# 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  
For the prisoner's dilemma, there are two possible strategies, Cooperation (C) and Defection (\(D\)), for two rational individuals. It is assumed that the \"reward\" payoff for cooperation is \(R^{\prime}\), the \"punishment\" payoff for mutual defection is \(P^{\prime}\), the \"temptation\" payoff for the defecting party in a mixed strategy is \(T^{\prime}\), and the \"sucker's\" payoff for a cooperator is \(S^{\prime}\). The payoffs satisfy \(T^{\prime}>R^{\prime}>P^{\prime}>S^{\prime}\). Let the game intensity be \(b>0\)  
Figure 2: Diagram of soft control of swarm behaviors. When \(R^{\prime}=1\), \(P^{\prime}=0\), \(T^{\prime}=1+b\), \(S^{\prime}=-b\), the payoff matrix for a single-shot game is shown in Table 1. When \(b\to 0\), the conflict between individual and collective interests is minimal. As \(b\) increases, the temptation to defect and the cooperation risk increase, and the conflict is deepened leading to a \"social dilemma\". That is to say, for a rational individual, defection becomes a strictly dominant strategy, while cooperation is a strictly dominated strategy. Under these assumptions, mutual defection is the Nash equilibrium strategy. However, cooperation yields a higher payoff than mutual defection. Studies show that repeated games facilitate the emergence of cooperative behaviors. In repeated games, an individual's strategy can be considered as a static Markov strategy. This means that the individual's current action depends solely on the opponent's action in the previous round, except for the first round. Let the elements of the static Markov strategy \(X_{i}=(p_{0},p_{c},\overline{p}_{c})\) be randomly valued \([0,1]\), where \(p_{0}\) is the probability of cooperating in the first round, and \(p_{c}\) and \(\overline{p}_{c}\) represent the cooperative probabilities when the game player chooses \(C\) or \(D\) in the previous round, respectively. Hence, the classic \"tit-for-tat\" strategy can be expressed as \(X_{s}=(1,1,0)\). From the perspective of strategy selection, both individuals actively seek cooperation while punishing defection to encourage cooperation in the next round. When both players adopt the \"tit-for-tat\" strategy, the overall cooperation rate is 1. In a complex network of the swarm system, the total payoff for an individual after \(L\) rounds with its neighbors can be expressed as:  
\[P_{i}(t)=\sum_{j\in\mathcal{N}_{i}}U_{i,j}(t) \tag{23}\]  
where \(\mathcal{N}_{i}\) is the neighbor set of an individual \(i\), and \(U_{i,j}(t)\) is the payoff of \(L\) rounds of repeated games between \(i\) and \(j\). Therefore, for an individual \(i\), the payoff of a single game can be expressed as:  
\[\overline{P}_{i}(t)=\frac{P_{i}(t)}{|\mathcal{N}_{i}|\cdot L}(|\mathcal{N}_{i }|\
eq 0) \tag{24}\]  
Combined with the framework of soft control, all individuals can be divided into ordinary ones and controlled ones, and the percentages of each are \(p_{norm}\) and \(p_{ctrl}\). When the swarm size remains constant, we have \(p_{norm}+p_{ctrl}=1\). Incorporating the game payoff, the discrete dynamic equation for the strategy update rule can be expressed as:  
\[\mathbf{X}_{i}(t+1)=f\big{(}\mathbf{X}_{j}(t),\overline{P}_{j}(t)\big{)},(i\in N,j\in \mathcal{N}_{i}) \tag{25}\]  
where \(\mathbf{X}_{i}(t+1)\) is the strategy of an individual \(i\) at time \(t+1\), and \(X_{j}(t)\) is the strategy of individual \(j\) at time \(t\). \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\mathbf{C}\) & \(\mathbf{D}\) \\ \hline \(C\) & \((1,1)\) & \((-b,1+b)\) \\ \(D\) & \((1+\text{b},-\text{b})\) & \((0,0)\) \\ \hline \end{tabular}
\end{table} 
Table 1: The payoff matrix for a single-shot game. For ordinary individuals, the updating rules are \"unconditional imitation\" and the \"Fermi rule\". When an individual \(i\) follows the \"unconditional imitation\" rule, it will select the neighbor with the highest payoff, and the strategy update is expressed as:  
\[f_{t}:j^{*}=\left\{