Influence of stochastic perturbation of both action updating and strategy updating in mixed-strategy $2\times 2$ games on evolution of cooperation

In a mixed-strategy game framework, each agent's strategy is defined by a real number; on the other hand, in a discrete strategy game framework, only binary strategies, either cooperation or defection, are allowed. In a spatial mixed-strategy game, with respect to the process for updating action (offer), either a synchronous or an asynchronous strategy update should be presumed. This study elucidates how stochastic perturbation that results from a synchronous or an asynchronous process for updating action significantly affects the enhancement of cooperation in an evolutionary process. Especially, when a synchronous process for updating action is assumed, the extent of cooperation increases with an increase in degree.

Figure 1

Comparison of fraction of cooperators for discrete strategy definition and mixed-strategy definition in the PD area on a regular lattice with $k=8$. The top panels show results when the strategy is updated synchronously, while the bottom panels show when it is updated asynchronously. In this case, for the moment to update the action for mixed-strategy definition, an asynchronous process for updating action is used.

Figure 2

Snapshots for action C (black solid square) and D (white square) in $D_g=0$ and $D_r=0.8$ (shown in Fig. 1 by the white circle on the horizontal axis). Panel (a) shows the action in a discrete strategy setting, where either $s_i=0$ or 1 is permitted as the strategy parameter. Panel (b) shows the action in a mixed-strategy setting with an asynchronous process for updating action. IM is employed for both simulations as a strategy update rule.

Figure 3

From left to right, the final fraction of cooperators on a regular lattice with $k=8$, 12, 24, and 48 with the mixed-strategy definition for agents' strategy. The panels in the top row are results in the case of an asynchronous process for updating action, and the panels in the bottom row are results in case of a synchronous process for updating action.

Figure 4

Snapshot for the average of each strategy value $s_i$ and its evolutionary course, which is defined by a real number in range $[0,1]$ on a regular lattice with $k=8$, for $D_g=D_r=0.3$, respectively. The synchronous process for updating action is employed. The number shown in the upper part of each snapshot corresponds to the generation of the evolutionary scenario. The red dotted lines also correspond to the generations for the snapshot. In the first step, D-type agents having low $s_i$ invade C-type agents who possess a relatively high value of $s_i$. However, as time advances, surviving C-type clusters can expand gradually and finally reach a frozen state.

Figure 5

Snapshot for the average of each strategy value $s_i$ and its evolutionary course, which is defined by a real number in range $[0,1]$ on a regular lattice with $k=192$, for $D_g=D_r=0.3$, respectively. The synchronous process for updating action is employed. The number shown in the upper part of each snapshot corresponds to the generation of the evolutionary scenario. The red dotted lines also correspond to the generations for the snapshot. In the first step, D-type (or those possessing a half-baked value of $s_i$) agents invade fiercely. However, in the second step, the surviving C-type cluster can expand rapidly.

Figure 6

Time series for the average payoff among agents, which is classified by strategy values possessed (interval depicted shown in legend) and maximum payoff among them. Panels (a1) and (a2) show the average payoff and maximum payoff for $k=8$, for $D_g=D_r=0.3$, and panels (b1) and (b2) show them for $k=192$, for $D_g=D_r=0.3$, respectively. Payoffs in these figures are normalized by the number of degrees of the network, $k$. A logarithmic axis is used when $k=8$ for simple visual information because we need long time steps to equilibrium. The synchronous process is used as a process for updating action.

Figure 7

The fraction of cooperators ${\rho }_c$ versus the degree of network $k$ for $D_g=D_r=0.3$. Each plot is obtained averaging over 100 independent realizations. The peak of ${\rho }_c$ appears around $k=48$. After that it starts to decrease in a linear fashion.

Figure 8

Schematic view for the evolution of spatial PD game with the concept of END and EXP. Enduring (END) period: Initial cooperators will be rapidly plundered by defectors, which leave only few cooperators by forming compact C clusters. Expanding (EXP) period: C clusters start to expand, since a cooperator on the clusters' border can attract a neighboring defector into the cluster. In the present framework, evolutionary dynamics can also be classified into these two types.

Figure 9

Probabilities that the special situation mentioned in Fig. 10 occurs in a cluster with degrees $k=8$, 24, 96, and 192. The inset shows magnification for the right bottom area. In the small-degree case, agents have a relatively high possibility of being the most profitable agent in the wide range of $s_x$.

Figure 10

Schematic view of special situation in the case of $k=8$, where a central agent in a cluster offers D, while all his or her neighbors offer C, even though all agents belonging to the cluster have identical strategy $s_x$.