Figure 1

(Color online) Asymptotic density of cooperators $x^{\ast }$ in a complete network with the replicator rule as update rule, starting with different initial densities of cooperators $x^0$, of values (a) 1/3, (b) 1/5, and (c) 2/3. Each unit square corresponds to a game: Harmony upper-left, Prisoner’s Dilemma lower-right, Stag Hunt lower-left, Snowdrift upper-right. The outcome is the same as that of the standard replicator dynamics on a well-mixed population (see text). Notice that Stag Hunt games are the only ones whose outcome depends on the initial condition, because of their bistable character. The values of the global measure of cooperation, obtained as the mean value over each game region, are shown next to each game square (above in the case of Harmony and Snowdrift games, and below for Stag Hunt games and Prisoner’s Dilemmas).Reuse & Permissions

Figure 2

Asymptotic density of cooperators $x^{\ast }$ in a square lattice with degree $k=8$, when the game is the Prisoner’s Dilemma defined by Eq. (8) $\left(ϵ=0\right)$, as studied by Nowak and May [6] (update rule: unconditional imitation). Note that the outcome with replicator dynamics on a well mixed population is $x^{\ast }=0$ for all the displayed range of the temptation parameter $T$. Notice also the singularity at $T=1.4$, with surrounding points located at $T=1.3999$ and $T=1.4001$ (see Fig. 7 and related discussion in Sec. 4). Lines are a guide to the eyes.Reuse & Permissions

Figure 3

Asymptotic density of cooperators $x^{\ast }$ in a square lattice with degree $k=8$, when the game is the Snowdrift defined by Eq. (9), as studied by Hauert and Doebeli [9] (update rule: replicator rule). Lines are a guide to the eyes. The result for a well mixed population is displayed as reference as a dashed line.Reuse & Permissions

Figure 4

Asymptotic density of cooperators $x^{\ast }$ in a square lattice with degree $k=8$, when the game is the Snowdrift defined by Eq. (9), as studied by Sysi-Aho and co-workers [26] (update rule: best response). Lines are a guide to the eyes. The result for a well mixed population is displayed as reference as a dashed line. Note how the promotion or inhibition of cooperation does not follow the same variation as a function of $T$ that in the case studied by Hauert and Doebeli [9] (Fig. 3).Reuse & Permissions

Figure 5

(Color online) Asymptotic density of cooperators $x^{\ast }$ in homogeneous random networks (upper row, A to C) compared to regular lattices (lower row, D to F), with degrees $k=4$ (A, D), 6 (B, E) and 8 (C, F). The update rule is the replicator rule and the initial density of cooperators is $x^0=0.5$. The values of the global index of cooperation in the upper row display the weak effect of homogeneous random networks [compare with Fig. 1b], which is opposite in Stag Hunt (lower-left square) and Snowdrift games (upper-right square). Comparing both rows, the differences are specially significant in Stag Hunt for $k=6$ and 8, revealing the strong promotion of cooperation caused by regular lattices with large clustering (see text). The influence on Harmony and Prisoner’s Dilemma games is negligible in all cases.Reuse & Permissions

Figure 6

(Color online) Asymptotic density of cooperators $x^{\ast }$ in homogeneous random networks (A to C) compared to regular lattices (D to F), with degrees $k=4$ (A, D), 6 (B, E) and 8 (C, F). The update rule is the Moran rule and the initial density of cooperators is $x^0=0.5$. With this update rule the effect is very small, in comparison with the other stochastic rules (see Fig. 5). In any case, the most relevant effect, albeit much weaker than in the other cases, is again a promotion of cooperation in Stag Hunt games with regular lattices (D to F).Reuse & Permissions

Figure 7

(Color online) Asymptotic density of cooperators $x^{\ast }$ in homogeneous random networks (upper row, A to C) compared to regular lattices (lower row, D to F), with degrees $k=4$ (A, D), 6 (B, E) and 8 (C, F). The update rule is unconditional imitation and the initial density of cooperators is $x^0=0.5$. Cooperation is fostered in Stag Hunt games in all cases. As in Fig. 5, the differences between homogeneous random networks and regular lattices of the same degree are significant only when the regular lattices have large clustering ($k=6$ and 8). Note the nonmonotonicity of the results, compared with the other update rules (see text). With this update rule the promotion of cooperation clearly extends to Snowdrift and Prisoner’s Dilemma games.Reuse & Permissions

Figure 8

(Color online) Asymptotic density of cooperators $x^{\ast }$ in regular lattices (left column) compared to small-world networks (right column), all with degree $k=8$. The update rules are: replicator rule (first row), multiple replicator rule (second row), Moran rule (third row), and unconditional imitation (fourth row). The initial density of cooperators is $x^0=0.5$. Notice that the panels A, E, and G are the same as, respectively, the panels 5 F, 6 F and 7 F, repeated here to facilitate the comparison. The evolutionary outcomes are practically identical for all the update rules, showing that the effect of small-world networks on the asymptotic state is due to the high clustering, also present in the regular lattices used to generate them. The probability of reshuffling used to generate the small-world networks was $p=0.01$ (see text).Reuse & Permissions

