The effects of aspiration under multiple strategy updating rules on cooperation in prisoner's dilemma game
Introduction
Cooperation exists widely in human society and nature, for example, ants live together, sacrifice each other, mutual benefit; upstream and downstream industry companies cooperate to complete product development. However, when the cooperation produces considerable benefits, the participants tend to give up cooperation to take advantage of the efforts of others under the regulation of selfish genes because of the high cost of cooperation [1,2]. Despite this unreasonable phenomenon, how cooperation prevails in social life and nature has aroused the interest of researchers [3,4]. Besides, Monte Carlo method in statistical physics is also used to study and analyze cooperative phenomena [5,6].
Recent studies on the emergence mechanism of cooperation mainly focus on network structure analysis. A lattice structure was proposed in Nowak and May's [7] seminal work to support cooperative behavior. By following this idea, works [8], [9], [10], [11], [12], [13], [14], [15], [16] have been devoted to the relationship between different types of complex networks and cooperation. In different ways, other studies have explored the impact of cooperation under different mechanisms, including co-evolution [17], [18], [19], [20], [21], [22], multi-strategy [23,24], aspiration [25,26,[30], [31], [32], [33]], social games exclusion [27] and strategy persistence [28,29]. In these studies, players heavily rely on the benefits from the game from their neighbors to decide their strategies for the next round. Meanwhile, the differences between individuals and the incomplete information would affect players’ strategies as well.
To overcome the limitation of external dependence in the evolution process as discussed above, two types of information resources are incorporated in the proposed work to achieve a more effective strategy evolution of the players. One is heterogeneous updating rules, the other is mutation iteration. Other studies have found that the cooperation level under the mixed mechanism is different from that under the single mechanism, such as Fang's research about dissimilarity-driven behavior [34]. It further proves that, in a well-mixed population, cooperation among imitators can be strengthened by an innovative rule combination, while in a structured population, such cooperation might be weakened instead [35], [36], [37], [38], [39].
Therefore, in the proposed model, the co-existence of the imitation dynamics [40] and aspiration dynamics are also considered. In this scheme, the updating rules of players are not fixed, which means that the players are able to dynamically change the updating rules of next round according to game returns of each round. Finally, the system will reach an equilibrium state in the continuous evolution. The proposed model has high flexibility and diversity of individual choice, and provides a new insight for the study of traditional renewal rules in the game.
Another information resource is mutation iteration [41], [42], [43]. Usually, in each round of the game, players' updating rules will change under certain conditions due to the random strategy chosen, which causes the widely used mutation strategy mechanism results in weaken the cooperation in payoff-based copying rule game. In this work, a different mechanism of rule updating by mutated individuals is employed, which can achieve a stronger stabilization performance on the cooperation rate of the population. The rest of the paper is organized as follows. The method proposed will be introduced in Section 2. Section 3 discusses the experiments and results. Section 4 concludes this work.
Section snippets
Model
To obtain the evolutionary outcomes, in the proposed work, we employed the model of weak prisoner's dilemma game, which is performed on 300 × 300 square lattices with periodic boundary conditions. Each node is assigned with a player, and all players have eight neighbors, with whom they participate in games. In the game, two players have the right to choose their own strategy, cooperation or defect. If the two players cooperate with each other, they will get R = 1, when they choose defection
Experiments
To evaluate the proposed work, the experiment of prisoner's dilemma is performed on 300 × 300 lattices with periodic boundary conditions. The frequency of cooperation Fc, an important metric, is obtained with 104 Monte Carlo steps by removing the unstable part anterior, and then take the average cooperation rate of the stable phase (the last 5000 steps) to make the result as accurate as possible.
Firstly, the relationship between the frequency of cooperation Fc and the temptation to defect b are
Conclusion
In this paper, we conduct Monte Carlo simulation to test their role in prisoner's dilemma by studying mutation iteration and heterogeneous strategy updating rules. The simulation results illustrate that the population proportion tends to be stable and finally reaches a certain proportion under the iterative condition of multiple update rules. In addition, such a mechanism is also able to resist the invasion of mutation rate and stabilize the cooperation rate of the whole population.
Different
Acknowledgments
This work is supported by National Natural Science Foundation of China 61801392, 61971352, Defense Industrial Technology Development Program (JCKY2018205B001), Equipment pre-research project (61400010207), as well as supported by the Fundamental Research Funds for the Central Universities (3102019ZY1004).
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