Dynamical reciprocity in interacting games: Numerical results and mechanism analysis

We study the evolution of two mutually interacting pairwise games on different topologies. On two-dimensional square lattices, we reveal that the game-game interaction can promote the cooperation prevalence in all cases, and the cooperation-defection phase transitions even become absent and fairly high cooperation is expected when the interaction becomes very strong. A mean-field theory is developed that points out dynamical routes arising therein. Detailed analysis shows indeed that there are rich categories of interactions in either the individual or bulk scenario: invasion, neutral, and catalyzed types; their combination puts cooperators at a persistent advantage position, which boosts the cooperation. The robustness of the revealed reciprocity is strengthened by the studies of model variants, including the public goods game, asymmetrical or time-varying interactions, games of different types, games with timescale separation, different updating rules, etc. The structural complexities of the underlying population, such as Newman-Watts small world networks, Erdős-Rényi random networks, and Barabási-Albert networks, also do not alter the working of the dynamical reciprocity. In particular, as the number of games engaged increases, the cooperation level continuously improves in general. However, our analysis shows that the dynamical reciprocity works only in structured populations, otherwise the game-game interaction has no any impact on the cooperation at all. In brief, our work uncovers a cooperation mechanism in the structured populations, which indicates the great potential for human cooperation since concurrent issues are so often seen in the real world.

Figure 1

Modeling two interacting games. (a) Consider a group of networked players: they play two games $G_{1,2}$ simultaneously, and two payoffs are obtained accordingly, which can be interpreted as the fitness in their evolution. (b) When two neighboring individuals, say, players $x$ and $y$, are to update their strategies with respect to a given game (e.g., game $G_1$ here), the update depends on not only the payoffs obtained in $G_1$ (with weight $1-\theta$) but also the one in the other game $G_2$ (with $\theta$), and this combination is termed the effective payoff; see Eq. (3).

Figure 2

The evolution of two symmetrically interacting PD games on the 2D square lattice. (a) The phase transitions of cooperation prevalence regarding game $G_1$ ($f_C^{G_1}=f_{CC}+f_{CD}$) for different interaction strengths $\theta$, and the temptation $b$ is the control parameter. (b) Time series for $b=1.1$. Note that $f_C^{G_2}\approx f_C^{G_1}$ due to the symmetry and is not shown. Parameters are $L=1024$ and $K=0.1$, the random initial condition for both games, and each point is averaged over 50 ensembles after transient in (a).

Figure 3

Color-coded fraction of cooperators regarding the first game ($f_C^{G_1}=f_{CC}+f_{CD}$) for the general pairwise game within the $S-T$ parameter space with $\theta =0$, 0.5, and 1 on the 2D square lattice, respectively shown in (a)–(c). Due to the symmetry, $f_C^{G_2}\approx f_C^{G_1}$. Four quadrants correspond to four different games (defined in Sec. 2). Parameters: $R=1$, $P=0$, $L=128$.

Figure 4

Color-coded fraction of cooperation prevalence $f_C^{G_1}$ in $\theta -\widehat{r}$ parameter space for two symmetrically interacting public goods games. $\widehat{r}=r/(k+1)$ is the normalized gain factor, and $k+1$ is the number of games the individual is evolved. Also $f_C^{G_2}\approx f_C^{G_1}$ due to the symmetry. Here $L=128$ and $K=0.5$.

Figure 5

The cooperator fraction $f_C^{G_1}$ for the two interacting pairwise games in the well-mixed population within $S-T$ parameter space for $\theta =0.5$. The resulting cooperation prevalence exactly corresponds to the solution of single game; the less significant bistability in SH game is simply due to the random initial conditions, where $f_C\approx f_D\approx 1/2$. Due to the symmetry, $f_C^{G_1}\approx f_C^{G_2}$ in all cases, except very few case along the line $S=T-1$ in SH, depending on their initial conditions. Other parameters: $R=1$, $P=0$, $N=2^{14}$.

Figure 6

Time series of all four fractions for two symmetrically cross-played ($\theta =1$) PD games starting from random initial conditions. All six binary compositions are included: (a) CC-DD, (b) CC-CD, (c) CC-DC, (d) CD-DD, (e) DC-DD, (f) DC-DC. (a), (b)–(e), and (f) correspond to category (i)–(iii), respectively. Parameters: $S=0$, $T=1.1$, $R=1$, $P=0$, $L=1024$ for the 2D square lattice.

Figure 7

Time series of all four fractions for two symmetrically cross-played ($\theta =1$) PD games starting from half-half patched initial conditions. (a) CC-DD, (b) CC-CD, (c) CC-DC, (d) CD-DD, (e) DC-DD, (f) CD-DC. (a), (b)–(e), and (f) correspond to category (i)–(iii), respectively. Parameters: $S=0$, $T=1.1$, $R=1$, $P=0$, $L=1024$ for the 2D square lattice.

Figure 8

Time evolution of interacting pair proportions for two symmetrically interacting PD games. Both random (a, c) and half-half patched (b, d) initial conditions are considered. Panels (a) and (b) and panels (c) and (d) correspond to $\theta =0.5$ and 1, respectively. $P_r=P_{CD-DC}/P_{CC-DD}$ to compares the relative proportion for CD–DC and CD–DD pairs. Parameters: $S=0$, $T=1.1$, $R=1$, $P=0$, $L=1024$ for the 2D square lattice.

