1
|
Takiguchi Y, Nemoto K. Approximate master equations for the spatial public goods game. Phys Rev E 2024; 109:024304. [PMID: 38491662 DOI: 10.1103/physreve.109.024304] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2023] [Accepted: 01/18/2024] [Indexed: 03/18/2024]
Abstract
The spatial public goods game has been used to examine factors that promote cooperation. Owing to the complexity of the dynamics of this game, previous studies on this model neglected analytical approaches and relied entirely on numerical calculations using the Monte Carlo (MC) simulations. In this paper, we present the approximate master equations (AMEs) for this model. We report that the results obtained by the AMEs are mostly qualitatively consistent with those obtained by the MC simulations. Furthermore, we show that it is possible to obtain phase boundaries analytically in certain parameter regions. In the region where the noise in strategy decisions is very large, the phase boundary can be obtained analytically by considering perturbations from the steady state of the voter model. In the noiseless region, discontinuous phase transitions occur because of the characteristics of the function that represents strategy updating. Our approach is useful for clarifying the details of the mechanisms that promote cooperation and can be easily applied to other group interaction models.
Collapse
Affiliation(s)
- Yu Takiguchi
- Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
| | - Koji Nemoto
- Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
| |
Collapse
|
2
|
Guo H, Shen C, Zou R, Tao P, Shi Y, Wang Z, Xing J. Complex pathways to cooperation emergent from asymmetry in heterogeneous populations. CHAOS (WOODBURY, N.Y.) 2024; 34:023139. [PMID: 38416672 DOI: 10.1063/5.0188177] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/20/2023] [Accepted: 02/02/2024] [Indexed: 03/01/2024]
Abstract
Cooperation within asymmetric populations has garnered significant attention in evolutionary games. This paper explores cooperation evolution in populations with weak and strong players, using a game model where players choose between cooperation and defection. Asymmetry stems from different benefits for strong and weak cooperators, with their benefit ratio indicating the degree of asymmetry. Varied rankings of parameters including the asymmetry degree, cooperation costs, and benefits brought by weak players give rise to scenarios including the prisoner's dilemma (PDG) for both player types, the snowdrift game (SDG), and mixed PDG-SDG interactions. Our results indicate that in an infinite well-mixed population, defection remains the dominant strategy when strong players engage in the prisoner's dilemma game. However, if strong players play snowdrift games, global cooperation increases with the proportion of strong players. In this scenario, strong cooperators can prevail over strong defectors when the proportion of strong players is low, but the prevalence of cooperation among strong players decreases as their proportion increases. In contrast, within a square lattice, the optimum global cooperation emerges at intermediate proportions of strong players with moderate degrees of asymmetry. Additionally, weak players protect cooperative clusters from exploitation by strong defectors. This study highlights the complex dynamics of cooperation in asymmetric interactions, contributing to the theory of cooperation in asymmetric games.
