Role of generosity and forgiveness: Return to a cooperative society

One's reputation in human society depends on what and how one did in the past. If the reputation of a counterpart is too bad, we often avoid interacting with the individual. We introduce a selective cooperator called the goodie, who participates in the prisoner's dilemma game dependent on the opponent's reputation, and study its role in forming a cooperative society. We observe enhanced cooperation when goodies have a small but nonzero probability of playing the game with an individual who defected in previous rounds. Our finding implies that even this small generosity of goodies can provide defectors chances of encountering the better world of cooperation, encouraging them to escape from their isolated world of selfishness.

Figure 1

Time evolution of goodie and cheater frequencies, $x_{\mathrm{G}}$ and $x_{\mathrm{H}}$, for (a) $\tau =1$, (b) $\tau =2$, and (c) $\tau =4$ with $\eta =0.2$. In (a)–(c), we use $c=0.2$, $\mu =0.001$, $\epsilon =0$, and $x_{{}^{\mathrm{G}}}^0=0.5$. Populations evolve quite differently for (a) $\tau =1$ and (b, c) $\tau \ge 2$. After a sufficiently long time, cheaters dominate for $\tau =1$, while goodies dominate for $\tau \ge 2$. Insets in (a)–(c): To clearly show the initial stage of time evolution, we present $x_{\mathrm{G}}$ and $x_{\mathrm{H}}$ in semilog plots here. (d) Stationary goodie frequencies $x_{\mathrm{G}}^{\infty }$ are plotted versus $\eta$ for various $\tau$. The $x_{\mathrm{G}}^{\infty }\approx 1$ for $\eta <{\eta }_c$ and $x_{\mathrm{G}}^{\infty }\approx 0$ otherwise, exhibiting sharp transitions at finite values of ${\eta }_c$ for $\tau \ge 2$. However, $x_{\mathrm{G}}^{\infty }=0$ for the whole range of $\eta$ for $\tau =1$. Inset: ${\eta }_c$ vs $\tau$. For $\tau =1$, ${\eta }_c$ is 0, but for $\tau \ge 2$, ${\eta }_c$ is around 0.7.

Figure 2

Schematic of transitions between states with $\tau =1$ in the presence of a mutation with $\epsilon =0$. Gray arrows indicate transitions that occur when the individual changes his or her last strategy; yellow arrows, itransitions when the individual keeps his or her last strategy. Dotted lines denote transitions which occur only through mutations. Where transitions are possible by learning the partner's (higher payoff) strategy, they are represented by thick solid lines. Once GG becomes GH, it easily becomes HH, while a mutant HG from HH likely returns to HH.

Figure 3

(a) Evolution of goodies for $\tau =1$ from the initial goodie frequency, $x_{{}^{\mathrm{G}}}^0=0.2$. The time evolution of $x_{\mathrm{G}}$ with $\epsilon =0.01$ is shown by the thick blue line. For comparison, $x_{\mathrm{G}}$ with $\epsilon =0$ is also shown by the thin red line. (b) The frequencies of four states, $x_{\mathrm{HH}}$, $x_{\mathrm{HG}}$, $x_{\mathrm{GH}}$, and $x_{\mathrm{GG}}$, in the population in (a) are plotted versus time on a log-log scale. At $t=1$, $x_{\mathrm{HH}}$ becomes almost 1 and stays there until $t_q\approx {10}^3$ (quasisteady state). Then it drops to its stationary value, ${10}^{-6}$ (out of plot range), while $x_{\mathrm{GG}}$ soars to its stationary value around 1. Here, $c=0.2$, $\eta =0.5$, and $\mu =0.001$ are used.

Figure 4

Stationary-state goodie frequencies for (a) $c=0.2$ and (b) $c=0.8$ with $\epsilon =0.01$, $\mu =0.001$, and initial frequencies $x_{\mathrm{HH}}=0.499$, $x_{\mathrm{HG}}=0.001$, $x_{\mathrm{GH}}=0.001$, and $x_{\mathrm{GG}}=0.499$. Numerical results are consistent with the quasisteady-state approximation, which predicts the evolution of goodies for $\eta <{\eta }_c=0.8$ (a) and $\eta <{\eta }_{\mathrm{G}}\approx 0.4$ (b).

Figure 5

(a) Evolution of frequencies of eight states for $\tau =2$ and $\eta =0.5$. (b) Goodie frequencies in the stationary state versus $\eta$ for $\tau =2$, 3, and 4 with zero generosity and a finite mutation probability. The subscript $S$ in (a) is the decimal number representation of Eq. (6); i.e., $S=0$, 1, $\cdots$, 7 for HHH, HHG, $\cdots$, GGG, respectively. Initial frequencies are given as $x_S=\frac{1}{2}\left[1-\delta (N-2)\right]$ for persistent states of $S=0$ and $S=N-1$ and as $x_S=\delta =0.001$ otherwise. For all cases, $c=0.2$, $\epsilon =0$, and $\mu =0.001$ are used.

Figure 6

Frequencies of goodies and cheaters ($x_{\mathrm{G}}$ and $x_{\mathrm{H}}$) versus time $t$ for $\tau =1$ with the initial goodie frequency $x_{{}^{\mathrm{G}}}^0=0.5$. Blue circles and red squares represent $x_{\mathrm{G}}$ and $x_{\mathrm{H}}$, respectively. (a) For $\eta =0.2$, goodies eventually flourish ($x_{\mathrm{G}}>x_{\mathrm{H}}$), while (b) for $\eta =0.8$, cheaters dominate the system after a transient time. Here, we have used $c=0.2$, $\mu =0$, and $\epsilon =0$.

Figure 7

(a) Goodie frequency $x_{\mathrm{G}}^{\infty }$ in the stationary state for $c=0.2$, $\epsilon =0$, and $\mu =0$ for three different initial goodie frequencies, $x_{{}^{\mathrm{G}}}^0=0.3$, 0.5, and 0.7. The final goodie frequency $x_{\mathrm{G}}^{\infty }$ undergoes a series of sharp drops as $\eta$ is varied. (b) Magnification of (a) for $x_{{}^{\mathrm{G}}}^0=0.5$ showing its second and subsequent drops.

Figure 8

Possible transitions between two states of unit memory. Yellow arrows indicate the transition when a player keeps his or her last strategy; gray arrows, the transition when the player changes the strategy. Here, $w_i$ represents the strategy change probability of state ${\mathrm{S}}_i$.

Figure 9

Goodie frequencies $x_G$ (a) versus $g$ at $c=0.2$ and (b) versus $c$ at $g=0.2$. Note that the condition $g+c<b$ must be satisfied. Squares are from simulation results and agree well with the solid gray line $g/(c+g)$ obtained from analytic calculation (see text). We have used the initial goodie frequency $x_{{}^{\mathrm{G}}}^0=0.5$.