Figure 9

(Color online) Asymptotic density of cooperators $x^{\ast }$ in regular lattices of degree $k=8$, with synchronous update (left column) compared to asynchronous (right column). The update rules are: replicator rule (first row), multiple replicator rule (second row), Moran rule (third row) and unconditional imitation (fourth row). The initial density of cooperators is $x^0=0.5$. As in Fig. 8, the panels A, E and G are the same as, respectively, the panels 5 F, 6 F and 7 F. With the replicator and multiple replicator rules the evolutionary outcome is very similar, whereas some differences appear in the case of the Moran rule. With unconditional imitation the results are also quite similar, but there are differences for some points, specially those in the Snowdrift square with $S\lesssim 0.3$ and $T>5/3\approx 1.67$. The particular game studied by [36], which reported an inhibition of cooperation due to the asynchronous update, belongs to this region.Reuse & Permissions

Figure 10

(Color online) Time evolution of the density of cooperators $x$ in regular lattices of degree $k=8$, for typical realizations of Stag Hunt (left column) and Snowdrift games (right column). The update rules are the replicator rule (upper row) and unconditional imitation (lower row). The realizations with synchronous update are represented by continuous lines and filled symbols, whereas the asynchronous ones appear with dashed lines and empty symbols. Colors and symbols are as indicated in the legends. The Stag Hunt games for the replicator rule (a) are: a, $S=-0.4$, $T=0.4$; b, $S=-0.5$, $T=0.5$; c, $S=-0.6$, $T=0.6$; d, $S=-0.7$, $T=0.7$; e, $S=-0.8$, $T=0.8$. For unconditional imitation the Stag Hunt games (c) are: a, $S=-0.6$, $T=0.6$; b, $S=-0.7$, $T=0.7$; c, $S=-0.8$, $T=0.8$; d, $S=-0.9$, $T=0.9$; e, $S=-1.0$, $T=1.0$. The Snowdrift games are, for both update rules (B, D): a, $S=0.9$, $T=1.1$; b, $S=0.7$, $T=1.3$; c, $S=0.5$, $T=1.5$; d, $S=0.3$, $T=1.7$; e, $S=0.1$, $T=1.9$. The initial density of cooperators is $x^0=0.5$ in all cases. The time scale of the asynchronous realizations has been rescaled by the size of the population $N=10000$, so that for both kinds of update a time step represents the same number of update events in the population. Figures A and B show that, in the case of the stochastic rule, not only the outcome but also the time evolution is independent of the kind of update. With unconditional imitation the results are also very similar for Stag Hunt (c), but not so much in Snowdrift (d), displaying the influence of the type of updating in this case. Notice also that unconditional imitation causes a much faster evolution than the replicator rule.Reuse & Permissions

Figure 11

(Color online) Asymptotic density of cooperators $x^{\ast }$ in regular lattices of degree $k=8$, for the Fermi update rule with $\beta$ equal to (a) 10, (b) 1, and (c) 0.1. The initial density of cooperators is $x^0=0.5$. For high $\beta$ the result is quite similar to that obtained with the replicator rule [Fig. 5F]. As $\beta$ decreases, or equivalently for weaker intensities of selection, the influence becomes smaller and more symmetrical between Stag Hunt and Snowdrift games.Reuse & Permissions

Figure 12

(Color online) Snapshots of the evolution of a population on a regular lattice of degree $k=8$, playing a Stag Hunt game ($S=-0.65$ and $T=0.65$). Cooperators are displayed in red (light gray) and defectors in blue (dark gray). The update rule is the replicator rule and the initial density of cooperators is $x^0=0.5$. The upper-left label shows the time step $t$. During the initial steps, cooperators with low local density of cooperators $\widehat{x}$ disappear, meanwhile those with high local density grow into the clusters that eventually take up the full population.Reuse & Permissions

Figure 13

(Color online) Snapshots of the evolution of a population on a regular lattice of degree $k=8$, playing a Stag Hunt game ($S=-0.65$ and $T=0.65$). Cooperators and defectors are displayed as in Fig. 12. The update rule is unconditional imitation and the initial density of cooperators is $x^0=1/3$ (this lower value than that of Fig. 12 has been used to make the evolution longer and thus more easily observable). The upper-left label shows the time step $t$. As with the replicator rule, during the initial time steps clusters emerge from cooperators with high local density of cooperators $\widehat{x}$. In this case, however, the interfaces advance deterministically at each time step, thus producing a much more rapid evolution (compare the time labels with those of Fig. 12).Reuse & Permissions

Figure 14

(Color online) Asymptotic density of cooperators $x^{\ast }$ in a regular lattice of degree $k=8$, for different initial densities of cooperators $x^0=1/3$ (A, D), 1/2 (B, E) and 2/3 (C, F). The update rules are the replicator rule (upper row, A to C) and unconditional imitation (lower row, D to F). With the replicator rule, the evolutionary outcome in Stag Hunt depends on the initial condition, as is revealed by the displacement of the transition line between full cooperation and full defection. However, with unconditional imitation this transition line remains in the same position, thus showing the insensitivity to the initial condition. In this case, the outcome is determined by the presence of small clusters of cooperators in the starting random population, which is ensured for a large range of values of the initial densities of cooperators $x^0$.Reuse & Permissions