Heterogeneous update mechanisms in evolutionary games: Mixing innovative and imitative dynamics

Innovation and evolution are two processes of paramount relevance for social and biological systems. In general, the former allows the introduction of elements of novelty, while the latter is responsible for the motion of a system in its phase space. Often, these processes are strongly related, since an innovation can trigger the evolution, and the latter can provide the optimal conditions for the emergence of innovations. Both processes can be studied by using the framework of evolutionary game theory, where evolution constitutes an intrinsic mechanism. At the same time, the concept of innovation requires an opportune mathematical representation. Notably, innovation can be modeled as a strategy, or it can constitute the underlying mechanism that allows agents to change strategy. Here, we analyze the second case, investigating the behavior of a heterogeneous population, composed of imitative and innovative agents. Imitative agents change strategy only by imitating that of their neighbors, whereas innovative ones change strategy without the need for a copying source. The proposed model is analyzed by means of analytical calculations and numerical simulations in different topologies. Remarkably, results indicate that the mixing of mechanisms can be detrimental to cooperation near phase transitions. In those regions, the spatial reciprocity from imitative mechanisms is destroyed by innovative agents, leading to the downfall of cooperation. Our investigation sheds some light on the complex dynamics emerging from the heterogeneity of strategy revision methods, highlighting the role of innovation in evolutionary games.

Figure 1

Asymptotic cooperation fraction ($\rho$) vs temptation to defect ($T$) in the weak Prisoner's dilemma ($S=0$) in the mean-field, well-mixed population for the innovative ($\alpha =1$) and imitative ($\alpha =0$) model, compared with the mixed population of half innovative and half imitative agents ($\alpha =0.5$).

Figure 2

Asymptotic cooperation level, $\rho$, vs fraction of innovators, $\alpha$, here $S=0$. Results obtained from the mean-field equation in the well-mixed population. The increasing of innovators is beneficial for cooperation in the weak Prisoner's Dilemma ($T>1$), while detrimental to it in the region $T<1$. In $T=1$ all models have the same asymptotic behavior.

Figure 3

Asymptotic cooperation level ($\rho$) vs temptation to defect ($T$) in a square lattice for the weak Prisoner's Dilemma. Each line corresponds to a different fraction of innovators ($\alpha$). The inset shows the $T$ region where the heterogeneous population, $\alpha =0.5$, has a cooperation level lower than any pure population.

Figure 4

Asymptotic cooperation level, $\rho$, for an increasing fraction of innovators ($\alpha$) in the square lattice. The behavior is quite different from the well-mixed population. In the region $T>1.04$, it is always better to have a fully innovative population. But for $0.8<T<1.04$, there is always a minimum value of $\rho$ for mixed populations.

Figure 5

Asymptotic cooperation level ($\rho$) in a triangular lattice. In (a) we show the dependence in $T$, while in (b) we show its dependence in $\alpha$. As we can see, the mixing of update rules can lead to a drop in cooperation for $0.95<T<1.25$. The behavior is qualitatively similar to the square lattice.

Figure 6

Asymptotic cooperation level ($\rho$) as we increase $T$ for a random network in (a) and a scale-free network in (b). The drop in cooperation for the mixed population, compared to pure populations, happens in both topologies.

Figure 7

Asymptotic cooperation level (colors) in the whole $T\text{−}S$ parameter space for the square lattice. We have (a) the pure imitation model ($\alpha =0$), (b) the mixture of imitation and Logit dynamics ($\alpha =0.5$), and (c) the pure Logit ($\alpha =1$) model. The main differences happen in the SH and SD regions.

Figure 8

Difference in asymptotic cooperation level (colors) between the mixed model ($\alpha =0.2$) and the average between the Logit and imitation model. The mixed model behaves differently from just the average of the two models, mainly in the SD and SH regions.

Figure 9

Asymptotic cooperation and defector fraction for the two types of agents, innovative and imitative, in the square lattice. Here $S=0$ and $\alpha =0.5$. While innovative cooperators ($C_{\text{inv}}$) can survive for high $T$, conversely it is the imitative defectors ($D_{\text{imt}}$) that fare better when $T>1$. A similar behavior occurs for all $\alpha$ values.

Figure 10

Asymptotic fraction of the four types of agents (imitative and innovative cooperators and defectors) as $\alpha$ is increased for (a) $T=0.8$, (b) $T=1$, and (c) $T=1.05$. The behavior is linear with $\alpha$ in (a) and (b). However, For $T=1$ there is a nonlinear behavior, with an increase of $D_{\text{imt}}$ until $\alpha =0.2$.

Figure 11

Ratio between the normalized fraction of innovative and imitative agents in the square lattice for three different values of $\alpha$. The ratio of cooperators is shown in black, and defectors are in red. Notice that ${\varphi }_C$ grows with $T$, while ${\varphi }_D$ is always smaller than 1.

Figure 12

Snapshots of the square lattice with $T=1$ for (a) $\alpha =0.1$ and (b) $\alpha =0.7$. As we mix innovators, the imitative cooperator islands get dissolved, leading to a drop in total cooperation. Cooperators are shown in blue and defectors in red.