Collapse
Affiliation(s)
- Hao Guo
- Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China
| | - Chen Shen
- Faculty of Engineering Sciences, Kyushu University, Fukuoka 816-8580, Japan
| | - Rongcheng Zou
- School of Automation, Northwestern Polytechnical University, Xi'an 710072, China
| | - Pin Tao
- Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China
| | - Yuanchun Shi
- Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China
| | - Zhen Wang
- School of Artificial Intelligence, Optics and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an 710072, China
| | - Junliang Xing
- Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China
| |
Collapse
|
3
|
Li X, Wang W, Ma Y, An X, Wang T, Shi L. Tax thresholds yield multiple optimal cooperation levels in the spatial public goods game. CHAOS (WOODBURY, N.Y.) 2023; 33:123119. [PMID: 38085227 DOI: 10.1063/5.0180979] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2023] [Accepted: 11/13/2023] [Indexed: 12/18/2023]
Abstract
Income redistribution, which involves transferring income from certain individuals to others, plays a crucial role in human societies. Previous research has indicated that tax-based redistribution can promote cooperation by enhancing incentives for cooperators. In such a tax system, all individuals, irrespective of their income levels, contribute to the tax system, and the tax revenue is subsequently redistributed to everyone. In this study, we relax this assumption by introducing a tax threshold, signifying that only individuals with incomes exceeding the threshold will be subject to taxation. In particular, we employ the spatial public goods game to investigate the influence of tax rates-the percentage of income allocated to tax-and tax thresholds, which determine the income level at which individuals become taxable, on the evolution of cooperation. Our extensive numerical simulations disclose that tax thresholds produce complex outcomes for the evolution of cooperation, depending on tax rates. Notably, at low tax rates (i.e., below 0.41), as the tax threshold increases, discontinuous phase transitions in cooperation performance suggest the presence of multiple intervals of effective tax thresholds that promote peak cooperation levels. Nevertheless, irrespective of the chosen tax rate, once the tax threshold surpasses a critical threshold, the redistribution mechanism fails, causing the collapse of cooperation. Evolutionary snapshots show that self-organized redistribution forms an intermediary layer on the peripheries of cooperative clusters, effectively shielding cooperators from potential defectors. Quantitative analyses shed light on how self-organized redistribution narrows the income gap between cooperators and defectors through precise identification of tax-exempt entities, thereby amplifying the cooperative advantage. Collectively, these findings enhance our comprehension of how income redistribution influences cooperation, highlighting the pivotal role of tax thresholds.
Collapse
Affiliation(s)
- Xiaogang Li
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China
| | - Wei Wang
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China
| | - Yongjuan Ma
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China
| | - Xingyu An
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China
| | - Ting Wang
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China
| | - Lei Shi
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China
- Interdisciplinary Research Institute of Data Science, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China
| |
Collapse
|
4
|
Gao B, Tao K, Mu C, Chu C, Li H. Asymmetry of individual activity promotes cooperation in the spatial prisoner's dilemma game. CHAOS (WOODBURY, N.Y.) 2023; 33:093143. [PMID: 37756610 DOI: 10.1063/5.0160264] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2023] [Accepted: 09/07/2023] [Indexed: 09/29/2023]
Abstract
We consider an aspiration-based asymmetric individual activity co-evolutionary prisoner's dilemma game model on the square lattice. In detail, each player has an individual weight that evolves with its strategy. We introduce an asymmetric setting that only μ fractions of players in the network chosen to be active can update their individual weights according to whether their payoffs satisfy their aspirations. Therefore, our individual weights can be regarded as a type of intrinsic motivation satisfaction. Through Monte Carlo simulations on the square lattice, our co-evolution mechanism has been proven to significantly promote cooperation. Furthermore, we find that the moderate μ leads to the highest cooperation level. By dividing the players into four types according to their strategies and weights, we prove that the inversion of the dominant relationship between cooperators and defectors with low weights leads to a conversion cycle of the four types of players, which enhances spatial reciprocity to promote cooperation. Moderate μ results in an appropriate number of low-weight players in this cycle. Interestingly, our results show that higher weight heterogeneity does not lead to higher levels of cooperation, which is contrary to the intuition formed in the previous work.
Collapse
Affiliation(s)
- Bo Gao
- School of Computer Information Management, Inner Mongolia University of Finance and Economics, Hohhot, Inner Mongolia 010070, China
| | - Kexin Tao
- School of Business Administration, Inner Mongolia University of Finance and Economics, Hohhot, Inner Mongolia 010070, China
| | - Chunjiang Mu
- School of Cybersecurity, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China
| | - Chen Chu
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
| | - Hao Li
- School of Cybersecurity, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China
| |
Collapse
|
5
|
David-Barrett T. Human group size puzzle: why it is odd that we live in large societies. ROYAL SOCIETY OPEN SCIENCE 2023; 10:230559. [PMID: 37593705 PMCID: PMC10427830 DOI: 10.1098/rsos.230559] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/28/2023] [Accepted: 07/21/2023] [Indexed: 08/19/2023]
Abstract
Human groups tend to be much larger than those of non-human primates. This is a puzzle. When ecological factors do not limit primate group size, the problem of coordination creates an upper threshold even when cooperation is guaranteed. This paper offers a model of group coordination towards behavioural synchrony to spell out the mechanics of group size limits, and thus shows why it is odd that humans live in large societies. The findings suggest that many of our species' evolved social behaviours and culturally maintained social technologies emerged as solutions to this problem.
