Evolution of cooperative imitators in social networks

Many evolutionary game models for network reciprocity are based on an imitation dynamics, yet how semirational imitators prevail has seldom been explained. Here we use a model to investigate the coevolutionary dynamics of cooperation and partnership adjustment in a polygenic population of semirational imitators and rational payoff maximizers. A rational individual chooses a strategy best responding to its neighbors when updating strategy and switches to a new partner who can bring it the maximal payoff from all candidates when adjusting the partnership. In contrast, a semirational individual imitates its neighbor's strategy directly and adjusts its partnership based upon a simple reputation rule. Individual-based simulations show that cooperation cannot evolve in a population of all best responders even if they can switch their partners to somebody who can reward them best in game playing. However, when imitators exist, a stable community that consists of cooperative imitators emerges. Further, we show that a birth-death selection mechanism can eliminate all best responders, cultivating a social regime of all cooperative imitators. Compared with parallel simulations that assume fixed networks, cooperative imitators are evolutionarily favored, provided they are able to adjust their partners.

Figure 1

A algorithmic diagram of this model. This is a complete algorithm of the coevolution of strategy update and partnership adjustment. When $p=0$, it is an evolutionary game model on a fixed network.

Figure 2

A sampled coevolution in situation 2. (a) Fractions of strategies are represented versus iteration step. (b) The partnership network at the end of the coevolution period. Cyan denotes the IC strategy, yellow denotes the ID strategy, green denotes the BC strategy, and red denotes the BD strategy. The initial percentage of each strategy is approximately $25\%$. (a) ${\rho }_{IC}$ first decreases and then increases until reaching approximately $40\%$, whereas ${\rho }_{BD}$ consistently increases until reaching approximately $50\%$. The BC strategy disappears, indicating that it is the inferior strategy. (b) The initial network disintegrates into isolated components. The largest connected component is composed of IC agents, while other components are typically dominated by BD agents, with a small number of ID agents. Where the I allele exists, IC agents can form a stable community to resist exploitation by other agents. A small population of $M=400$ is adopted to aid visualization of network structure, and node size is scaled in proportion to degree. Other parameters are $b=1.1,p=0.2$. Partnership networks were plotted using Pajek [54].

Figure 3

Distribution of four strategies at the equilibrium stage in situation 2. Panels (a) to (d) correspond to IC, ID, BC, and BD, respectively. Each strategy's distribution is shown as a function of both the advantage of defectors against cooperators and partner-switching probability $(p,b)$. ${\rho }_{BC}$ and ${\rho }_{BD}$ are insensitive to parameter variation, whereas ${\rho }_{IC}$ and ${\rho }_{ID}$ are sensitive. $M=1600$.

Figure 4

Equilibrium frequency of cooperation ${\rho }_C$ as a function of $(p,b)$ in situation 2. Data are the same as those of Fig. 3. ${\rho }_C$ does not increase monotonously with $p$. The highest level of ${\rho }_C$ occurs at an intermediate level of $p$.

Figure 5

A sampled coevolution in situation 3. (a) Fractions of strategies are represented versus iteration step. (b) The partnership network at the end of the coevolution period. A parallel simulation to that in Fig. 2 is shown. In (a), ${\rho }_{IC}$ increases to $100\%$. In (b), the partnership network evolves into a fully-connected heterogeneous one. Compared with Fig. 2, this figure reveals that a selection eliminates the BD agents, thereby fostering a perfect social regime.

Figure 6

Distribution of the four strategies at the equilibrium stage in situation 3. Panels (a) to (d) correspond to IC, ID, BC, and BD, respectively. Each strategy's frequency is shown as a function of both the partner-switching probability and the advantage of defectors against cooperators $(p,b)$. Compared with the simulation results for situation 2, either BC or BD vanishes from the population. $M=1600$.

Figure 7

The emergence of cooperative imitators during coevolution. (a) Fractions of strategies are represented versus iteration step. (b) The partnership network at the end of the coevolution period. The parameters are set as in situation 2. The initial fraction of IC strategy was 0, whereas the initial fractions of the other strategies were set at random. As coevolution proceeds, ${\rho }_{IC}$ increases to a certain level. Although there is no mutation, some noncooperative imitators can become cooperative ones. The equilibrium partnership network exhibits community structure. Parameters are $b=1.4,p=0.8,M=400$.

Figure 8

The minimum initial IC fraction $\left({\overline{\rho }}_{IC}\right)$ for invasions with $90\%$ probability of success as a function of $(p,b)$ in situation 2. An increase in partner-switching probability has a strong effect on the fraction of IC agents. Under the arduous condition where the advantage of defectors against cooperators $\left(b\right)$ is high, a high partnership-adjustment frequency can help a very small number of cooperative imitators survive and grow into a large cluster.