0.1 Overview

Our experiment is performed on the 100 × 100 square lattice with periodic boundary conditions. In initial, cooperators and defectors are distributed uniformly at random occupying half of the square lattice respectively, and all the nodes are given the same initial aspiration A. As the main parameters, we consider the initial aspiration level A and the temptation to defect b. Fig 1 presents the fraction $$ r_{\mathcal{C}}$$ of cooperators when stable as a function of the b for different aspiration levels. It is found that the aspiration level has a significant influence on $$ r_{\mathcal{C}}$$. Three different phases, Stable Coexistence under Low Aspiration, Dependent Coexistence under Moderate aspiration and Defection Explosion under High Aspiration could be observed for different values of A. For small values of A (A = 0 and A = 0.8), $$ r_{\mathcal{C}}$$ is equal/close to 0.5 no matter what value of b is. For large values of A (A ≥ 2.4), cooperators could hardly survive for any b > 1.0. An interesting phenomenon is discovered for moderate values of A (A = 1.6) that when b is lower than 1.6, cooperators could not survive and $$ r_{\mathcal{C}}$$ almost equals 0, but when b is higher than 1.6, cooperation could survive and $$ r_{\mathcal{C}}$$ is bigger than 0.2. According to common sense, the higher b is, the harder cooperators survive, which is inconsistent with our experimental results. Above unusual phenomenon is the focus in our paper.

Fig 1

Average fractions of cooperation when stable as a function of b for different values of the initial aspiration A, as obtained by means of simulations on square lattices.

0.2 Stable Coexistence under low aspiration (A ≤ 1.0)

For small values of A, individual’s aspiration is easy to be satisfied so cooperators and defectors can coexist. For A = 0, all the nodes are satisfied and never change their strategies, so $$ r_{\mathcal{C}}$$ will always keep 0.5. And for A = 0.8, there are only a few nodes whose four neighbors are all defectors will be dissatisfied: if it is a cooperator, it will change its strategy to $$ \mathcal{D}$$; if it is a defector, it will be always dissatisfied but can’t change its strategy since all his neighbors’ strategies are the same. At the beginning, $$ r_0=r_{\mathcal{D}0}=0.5$$, so the frequency of cooperators with four neighbors being all defectors could be calculated approximately:

0.3 Dependent Coexistence under moderate aspiration (1.0 < A ≤ 2.0)

For moderate values of A, cooperators can’t survive for small values of b. Fig 2 shows the spatial distributions of strategies and aspirations at different time steps t for A = 1.6 and b = 1.2. The evolution process can be divided into the following stages:

At first, every node with strictly less than two $$ \mathcal{C}$$ neighbors is dissatisfied. During this phase, defectors expand quickly and cooperators try to endure defectors’ invasion, which is the so-called END period [40–42]. $$ r_{\mathcal{C}}$$ decreases and the aspirations of all the nodes are lower than 2.0.

When t = 10, some cooperators still survive by forming some clusters in the END period. The defectors neighboring with the clusters are dissatisfied and have lower payoffs than their $$ \mathcal{C}$$ neighbors, so these clusters could expand by converting the neighboring defectors to cooperators, which is called the EXP period [40–42]. Dissatisfied defectors may evolve into cooperators gradually. Meanwhile, the aspirations of these cooperators gradually rise up.

When t = 100, cooperators have expanded fully during the EXP period and $$ r_{\mathcal{C}}$$ reaches a maximum. However, during the process of cooperation’s expansion, cooperations’ aspirations rise up gradually. The aspirations of the nodes on the boundary of the cooperators’ clusters are higher than 2.0 now. These cooperators are no longer satisfied if they neighbor with two defectors so they will re-evolve into defectors. This leads to a repeated evolution process on the boundaries of the cooperators’ clusters where nodes evolve into cooperators and defectors repeatedly. At the same time, nodes inside the cooperators’ clusters are still satisfied so they keep their strategies and their aspirations rise up gradually.

When t = 200, defectors gradually penetrated into the cooperators’ clusters. These cooperators’ aspirations are now close to 4.0 so once they neighbor with a defector, they will be dissatisfied and also change into defectors gradually. As a result, chain phenomenon happens that defectors almost occupy the entire network and the cooperators almost disappear rapidly.

Fig 2

Snapshots of typical distributions of strategies and aspirations at different time steps t for A = 1.6 and b = 1.2.