Collapse
|
6
|
Flores LS, Vainstein MH, Fernandes HCM, Amaral MA. Heterogeneous contributions can jeopardize cooperation in the public goods game. Phys Rev E 2023; 108:024111. [PMID: 37723706 DOI: 10.1103/physreve.108.024111] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2023] [Accepted: 07/12/2023] [Indexed: 09/20/2023]
Abstract
When studying social dilemma games, a crucial question arises regarding the impact of general heterogeneity on cooperation, which has been shown to have positive effects in numerous studies. Here, we demonstrate that heterogeneity in the contribution value for the focal public goods game can jeopardize cooperation. We show that there is an optimal contribution value in the homogeneous case that most benefits cooperation depending on the lattice. In a heterogeneous scenario, where strategy and contribution coevolve, cooperators making contributions higher than the optimal value end up harming those who contribute less. This effect is notably detrimental to cooperation in the square lattice with von Neumann neighborhood, while it can have no impact in other lattices. Furthermore, in parameter regions where a higher-contributing cooperator cannot normally survive alone, the exploitation of lower-value contribution cooperators allows their survival, resembling a parasitic behavior. To obtain these results, we examined the effect of various distributions for the contribution values in the initial condition and we conducted Monte Carlo simulations.
Collapse
Affiliation(s)
- Lucas S Flores
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, Rio Grande do Sul, Brazil
| | - Mendeli H Vainstein
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, Rio Grande do Sul, Brazil
| | - Heitor C M Fernandes
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, Rio Grande do Sul, Brazil
| | - Marco A Amaral
- Instituto de Humanidades, Artes e Ciências, Universidade Federal do Sul da Bahia, CEP 45638-000, Teixeira de Freitas, Bahia, Brazil
| |
Collapse
|
7
|
Yao Y, Zeng Z, Pi B, Feng M. Inhibition and activation of interactions in networked weak prisoner's dilemma. CHAOS (WOODBURY, N.Y.) 2023; 33:2894480. [PMID: 37276564 DOI: 10.1063/5.0146999] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2023] [Accepted: 05/18/2023] [Indexed: 06/07/2023]
Abstract
In the framework of the coevolution dynamics of the weak prisoner's dilemma, inspired by prior empirical research, we present a coevolutionary model with local network dynamics in a static network framework. Viewing the edges of the network as social interactions between individuals, when individuals play the weak prisoner's dilemma game, they accumulate both payoffs and social interaction willingness based on a payoff matrix of the social interaction willingness we constructed. The edges are then inhibiting or activating based on the social interaction willingness of the two individuals, and individuals only interact with others through activated edges, resulting in local network dynamics in a static network framework. Individuals who receive more cooperation will be more likely to activate the edges around them, meaning they will participate in more social interactions. Conversely, individuals who receive more defects will do the opposite. Specifically, we investigate the evolutionary dynamics of cooperation under different levels of sensitivity to social interaction willingness and the temptation to defect. Through the simulation, we find that sparse cooperator clusters can expand greatly when social interaction sensitivity and temptation to defect are low. In contrast, dense cooperator clusters form rapidly in a high social interaction sensitivity, which protects the cooperation from high temptation.
