reach a majority in the imitation-only case with M = 0. For all payoffs P and S corresponding to a prisoner's dilemma, cooperators are clearly in the minority, as expected. However, taking into account success-driven migration changes the situation in a pronounced way: For a mobility range M = 1, the additional area with >50% cooperators is represented by dark blue, the further extended area of prevailing cooperation for M = 2 by green, and for M = 5 by yellow. If M = 5, defectors are in the majority only for parameter combinations falling into the red area. This demonstrates that success-driven migration can promote predominant cooperation in considerable areas, where defection would prevail without migration. For larger interaction neighborhoods m, e.g., m = 8, the area of prevalent cooperation is further increased overall (data not shown). Note that the irregular shape of the separating lines is no artifact of the computer simulation or initial conditions. It results by superposition of the areas defined by Eq. 1 (see A).