(a) represents strategies, where cooperators are depicted white and defectors are depicted black. (b) represents aspirations. The steps of them are t = 0, 10, 100, 200, 500 and 1000 respectively.

One can see that although cooperators can survive in the END period and expand in the EXP period by forming clusters, defectors finally occupy the network when it is stable. The network reciprocity is undermined by dynamic aspirations. In dynamic aspiration models, cooperators’ aspirations will become too high to endure defectors’ re-expansion because of the long-term satisfaction. which is different from the fixed aspiration model. Fig 3 shows the probability that cooperators could survive as a function of the cooperators’ initial proportion $$ r_{\mathcal{C}0}$$ for A = 1.6 and b = 1.2. For every different $$ r_{\mathcal{C}0}$$, 100 independent experiments is performed. Under these parameters, cooperators are possible to survive only when $$ r_{\mathcal{C}0}>0.95$$, and sure to survive only when $$ r_{\mathcal{C}0}>0.99$$. In other words, defectors could expand and occupy the network finally even if they are very few initially. This result is different from the result of fixed aspiration model and it may be caused by some special local structures which could lead to the high aspirations of cooperators.

Fig 3

The probability that cooperators can survive as a function of the cooperators’ initial proportion $$ r_{\mathcal{C}0}$$ for A = 1.6 and b = 1.2.

Cooperators are easier to survive when $$ r_{\mathcal{C}0}$$ is higher. They are possible to survive only when $$ r_{\mathcal{C}0}>0.95$$, and sure to survive only when $$ r_{\mathcal{C}0}>0.99$$.

Fig 4 shows all possible local structures in the network for A = 1.6 and b = 1.2. When a node has two or less $$ \mathcal{D}$$ neighbors, it is satisfied and won’t change its strategy and its aspiration will keep lower than its payoff. When a node has four $$ \mathcal{D}$$ neighbors, it is always dissatisfied. But since all its neighbors are defectors, it can only be a defector by copying neighbors’ strategies and won’t change its strategy any more, and its aspiration will keep decreasing.

Fig 4

The local structures of strategies for A = 1.6 and b = 1.2.

Each square corresponds to a single player, where cooperators are depicted blue and defectors are depicted red. Value denoted in the center square is the individual’s payoff. Smiling face represents satisfaction while crying face represents dissatisfaction.

Now we consider the structure that a node has three $$ \mathcal{D}$$ neighbors and one $$ \mathcal{C}$$ neighbor. Fig 5 shows the detailed principle for defectors’ expanding under this initial structure. The evolution process can be divided into the following stages:

When t = 0, the only one node dissatisfied is node X because its aspiration is 1.0. Since it has three D neighbors and one $$ \mathcal{C}$$ neighbor, X will evolve into cooperator and defector repeatedly. We call such node as an Infector. As a result, node Y’s payoff is sometimes 4.8 and sometimes 3.6, which we called an Infected node, and Y’s aspiration at step t can be got by the recursive equation:



With t growing, we can easily prove that A_Y will be higher than 3.6. Next time when X evolves into a defector, P_Y = 3.6 < A_Y, so Y is dissatisfied. Y has three $$ \mathcal{C}$$ neighbors, so it will be easy to evolve into a cooperator. But when Y evolves into cooperator, its payoff will decrease and it is still dissatisfied, so Y will also evolve into cooperator and defector repeatedly. In other words, the Infected node Y also becomes an Infector. For the same reason, node Z’s payoff is sometimes 4.0 and sometimes 3.0, Z becomes an Infected node and Z’s aspiration at step t can be got by the recursive equation:



With t further growing, we can easily prove that A_Z will be higher than 3.0. Next time when Y evolves into a defector, P_Z = 3.0 < A_Z, so Z is dissatisfied and may evolve into a defector in a few steps. Now Z’s other neighbors’ aspirations are all near to 4.0, which we call High-risk cooperator. Once Z evolves into a defector, their payoffs decrease to 3.0 so they are dissatisfied and may evolve into defectors, too.

Furthermore, almost every node’s aspiration in the network is near to 4.0 because their payoffs have been keeping 4.0 for a long time. In other word, all the cooperators in the network have became High-risk cooperators. As a result, for each cooperator i, once one of i’s neighbors evolves into a defector, i may evolve into a defector soon, which is a chain phenomenon and causes defectors’ expanding.

