Acculturation and the evolution of cooperation in spatial public goods games

Abstract

Cooperation is one of the foundations of human society. Many solutions to cooperation problems have been developed and culturally transmitted across generations. Since immigration can play a role in nourishing or disrupting cooperation in societies, we must understand how the newcomers’ culture interacts with the hosting culture. Here, we investigate the effect of different acculturation settings on the evolution of cooperation in spatial public goods games with the immigration of defectors and efficient cooperators. Here, immigrants may be socially influenced, or not, by the native culture according to four acculturation settings: integration, where immigrants imitate both immigrants and natives; marginalization, where immigrants do not imitate either natives nor other immigrants; assimilation, where immigrants only imitate natives; and separation, where immigrants only imitate other immigrants. We found that cooperation is greatly facilitated and reaches a peak for moderate values of the migration rate under any acculturation setting. Most interestingly, we found that the main acculturation factor driving the highest levels of cooperation is that immigrants do not avoid social influence from their fellow immigrants. We also show that integration may not promote the highest level of native cooperation if the benefit of cooperation is low.

Graphic abstract

Appendix A: computer simulation method

The simulations are run in a square lattice of size $N=100\times 100$ starting fully occupied with a fraction of 0.2 defectors and 0.8 standard cooperators. Note that the initial state does not affect the equilibrium states of the system. In each Monte Carlo step (MCS), the following steps are repeated $N_o$ times, with $N_o$ being the number of occupied sites. First, a player i is randomly chosen to imitate an individual j in the neighborhood with a probability

$$\begin{aligned} p_{i \rightarrow j}= \max \left\{ \frac{\Pi _{j}-\Pi _{i}}{\Delta \Pi _{\max }},0 \right\} , \end{aligned}$$

(A1)

where $\Delta \Pi _{\max }$ is the maximum payoff difference considering all possible combinations of allowed payoffs, which is included to normalize the probabilities [23]. Here, $\Pi _i$ is the total payoff of player i, obtained by summing the payoff from all games that player i participates. Notice that the probability $p_{i \rightarrow j}$ does not take into account irrationality. The Fermi imitation rule [23], on the other hand, is more realistic and considers irrationality. The conclusions of our model remain the same if we use the Fermi imitation rule.

After the imitation step, another individual and one of its first-neighbor sites are randomly chosen. If the neighbor site is empty, the individual reproduces with probability $\beta$. Third, another individual is chosen randomly and dies with probability $\gamma \beta$. Last, for the immigration step, a site is chosen randomly and, if it is empty, it receives an immigrant with probability $\min \{1,\mu /\rho _o\}$, where $\rho _o$ is the fraction of occupied sites and $\mu$ is the immigration coefficient. As long as $\mu \le \rho _o$, the factor $\rho _o$ guarantees that at each MCS the number of incoming immigrants is, on average, equal to $(\mu /\rho _o)(1-\rho _o)N_o=\mu (1-\rho _o)N=\mu N_v$, where $N_v$ is the number of vacant sites.

We consider a transient time around $10^4$ to $10^5$ MCS, after which we average the measures over the last 1000 steps. The results are further averaged over 100 independent samples. In the current paper, we fixed $\alpha =4, \gamma =0.005, \beta =0.2$, unless stated otherwise. A further investigation on the effects of varying these parameters in a scenario without acculturation can be found in Ref. [37].