Promotion of cooperation based on swarm intelligence in spatial public goods games

doi:10.1016/j.amc.2017.10.022

Applied Mathematics and Computation

Volume 320, 1 March 2018, Pages 614-620

https://doi.org/10.1016/j.amc.2017.10.022 Get rights and content

Highlights

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We introduce the swarm intelligence methods into the evolutionary dynamics.
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We integrate the best personal strategy in the past with the best current strategy in the neighborhood.
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The cooperation level is enhanced markedly by the swarm intelligence.

Abstract

In this paper, we introduce the swarm intelligence methods into the evolutionary dynamics, and have studied the impact of swarm intelligent algorithm on the evolution of cooperation among selfish individuals in the continuous version of spatial public goods games (PGG). We update an individual’s strategy according to the memory which records the most successful individual strategy in the past (referred to as its personal best strategy) as well as the knowledge of the best current strategy found by its nearest neighbors (referred to as the neighborhood best strategy). Through extensive simulations, we find that the introduction of swarm intelligence into PGG can promote cooperation strongly. Other pertinent quantities such as the time evolution of cooperator density, the spatial distribution of strategies and the updated velocities are also investigated.

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 $k_{x} + 1$ PGG-groups centered at x and its k_x 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 ρ_c, defined as $ρ_{c} = L^{- 2} * \sum_{i = 0}^{L^{2}} s_{i},$ where s_i denotes the strategy of player i when the system reaches dynamical equilibrium. In all of the following simulations, ρ_c 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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Promotion of cooperation based on swarm intelligence in spatial public goods games

Highlights

Abstract

Introduction

Section snippets

Models and methods

Results and discussions

Conclusion

Acknowledgments

Phys. Life Rev.

BioSystems

Appl. Math. Comput.

J. Theor. Biol.

J. Theor. Biol

Phys. A

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Phys. A

Phys. Life Rev.

Phys. Lett. A

Phys. Rep.

Phys. Rep.

Appl. Math. Comput.

Phys. Lett. A

Trends Ecol. Evol.

J. Theor. Biol.

Game theory and its applications in the social and biological sciences

J. Behav. Decis. Making

The evolution of cooperation

Science

Evolutionary games and spatial chaos

Nature

Public goods provision in an experimental environment

J. Public Econ.

Public Goods: A Survey of Experimental Research

Sustaining cooperation in laboratory public goods experiments: a selective survey of the literature

Exp. Econ.

Diversity of cooperation in the tragedy of the commons

Biol. Theory

Phase transitions and volunteering in spatial public goods games

Phys. Rev. Lett.

Topology-independent impact of noise on cooperation in spatial public goods games

Phys. Rev. E

Success-driven distribution of public goods promotes cooperation but preserves defection

Phys. Rev. E

Individual wealth-based selection supports cooperation in spatial public goods games

Sci. Rep.

Social dilemma alleviated by sharing the gains with immediate neighbors

Phys. Rev. E

Cooperation percolation in spatial prisoner’s dilemma game

New J. Phys.

Diversity of reproduction time scale promotes cooperation in spatial prisoner’s dilemma games

Phys. Rev. E

Evolutionary dynamics of group interactions on structured populations: a review

J. R. Soc. Interface

Spatial structure often inhibits the evolution of cooperation in the snowdrift game

Nature

A double-edged sword: Benefits and pitfalls of heterogeneous punishment in evolutionary inspection games

Sci. Rep.

Different perceptions of social dilemmas: evolutionary multigames in structured populations

Phys. Rev. E