The effects of aspiration under multiple strategy updating rules on cooperation in prisoner's dilemma game

Abstract

In the traditional game theory, a widely accepted assumption is that the participants are completely rational and under the condition of complete information. But in the realistic economy experiences, the participants’ conditions of complete rationality and information are usually hard to be fulfilled. The mutations, un-fixed mechanisms, and policy variations usually challenge the process of natural game evolution. In this work, we mainly study how to make the iterative transformation of policy updating rules to maximize their own benefits in the case of heterogeneous updating rules. In particular, a novel mutation mechanism is developed to study the cooperation situation in natural selection by introducing two different proportion groups in the population, one is based on imitation, the other is based on aspiration level, and both proportions can transform each other under certain conditions at the end of each generation. The mutation of the individual mechanism may occur, it is found that the population proportion and individual rule mutation rate have a strong stabilizing effect on population cooperation under this mixed mechanism, and the lower aspiration level has a certain promoting effect on cooperation as well.

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.