We investigate the effects of Memory Step M and sucker's (S) reward function on the evolution of cooperation first. Figure 1(a) shows how the stationary frequency of cooperation Fc varies in the range of defection temptation \(b\) together with different Sucker(S)'s factor α at a fixed step size \(M=3\). When \(\alpha =0\), all players played the PDG in the weakest dilemma, and the frequency of cooperation decreased rapidly even the value of b is small. When \(0<\alpha \le 1\), there are two kinds of game models in the whole population, and the role of multi-game in the interactive diversity effectively promotes cooperation.
At the same time, the player who can keep his strategy unchanged for \(3\) steps can switch from PDG to SDG, which is to avoid a higher social dilemma. In particular, the higher the \(\alpha\), the higher the frequency of cooperators. Figure 1(b) shows how cooperation evolves in the range of \(b\) when \(\alpha =0.8\) for different \(M\). The whole evolutionary process is much more resistant to the temptation of defection \(b\). This is because the weak dilemma Snowdrift Game is introduced into the interactive diversity. When \(b\) starts at \(1.8\), the temptation effect starts to kick in. It can be seen from the figure that when M is small, the cooperation frequency is the highest. However, it is difficult for agents to keep their strategies unchanged for a long time. With the increase of \(M\), a large number of participants will fall into the high social dilemma game inevitably (Continuous PDG), which is not conducive to the survival of cooperation. The larger the value of \(M\), the more likely of population falls into a high social dilemma PDG.
To observe the influence of suckers (\(S\)) \(\alpha\) and the defection of temptation (\(T\)) \(b\) on the evolution of cooperation respectively and the influence of the two together on the evolution of cooperation. We have drawn three different color maps, \(M-\alpha\), \(M-b\), and \(b-\alpha\). In Fig. 2. Figure \(M-\alpha\) shows the effect of different Memory Step \(M\) and different Suckers (S) \(\alpha\) on cooperation rate, and the curve of the cooperation rate change for all \(M\)can also be seen in the figure. When \(M=1\), the social dilemma is the weakest in the whole game process. As M increases, more and more players fall into the Prisoner's Dilemma Game, the social dilemma gradually becomes bigger and bigger, but the cooperation rate decreases gradually. Figure \(M-b\) is about the influence of different Memory Step \(M\) and different defection temptation \(b\) on cooperation rate. For all the values of \(M\), the resistance of the whole group to the defection temptation of \(b\) first increases and then decreases regardless \(b\).
When \(M=1\), the resistance of the whole group to the defection temptation of \(b\) is the highest. As the value of \(M\) increases, the resistance of the whole group to the defection temptation of \(b\) gradually decreases and finally stabilizes at about \(b =1.8\). Figure b-α shows the effect of the co-action of defection temptation \(b\) and Suckers (S) \(\alpha\) on the evolution of cooperation. As shown in the figure, with the combined action of the two values, four phases can be found: the full \(C\) state, the mixed \(C+D\) state, the mixed \(D+C\) state, and the full \(D\) state. It can be seen that when \(b<1.8\), no matter what \(\alpha\) is, the partner is dominant owing to the effect of Memory Step \(M\)(SDG is introduced). Besides, these observations indicate that in multi-game with interactive diversity, introducing Memory Step not only improves the survival ability of the cooperators but also introduces enhanced network reciprocity into the model. Then, we will explain these findings in detail.
Figure 4 shows the overall evolution of cooperation under different Memory Step M. Where, \({P}_{C}\) and \({P}_{D}\) represent the survival rates of cooperators and defectors in the game process, respectively. The upper part of the figure is the survival rate of defectors, and the lower part is the survival rate of cooperators, \(M= 0, 3, 12, 40.\) Meanwhile, to more clearly illustrate the existing threshold of Memory Step \(M\), we set a larger \(M=500\). It can be seen from the figure, no matter what the value of M is, the survival rate of the cooperators first decreases and then increases, while the survival rate of the corresponding defectors first increases and then decreases. For different \(M\), the survival rate of cooperators becomes lower and lower when they finally stabilize. It indicates that when \(M\) does not exceed the threshold value, with the increase of \(M\), the influence on cooperation becomes smaller and smaller, and the survival rate of cooperators becomes lower and lower. When the threshold value of \(M\) is exceeded, the influence on cooperation is the least, and the survival rate of the cooperators is the lowest. After a long time of evolution, only the defectors exist in the final game group.
This shows that in multi-game based on interactive diversity, the introduction of Memory Step has an obvious effect on cooperation. However, players in the game group would only maintain their strategies for a certain number of games, and once the number of moves exceeds this number, their strategies will change, that is, the Memory Step \(M\) exists a limitation.
As shown in Fig. 5, four snapshots from top to bottom are \(M= \text{0,3},\text{12,500}\). when \(M=0\), the game steps from left to right are 0, 20, 140, 1050; when M=3, the game steps from left to right are 0, 50, 1070, 2800; when \(M=12\), the game steps from left to right are 0, 260, 3200, 7000; when \(M=500\), the game steps from left to right are 0, 350, 5000, 10000. For the first three groups of images, we find that no matter what the value of \(M\) is and how the number of game steps changes, the number of cooperators always decreases first and then increases. Compared with the first three pictures, in the fourth picture, it can be seen that the number of our cooperators become less and less. Eventually, the players become the traitors who play PDG and will be satisfied with the current policy. And, the game becomes that only defectors continue to play SDG. This indicates that Memory Step \(M\) has some limitations. The closer to the threshold \(M\), betrayal is the best choice, and everyone will become a traitor in the end. Interestingly, from all the images, it can be also found that PDG players are easily satisfied within the historical step -- keeping the strategy unchanged. As the game goes on, more and more players in the games would play SDG. During the game, players form clusters. From (1,2), (2,2), and (1,3), (2,3), the defection in SDG games will slowly erode collaborators. From (2,2), (3,2), and (2,3), (3,3), the cooperators in SDG will gradually form clusters and resist defectors, which make defectors gradually develop toward cooperation as well. This indicates that the cooperators will form clusters and make them more compact and difficult for the defectors to attack. The reason behind this is that the payoff for players at the cluster boundaries is high.