Figure 9

Cluster size distribution for two symmetrically interacting PD games starting from random initial conditions. (a) and (b) are time evolution of the average cluster size of the four species for $\theta =0.5$ and 1, respectively. (c) and (d) are probability function distributions of cluster size at $t=1000$ for $\theta =0.5$ and 1, respectively. In (d), there is a giant CC cluster with the size comparable to the population size ($\sim N$), which is not shown. Parameters: $S=0$, $T=1.1$, $R=1$, $P=0$, $L=1024$ for the 2D square lattice.

Figure 10

Cluster compactness analysis. (a, b) Time evolution of compactness of the four species for $\theta =0.5$ and 1, respectively. (c, d) Probability function distributions of cluster compactness at $t=1000$ for $\theta =0.5$ and 1, respectively. Same settings as in Fig. 9.

Figure 11

Color-coded cooperation prevalences for two asymmetrically interacting PD games on a 2D square lattice within ${\theta }_1-{\theta }_2$ parameter space; (a) and (b) are for game $G_{1,2}$, respectively. Other parameters: $S=0$, $T=1.1$, $R=1$, $P=0$, $L=128$.

Figure 12

Color-coded cooperation prevalence in $S-T$ parameter space for two interacting games; one is PD and the other is SD, on the 2D square lattice, for $\theta =0,0.5,1$ (a–c). They have opposite sign in $S$, e.g., the SD at top left corner $(T=1,S=1)$ interacts with PD with $(T=1,S=-1)$ at the bottom left. Other parameters: $R=1$, $P=0$, and $L=128$ for the lattice.

Figure 13

The impact of timescale separation of the two interacting PD games. The cooperation prevalence $f_c$ vs the timescale ratio $T_r$ for $\theta =0.5$ (a) and $\theta =1$ (b). No cooperation is seen for $\theta =0$ for the given parameters. Other parameters: $S=0$, $T=1.1$, $R=1$, $P=0$, $L=1024$ for the 2D square lattice.

Figure 14

Cooperation phase diagram for $f_C^{G_1}$ with other four update rules with AU, for $\theta =0,0.5,1$. The update rules are the following: replicator rule (first row, a–c), multiple replicator rule (second row, d–f), Moran rule (third row, g–i), and unconditional imitation (bottom row, j–l). Random initial conditions are adopted. The four numbers are the average cooperation prevalence for the corresponding quadrants. $f_C^{G_2}\approx f_C^{G_1}$ due to the symmetrical settings. Parameters: $R=1$, $P=0$, and $L=128$ for the 2D square lattice.

Figure 15

Cooperation phase diagram with other four update rules but with SU, for three interaction strengths $\theta =0,0.5,1$. Other settings are exactly the same as in Fig. 14.

Figure 16

Two symmetrically interacting PD games on Newman-Watts small-world networks. (a) The cooperation prevalence $f_c$ vs the temptation $T=b$ for a couple of interaction strengths $\theta$. (b) $f_c$ vs $\varphi$ for three temptation values $b$ with $\theta =0.5$. $f_C^{G_2}\approx f_C^{G_1}$ due to the symmetrical settings. Parameters: $P=S=0$, $R=1$ for both games, the network size $N=2^{20}$ with $\varphi =0.01$.

Figure 17

Two symmetrically interacting PD games on Erdős-Rényi random networks. (a) The cooperation prevalence $f_c$ vs the temptation $T=b$ for a couple of interaction strengths $\theta$. (b) $f_c$ vs the average degree for three temptation values $b$ with $\theta =0.5$. $f_C^{G_2}\approx f_C^{G_1}$ due to the symmetrical settings. Parameters: $P=S=0$, $R=1$ for both games, the network size $N=2^{20}$ with $\langle k\rangle =4$.

Figure 18

Two symmetrically interacting PD games on BA scale-free networks. The cooperation prevalence $f_c$ vs the temptation $T=b$ for a couple of interaction strengths $\theta$. $f_C^{G_2}\approx f_C^{G_1}$ due to the symmetrical settings. Parameters: $P=S=0$, $R=1$ for both games, the network size $N=2^{20}$ with $\langle k\rangle =4$. $K=0.025$ due to the average payoff scheme; the flipping noise $p_f=0.01$.

Figure 19

Three interacting games. (a) Phase transitions of the three cooperation prevalences vs the interaction strength ${\theta }^{\prime }$ by assuming ${\theta }_1^{\prime }={\theta }_2^{\prime }={\theta }^{\prime }$. (b) Color-coded fraction of cooperators regarding game $G_1$ ($f_C^{G_1}={\sum }_{X,Y\in {\mathbb{S}}_1}{f}_{CXY}$) within the interaction space ${\theta }_1^{\prime }\text{−}{\theta }_2^{\prime }$. Due to the restriction of ${\theta }_1^{\prime }+{\theta }_2^{\prime }\le 1$, the upper right half is unphysical. Note that the cooperation prevalence for the three games $f_C^{G_{1,2,3}}$ is approximately identical due to the symmetrical setting, as shown in (a). Other parameters: $S=0$, $R=1$, $P=0$, $T=1.1$, $L=1024$ for the 2D square lattice.