Fig 5

The detailed principle for defectors’ expanding for A = 1.6 and b = 1.2.

A node is surrounded by three defectors and one cooperator initially. Smiling face represents satisfaction while crying face represents dissatisfaction.

Fig 6 shows the spatial distributions of strategies and aspirations at different time steps t for A = 1.6, b = 1.2 with the above initial structure, from which we can also get the expansion trajectory by the aspiration distribution.

Fig 6

Snapshots of typical distributions of strategies and aspirations at different time steps t under the initial structure shown in Fig 4 for A = 1.6 and b = 1.2.

(a) represents strategies, where cooperators are depicted white and defectors are depicted black. (b) represents aspirations. The steps of them are t = 0, 10, 100, 200, 500 and 1000 respectively.

In the network with random setup, during the END and EXP period, cooperators will survive and expand by the mechanism of network reciprocity. But once there is at least one Infector who has three $$ \mathcal{D}$$ neighbors and one $$ \mathcal{C}$$ neighbor initially, defectors will re-expand to the whole network. If $$ r_{\mathcal{D}0}>0.05$$, such an Infector almost certainly exists, so defectors almost certainly expand to the whole network. Cooperators can survive only when there is no Infector initially in the network.

However, for large values of b, cooperators can partially survive. Fig 7 shows all possible local structures in the network for A = 1.6 and b = 1.7. Compared to Fig 4, if a cooperator has three $$ \mathcal{D}$$ neighbors and one $$ \mathcal{C}$$ neighbor, it will be dissatisfied and may evolve into a defector. But once it evolves into a defector, it becomes satisfied and doesn’t change any longer, and the network becomes stable. In other word, there is no Infector existing in the network. Fig 8 shows the spatial distributions of strategies and aspirations at different time steps t for A = 1.6, b = 1.7 with the above initial structure. Cooperators can survive by forming clusters, but as the values of b is higher, these clusters couldn’t expand. The above evolution process only goes through the END period and then the network has been stable.

Fig 7

The local structures of strategies for A = 1.6 and b = 1.7.

Each square corresponds to a single player, where cooperators are depicted blue and defectors are depicted red. Value denoted in the center square is the individual’s payoff. Smiling face represents satisfaction while crying face represents dissatisfaction.

Fig 8

Snapshots of typical distributions of strategies and aspirations at different time steps t for A = 1.6 and b = 1.7.

(a) represents strategies, where cooperators are depicted white and defectors are depicted black. (b) represents aspirations. The steps of them are t = 0, 10, 100, 200, 500 and 1000 respectively.

From the above we know the main difference between b < 1.6 and b ≥ 1.6 for A = 1.6 is whether a defector who has three $$ \mathcal{D}$$ neighbors and one $$ \mathcal{C}$$ neighbor is satisfied, more simply, whether an Infector exists. For b < 1.6, P = b < A, the defector is dissatisfied and will evolve into cooperator and defector repeatedly and it is an Infector, which causes the chain phenomenon leading to defectors’ expansion shown in Fig 5. But for b ≥ 1.6, P = b ≥ A, the defector is satisfied and the network will be stable soon so that cooperators can survive in the end. In fact, for all the A ≤ 2.0, the conclusion is the same and phase transition happens in b = A because A is the highest values of b which allows Infectors to exist.

To conclude, for moderate values of A, cooperators will survive and expand in the early stages of evolution when b is lower than A, which are END and EXP periods respectively. But according to our results, the existence of Infectors may lead to defectors’ re-expansion. The core reason for this phenomenon is that the cooperators increase their aspirations excessively and become the so-called High-risk cooperators, which needs to be vigilant in the evolution process of cooperation.

0.4 Defection Explosion under High Aspiration (A > 2.0)

For A = 2.4, cooperators can survive only when $$ r_{\mathcal{C}0}$$ is high enough for any b > 1, on the contrary, only a few defectors initially can lead cooperators to die out. Fig 9 shows all possible local structures in the network for A = 2.4 and b = 1.7. If a cooperator has two $$ \mathcal{D}$$ neighbors, it is dissatisfied and evolves into a defector soon, and then it is satisfied. If a cooperator or defector X has three $$ \mathcal{D}$$ neighbors, it is dissatisfied and evolves into cooperator and defector repeatedly. X’s payoff is 1.0 when it is a cooperator and 1.7 when it is a defector. X’s aspiration at step t can be got by the recursive equation:

Fig 9

The local structures of strategies for A = 2.4 and b = 1.7.