Collapse
Affiliation(s)
- Yichao Yao
- College of Artificial Intelligence, Southwest University, Chongqing 400715, China
| | - Ziyan Zeng
- College of Artificial Intelligence, Southwest University, Chongqing 400715, China
| | - Bin Pi
- College of Artificial Intelligence, Southwest University, Chongqing 400715, China
| | - Minyu Feng
- College of Artificial Intelligence, Southwest University, Chongqing 400715, China
| |
Collapse
|
8
|
David-Barrett T. Clustering drives cooperation on reputation networks, all else fixed. ROYAL SOCIETY OPEN SCIENCE 2023; 10:230046. [PMID: 37122944 PMCID: PMC10130726 DOI: 10.1098/rsos.230046] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/16/2023] [Accepted: 04/03/2023] [Indexed: 05/03/2023]
Abstract
Reputation-based cooperation on social networks offers a causal mechanism between graph properties and social trust. Using a simple model, this paper demonstrates the underlying mechanism in a way that is accessible to scientists not specializing in networks or mathematics. The paper shows that when the size and degree of the network is fixed (i.e. all graphs have the same number of agents, who all have the same number of connections), it is the clustering coefficient that drives differences in how cooperative social networks are.
Collapse
Affiliation(s)
- Tamas David-Barrett
- Trinity College, University of Oxford, Oxford OX1 3BH, UK
- Population Studies Institute, Helsinki 00101, Finland
| |
Collapse
|
9
|
Chen H, Wang L, Wang X. A combat game model with inter-network confrontation and intra-network cooperation. CHAOS (WOODBURY, N.Y.) 2023; 33:033123. [PMID: 37003839 DOI: 10.1063/5.0137338] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/19/2023]
Abstract
Inter-network combat and intra-network cooperation among structured systems are likely to have been recurrent features of human evolutionary history; however, little research has investigated the combat mechanism between structured systems that the adversarial interactions will cause the disability of agents and agents are prone to seek cooperation with neighbors. Hence, the current study has proposed a two-network combat game model and designed the corresponding rules of how to attack, how to be disabled, how to cooperate, and how to win. First, within the framework of our model, we have simulated the combat among four common network structures-the Erdős-Rényi (ER) random network, the grid graph, the small-world network, and the scale-free network. We found that the grid network always holds the highest winning percentage, while the ER random graph is most likely to lose when combating with the other three network structures. For each structure, we have also simulated the combat between the same network structures with different generating parameters. The simulations reveal that the small-world property and heterogeneity can promote winning a combat. Besides, by broadening and deepening cooperation, we have found that broader cooperation helps defeat the opposite system on grid and scale-free networks, yet hinders it on ER and Watts-Strogatz (WS) networks, while deeper cooperation can benefit to winning except on scale-free networks. These findings inform our understanding of the effects of structure and cooperation in a combat.
Collapse
Affiliation(s)
- Hao Chen
- Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, People's Republic of China
| | - Lin Wang
- Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, People's Republic of China
| | - Xiaofan Wang
- Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, People's Republic of China
| |
Collapse
|
10
|
He X, Li G, Du H. Conformity effect on the evolution of cooperation in signed networks. CHAOS (WOODBURY, N.Y.) 2023; 33:023114. [PMID: 36859219 DOI: 10.1063/5.0101350] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2022] [Accepted: 01/16/2023] [Indexed: 06/18/2023]
Abstract
Human behaviors are often subject to conformity, but little research attention has been paid to social dilemmas in which players are assumed to only pursue the maximization of their payoffs. The present study proposed a generalized prisoner dilemma model in a signed network considering conformity. Simulation shows that conformity helps promote the imitation of cooperative behavior when positive edges dominate the network, while negative edges may impede conformity from fostering cooperation. The logic of homophily and xenophobia allows for the coexistence of cooperators and defectors and guides the evolution toward the equality of the two strategies. We also find that cooperation prevails when individuals have a higher probability of adjusting their relation signs, but conformity may mediate the effect of network adaptation. From a population-wide view, network adaptation and conformity are capable of forming the structures of attractors or repellers.
