Emergent Hierarchical Structures in Multiadaptive Games

We investigate a game-theoretic model of a social system where both the rules of the game and the interaction structure are shaped by the behavior of the agents. We call this type of model, with several types of feedback couplings from the behavior of the agents to their environment, a multiadaptive game. Our model has a complex behavior with several regimes of different dynamic behavior accompanied by different network topological properties. Some of these regimes are characterized by heterogeneous, hierarchical interaction networks, where cooperation and network topology coemerge from the dynamics.

Figure 1

Parameter dependence of the game reflected in the temptation and average cooperator density. Panel (a) shows average density of cooperators $\overline{\rho }$, as a function of the initial temptation $b_0$, with $\alpha =0.1$, 3, and 4. The bar represents points averaged over the last 500 (of ${10}^3$) steps. Panels (b) and (c) correspond to the time evolution of $\overline{\rho }$ and $\overline{b}$, respectively, for different values of $b_0$. Panel (d) shows the diagram over the three regions in $\alpha \mathrm{\text{−}}{b}_0$ space. The curves are averages over ${10}^4$ runs.

Figure 2

Correlations between the strategy and network structure. Circles (squares) correspond to the average density of cooperators (defectors) with degree $k$, ${\rho }_k$. Panel (a) is for $b_0=1.3$ (region I), (b) is for $b_0=3.5$ (region II), and (c) is for $b_0=7$ (region III). In panels  (b) and (c), the exponent of the power law is $2.7\pm 0.1$.

Figure 3

Structural properties of network in the steady state for different values of $b_0$. Panel (a) displays the cumulative degree distribution. The line for $b_0=7$ follows a decay like a sum of an exponential and stretched exponential function. Panel (b) shows the clustering coefficient $C$ as a function of degree $k$. The line marks a scaling with exponent $-1$.