Promotion of cooperation based on swarm intelligence in spatial public goods games
Introduction
Cooperative behavior is ubiquitous in both natural and social systems [1]. So far, the evolutionary game theory has provided a mathematical framework [2], [3] to address the emergence of cooperation. The public goods game (PGG) [4] has long been established as a paradigm in studying cooperative behavior. The original PGG is established as an archetypical context that succinctly captures a conflict between group interest and individual interests [5], [6]. In its simplest form, the game requires that players decide whether to contribute to a common pool or not. Regardless of the chosen strategy by the player himself, he receives an equal share of the public good which results from total contributions being multiplied by a fixed rate of return. Consequently, players are faced with the temptation of being free-riders and rational players invest nothing which causes the Tragedy of the Commons [7], [8], [9].
Considering the rapid development of complex network theory, much effort has been devoted to the evolutionary game on complex networks in the past decade [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. To explain the emergence of cooperation, researchers have proposed many important mechanisms, such as the direct and indirect reciprocity [23], [24], [25], memory effects [26], [27], noise [28], [29], [30], punishment and reward [31], [32], [33], [34], [35], migration [36], [37], [38], [39], social diversity [40], [41], [42], [43], volunteering [44], [45], [46], network reciprocity [47], [48] and aspiration [49], [50], [51], [52], [53], for the recent reviews about the evolution of cooperation, we recommend the readers to refer tox [54], [55], [56], [57] and references therein. Besides the above scenarios, the role of reputation mechanism is considered as an effective approach to help the cooperators to form the cooperative clusters to resist the invasion of defectors [58], [59], [60], [61], [62], [63]. Several typical examples include: Cuesta et al. [61] declared that the reputation really fosters the cooperation through experiments conducted within groups of humans playing an iterated prisoner’s dilemma on a dynamical network. Wang et al. [62] presented a new reputation inferring mechanism to consider the difference of the inferring ability among agents, through simulations they found that the introduction of reputation inferring could promote the cooperation strongly. Chen et al. [63] proposed a model of public goods game with adaptive reputation assortment simultaneously taking the individual reputation and strategy transfer difference into account, and the results demonstrated that the cooperative behaviors can be drastically promoted after the reputation assortment is introduced. However, in the above mentioned works, only the individuals’ past experiences are considered in the reputation mechanism. But in the complex real world systems, when individuals take strategies, they will not only be influenced by their past behaviors, but also consider the current environment. Inspired by the facts, here we propose a swarm intelligence method that take the historical behaviors and current environment into account. The swarm intelligence originates from the nature [64], [65], [66], [67], [68], swarm enables animals to solve problem that go beyond the capacity of single animal. Typical examples include termites swarm to build colonies, birds swarm to find food, bees swarm to reproduce. Essentially, the swarm intelligence is a mechanism that individuals interact with one another while learning from their own experience, and gradually move towards the goal. The swarm intelligence has been used in the prisoner’s dilemma game and the experimental results show that the swarm intelligence promotes the cooperation [69], [70], [71].
What is more, in most of the previous studies, agents are assumed to choose only between the two discrete strategies, namely to either purely cooperate or to purely defect. However, in many realistic cases, people are unlikely to stick with one simple pure strategy. In this paper, we will use the continuous version of the public goods game [72], [73]. A continuous variable from the unit interval [0,1] is introduced, which defines the fraction of the total cost a given player is willing to pay. The limits 0 and 1 recover the two pure strategies. It is important to study how players change their strategies, that is, increase or decrease their contributions to the common pool.
The following of this paper is organized as follows. In Section 2, we introduce the spatial public goods games model and the swarm intelligence. In Section 3, we show the experimental results and discussions. In Section 4, we summarize our findings.
Section snippets
Models and methods
Players are located on an L × L square lattice with periodic boundary conditions. Every player occupies a lattice point and has four neighboring points. During the evolutionary process, each player x participates in PGG-groups centered at x and its kx neighbors, where each group contains a central node and nodes connected to the central node. Initial, a strategy is randomly chosen for each player with uniform probability in the interval [0, 1]. At each time step, each player contributes a
Results and discussions
In the following studies, players are assumed to occupy nodes on a 200*200 square lattice, but the results remain valid also if we use larger lattices. The key quantity that characterizes the cooperative behavior of a system is the cooperation fraction ρc, defined as where si denotes the strategy of player i when the system reaches dynamical equilibrium. In all of the following simulations, ρc is obtained by averaging over the last 2000 generations of the entire 100,000
Conclusion
In conclusion, we investigated the effect of swarm intelligence method on the evolution of cooperation in the continuous version of spatial public goods games. Under the introduction of swarm intelligence, individuals update their strategies in the next time according to the integration of their personal best strategy in the past and the best current strategy in their neighborhood. Our results reveal that the cooperative behavior can be effectively produced and maintained when adopting swarm
Acknowledgments
This work is partly supported by the National Natural Science Foundation of China under Grant Nos. 61403083, 61672159 and 61773121, the Technology Innovation Platform Project of Fujian Province under Grant Nos. 2014H2005 and 2009J1007, the Fujian Collaborative Innovation Center for Big Data Application in Governments, the Fujian Engineering Research Center of Big Data Analysis and Processing, the training plan established in the year 2017 for Distinguished Young Scholars of Fujian Province, the
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2022, Applied Mathematics and ComputationCitation Excerpt :In PGG, selfish individuals are inclined to contribute nothing to the common pool and to be a free rider, but if nobody contributes, the group will fail in reaping benefits, and thus it leads to the tragedy of the commons[21–23]. In most existing researches, the common resource keeps basically constant[24–27]. However, in reality, human behavior affects and is also affected by the environment they are located in, which brings about the evolution of the resources.
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2022, Applied Mathematics and ComputationCitation Excerpt :However, these above strategy update rules focus on the case of discrete strategy but neglect the case of continuous strategy [59]. The scholars try to introduce PSO intelligence algorithm into the strategy update rule and investigate the effect of PSO learning rule on the evolution of cooperation in social dilemma with continuous strategy [60–68]. Inspired by PSO intelligence algorithm, players will consider both the performance of the opponents and their own previous experiences when they adjust their strategies [69,70].