Average level of cooperation in dependence on for different values of .

<p>It can be observed that while imitating the best performing player in the swarm () might be beneficial at low temptations to defect, imitating personal success () is definitively better for the evolution of cooperation in strongly defection-prone environments. Each data point is an average of the final outcome (stationary state) of the game over independent realizations. Lines connecting the symbols are just to guide the eye.</p

Distribution of strategies in the whole population, as obtained for different combinations of and .

<p>It can be observed that for the nature of the stochastic strategy prisoner's dilemma game is essentially completely overridden by the selfish drive of players to reach the highest current payoffs in the swarm, in turn virtually completely transforming the game to its two-strategy [only (full defection) or (full cooperation) strategies are present in the population] version. Conversely, for the full spectrum of available strategies is exploited to arrive at the final stationary state. Note that the horizontal axis displays the willingness to cooperate (defining the strategy of every player), while the vertical axis depicts the probability that this strategy is present in the population. Depicted results are averages of the final outcome (stationary state) over independent realizations.</p

Characteristic spatial distributions of strategies, as obtained for different combinations of and .

<p>As concluded from results depicted in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0021787#pone-0021787-g002" target="_blank">Fig. 2</a>, for low values of only the two “extreme” strategies (with rare exceptions) are adopted, while for high values of the whole array of available strategies comes into play. Moreover, it is interesting to observe that values of yield the well-known clustering of cooperators <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0021787#pone.0021787-Nowak3" target="_blank">[34]</a> on the square lattice, while the snapshots for seem to have these feature somewhat less pronounced, although still clearly inferable (note that the distinction of clusters is somewhat difficult due to the continuous array of possible strategies). This suggests that, besides the clustering of cooperators, additional mechanisms may underlie the survival of cooperators at high temptations to defect and within the present setup. The color encoding, as depicted right, indicates the values of for each individual player.</p

Characteristic spatial distributions of velocities, as obtained for different combinations of and .

<p>Top row depicts results for , while bottom row features results for . Irrespective of , it can be observed that for the whole population essentially becomes a swarm in that the velocities of all players are much the same and close to zero. The fact that the prevailing velocity is close to zero simply reflects that the stationary state has been reached by means of adaptive, locally-inspired and slow strategy changes (which are, however, very effective even if the temptations to defect are strong). For , however, only isolated clusters can be considered to act as swarms, while the majority of players cannot be associated with any kind of group dynamics and is simply caught in the futile pursuit for the highest, yet for the majority unattainable, payoffs. These results indicate that swarming is an important agonist that promotes cooperation at high temptations to defect (see results presented in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0021787#pone-0021787-g001" target="_blank">Fig. 1</a>). The color encoding, as depicted right, indicates the values of for each individual player, where was chosen sufficiently large such that the stationary state of the game has been reached. Importantly, we note that for the stationary state has in fact been reached, although at a given instance in time the average velocity in the population might be different from zero.</p

Equilibrium level of cooperation as predicted by the analytical pair approximation method.

<p>As before, the degree of cooperation achieved in a Prisoner's Dilemma game (PDG, left) and in a Snowdrift game (SDG, right) is shown as a function of the benefit of cooperation and the incongruence between the interaction and the learning network. Since the pair approximation method is based on scenario 2, the panels should be compared with the simulation results shown in the upper left panels of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090288#pone-0090288-g002" target="_blank">Figs 2</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0090288#pone-0090288-g003" target="_blank">3</a>, respectively.</p

Diagrams illustrating a potential learning event.

<p>In (a) the focal individual learns from an individual that is part of 's interaction network (). Since , both and have three other interaction partners, whose strategy ( or ) is indicated by , , and , , , respectively. In (b) learns from an individual that does not belong to 's interaction network (). Now both and have four different interaction partners.</p

Fraction of new links that are assigned to cooperators in dependence on the temptation to defect for different values of .

<p>It can be observed that the higher the temptation to defect , the lower the fraction of new links that are received by cooperators. As by results presented in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0026724#pone-0026724-g001" target="_blank">Fig. 1</a>, it can be concluded that intermediate values of are optimal for cooperators to expand their neighborhoods, although as before, here too this depends somewhat on the temptation to defect . Altogether, this leads to the conclusion that who (either cooperators or defectors) obtains the new links is crucial for the successful evolution of cooperation. Presented results are averages over 100 independent realizations obtained with the system size and . Lines connecting the symbols are just to guide the eye.</p

Degree of cooperation achieved in a Prisoner's Dilemma game (PDG, left) and in a Snowdrift game (SDG, right) as a function of the benefit of cooperation and the incongruence between the interaction and the learning network.

<p>The simulation are based on scenario 1, where interaction and learning network overlap and both are random-regular networks with degree .</p

Frequency of cooperation achieved in a Prisoner's Dilemma game as a function of the benefit of cooperation and the incongruence between the interaction and the external learning network.

<p>The simulations are based on scenario 2. Both networks can either be random-regular or scale-free. Cooperation is strongly favoured when the interaction network is scale-free (bottom row) and weakly favoured when the external learning network is scale-free (right column).</p

Fraction of cooperators in dependence on the temptation to defect for different values of .

<p>It can be observed that intermediate values of are optimal for the evolution of cooperation, albeit this depends somewhat on the temptation to defect . Presented results are averages over 100 independent realizations obtained with the system size and . Lines connecting the symbols are just to guide the eye.</p