Each square corresponds to a single player, where cooperators are depicted blue and defectors are depicted red. Value denoted in the center square is the individual’s payoff. Smiling face represents satisfaction while crying face represents dissatisfaction.

Fig 10 shows the structure which causes the defectors’ expansion. In initial, nodes X₁ and X₂ are dissatisfied and may evolve into defectors. Once X₁ evolves into a defector, nodes Y₁ and Y₂ become dissatisfied and may also evolve into defectors, so do the other five nodes. All the nine nodes are dissatisfied and evolve into cooperators and defectors repeatedly. In other word, they are all Infectors. As a result, colored cooperators’ aspiration will be higher than 3.0 as time goes so they are High-risk cooperators. Next time when one of the nodes evolves into a defector, the cooperator will be dissatisfied and evolve into a defector. Since the other nodes’ aspirations are close to 4.0 and have became High-risk cooperators, chain phenomenon occurs and defectors will occupy the whole network.

Fig 10

The detailed principle for defectors’ expanding for A = 2.4, b = 1.7.

The initial local structure is shown in (a). Smiling face represents satisfaction while crying face represents dissatisfaction.

When b is lower, defectors’ expansion requires more strict requirement. When b = 1.2, for the same structure shown in Fig 10, defectors can’t expand. We find that before cooperators’ aspirations are higher than 3.0, the nine nodes will be all satisfied in a step so that the network becomes stable with high probability. The lower b makes the network stable soon if there are only nine nodes participating in the evolution. Fig 11 shows the initial structure which causes the defectors’ expansion, and similar to Fig 10, sixteen nodes participate in the evolution. More Infectors make the evolutionary process last longer, so the colored nodes have enough time to increase their aspirations to higher than 3.0 and all of them will become High-risk cooperators. Fig 12 shows the spatial distributions of strategies and aspirations at different time steps t for A = 2.4, b = 1.7 with the initial structure shown in Fig 10. The situation of A = 2.4, b = 1.2 with the initial structure shown in Fig 11 is almost the same. In fact, the above conclusion is suitable for all the 2.0 < A ≤ 3.0. Defectors’ expansion requires more defectors’ gathering when b is lower or A is higher, vice versa.

Fig 11

The initial structure that causes defectors’ expansion for A = 2.4, b = 1.2.

Smiling face represents satisfaction while crying face represents dissatisfaction.

Fig 12

Snapshots of typical distributions of strategies and aspirations at different time steps t for A = 2.4 and b = 1.7 with the initial structure shown in Fig 10.

(a) represents strategies, where cooperators are depicted white and defectors are depicted black. (b) represents aspirations. The steps of them are t = 0, 10, 100, 200, 500 and 1000 respectively.

For A > 3.0, the nodes which have at least one $$ \mathcal{D}$$ neighbor are dissatisfied, in other word, all the cooperators are High-risk cooperators initially. so once there are at least one defector in the network, defectors will expand to the whole network soon for any b > 1. The higher b is, the faster defectors’ expansion will be. Fig 13 shows the spatial distributions of strategies and aspirations at different time steps t for A = 3.2, b = 1.2 with only one defector initially. Cooperators are almost impossible to survive under high aspirations.

Fig 13

Snapshots of typical distributions of strategies and aspirations at different time steps t for A = 3.2 and b = 1.2 with only one defector initially.

(a) represents strategies, where cooperators are depicted white and defectors are depicted black. (b) represents aspirations. The steps of them are t = 0, 10, 100, 200, 500 and 1000 respectively.

In dynamic aspiration model, three different phases could be observed. The phase under low aspiration is similar to the fixed aspiration model because most nodes are always satisfied and their aspirations are changed in a small range. However, dynamic aspiration model plays a critical role under moderate aspiration and high aspiration, where some nodes called Infector are dissatisfied no matter they are cooperators or defectors and their strategies are changed repeatedly. As a result, their neighbors’ payoff are changed and aspirations will be influenced by the evolution process, and these neighbors act as Infected nodes. Their aspirations become higher gradually but their payoffs changed repeatedly, which results in their dissatisfaction and Infected nodes will become Infectors. Once a High-risk cooperator node becomes dissatisfied, chain phenomenon happens in High-risk cooperators and defectors will expand fast.