# Information Propagation and Bionic Evolution Control of the _Sebar_ Model in a Swarm System
## 5 Bionic Information Control Based on Controllable Individuals in a Repeated Prisoner's Dilemma
### Emergence of Cooperative Behaviors in Repeated Prisoner's Dilemma
\begin{array}{l}\underset{j\in N_{i}}{\operatorname{argmax}}P _{j},if\left|\underset{j\in N_{i}}{\operatorname{max}}P_{j}\right|=1\\ \min\left\{ik\left|\dot{j}_{k}\in\underset{j\in N_{i}}{\operatorname{argmax}}P _{j},k\geq 2\right.\right\},else\end{array} \right. \tag{26}\]  
where \(\left|\underset{j\in N_{i}}{\operatorname{max}}P_{j}\right|\) is the neighbor with the highest payoff. When there is more than one neighbor that satisfies the condition, only the neighbor with the smallest index is selected. When an individual \(i\) follows the \"Fermi rule\", a neighbor is randomly selected to update its strategy with equal probability; as such,  
\[f_{II}:j^{*}\in\mathcal{N}_{i},if\leavevmode\
obreak\ rand\geq\frac{1}{1+ \exp\left[\left(\overline{P}_{i}(t)-\overline{P}_{j^{*}}(t)\right)/\zeta\right]} \tag{27}\]  
where \(\zeta\) represents the noise intensity, characterizing the uncertainty in the strategy selection. From the update strategies of ordinary individuals, it is known that strategy updates are only related to the current payoff. An individual usually considers past experiences when making important decisions [35; 36]. For example, by interacting and cooperating, some swarm systems like bird flocks use their memory to reach the food source in the shortest time [37]. Therefore, it is significant to introduce a memory mechanism into the complex system evolution. In addition, the memory mechanism also helps avoid unnecessary losses caused by occasional strategy selection errors, improves the robustness of the strategy selection process, and thus promotes the emergence of cooperative behavior within the cluster [38; 39]. For controllable individuals in the repeated prisoner's dilemma game, assume that the strategy of an individual \(i\) at time \(t\) is given by \(\mathbf{X}_{i}(t)=(p_{0},p_{c},\overline{p}_{c})\), the optimal strategy within the memory cycle \(t_{0}\) is \(\mathbf{X}_{i}^{best}\). When \(t_{0}=1\), the individual only remembers the optimal strategy from the previous step. When \(t_{0}=t\), the individual remembers the optimal strategy \(\mathbf{X}_{j^{*}}^{best}\) up to the current time \(t\). In addition, for controllable individuals, the strategy update can be viewed as an optimization process in the strategy space \([0,1]\in\mathbb{R}^{3}\), which is given by:  
\[\mathbf{X}_{i}(t+1)=\mathbf{X}_{i}(t)+\mathbf{V}_{i}(t+1) \tag{28}\] \[\mathbf{V}_{i}(t+1)=\mathbf{V}_{i}(t)+\omega\Big{(}\mathbf{X}_{i}^{best}-\mathbf{ X}_{i}(t)\Big{)}+(1-\omega)\Big{(}\mathbf{X}_{j^{*}}^{best}(t)-\mathbf{X}_{i}(t)\Big{)}\] (29) \[\omega=\left\{ \begin{array}{l}\omega_{\min}-(\omega_{\max}- \omega_{\min})\frac{\overline{p}_{i}(t)-\overline{p}_{avg}(t)}{\overline{p}_{ avg}(t)-\overline{p}_{i}(t)},\leavevmode\
obreak\ \text{if}\leavevmode\
obreak\ \overline{P}_{i}(t)\leq\overline{P}_{avg_{i}}(t)\\ \omega_{\max},\leavevmode\
obreak\ \text{else}\end{array} \right. \tag{30}\]  
where \(\mathbf{V}_{i}(t)\) represents the search speed, \(\omega\) adjusts the strategy weight between an individual and its neighbors, and \(\omega_{\min}\) and \(\omega_{\max}\) represent the minimum and maximum values of the weights, respectively. When \(\omega\to 0\), individuals tend to follow the strategy of the best-performing neighbor. When \(\omega\to 1\), individuals are more inclined to update strategies based on their own strategies within the memory. \(\overline{P}_{avg_{i}}(t)\) is the average single-game payoff of the neighbors.