Collapse
Affiliation(s)
- Xiaochen He
- School of Public Policy and Administration, Xi'an Jiaotong University, Xi'an, Shaanxi Province 710049, China
| | - Guangyu Li
- School of Public Policy and Administration, Xi'an Jiaotong University, Xi'an, Shaanxi Province 710049, China
| | - Haifeng Du
- School of Public Policy and Administration, Xi'an Jiaotong University, Xi'an, Shaanxi Province 710049, China
| |
Collapse
|
11
|
Shi Z, Wei W, Li B, Li C, Li H, Zheng Z. Two-layer network model of public goods games with intervention and corruption. CHAOS (WOODBURY, N.Y.) 2022; 32:063138. [PMID: 35778150 DOI: 10.1063/5.0088493] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/17/2022] [Accepted: 05/16/2022] [Indexed: 06/15/2023]
Abstract
Public goods games are widely used to model social dilemmas involving multiple agents. Though defection is the only rational choice for an individual in a public goods game, cooperative behavior is observed in a variety of social dilemmas, which is the subject of our research. Punishing defectors has been shown to be an effective mechanism for promoting cooperation, but it relies on the third-party umpire being fair. In this article, an umpire intervention model with corruption is proposed to explore the impact of corruption on punishment mechanisms. In our model, players and umpires operate in a multilayer network. The players play public goods games, which are overseen by umpires. Fair umpires punish defectors, whereas corrupt umpires take bribes from defectors rather than meting out a punishment. We separately explore the effects of the fraction of fair umpires ρ, the spatial distribution, and the fine cost α and bribe cost β. Our Monte Carlo simulation shows that the above factors have a significant impact on cooperation. Intervention by an umpire always improves social efficiency, even for an entirely corrupt system. Moreover, relatively developed systems can resist corruption. Staggered and centralized distributions always have opposite effects on cooperative behavior, and these effects depend on ρ and r. We also find that whether cooperators fully occupy the player layer depends only on whether β reaches a certain threshold.
Collapse
Affiliation(s)
- Zhenyu Shi
- School of Mathematical Sciences, Beihang University, Beijing 100191, China
| | - Wei Wei
- School of Mathematical Sciences, Beihang University, Beijing 100191, China
| | - Baifeng Li
- School of Mathematical Sciences, Beihang University, Beijing 100191, China
| | - Chao Li
- Department of Mathematics and Computer Science, Hengshui University, Hengshui 053000, China
| | - Haibin Li
- Key Laboratory of Mathematics Informatics Behavioral Semantics, Ministry of Education, Beijing 100191, China
| | - Zhiming Zheng
- School of Mathematical Sciences, Beihang University, Beijing 100191, China
| |
Collapse
|
12
|
Liang R, Wang Q, Zhang J, Zheng G, Ma L, Chen L. Dynamical reciprocity in interacting games: Numerical results and mechanism analysis. Phys Rev E 2022; 105:054302. [PMID: 35706290 DOI: 10.1103/physreve.105.054302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2021] [Accepted: 04/12/2022] [Indexed: 06/15/2023]
Abstract
We study the evolution of two mutually interacting pairwise games on different topologies. On two-dimensional square lattices, we reveal that the game-game interaction can promote the cooperation prevalence in all cases, and the cooperation-defection phase transitions even become absent and fairly high cooperation is expected when the interaction becomes very strong. A mean-field theory is developed that points out dynamical routes arising therein. Detailed analysis shows indeed that there are rich categories of interactions in either the individual or bulk scenario: invasion, neutral, and catalyzed types; their combination puts cooperators at a persistent advantage position, which boosts the cooperation. The robustness of the revealed reciprocity is strengthened by the studies of model variants, including the public goods game, asymmetrical or time-varying interactions, games of different types, games with timescale separation, different updating rules, etc. The structural complexities of the underlying population, such as Newman-Watts small world networks, Erdős-Rényi random networks, and Barabási-Albert networks, also do not alter the working of the dynamical reciprocity. In particular, as the number of games engaged increases, the cooperation level continuously improves in general. However, our analysis shows that the dynamical reciprocity works only in structured populations, otherwise the game-game interaction has no any impact on the cooperation at all. In brief, our work uncovers a cooperation mechanism in the structured populations, which indicates the great potential for human cooperation since concurrent issues are so often seen in the real world.
Collapse
Affiliation(s)
- Rizhou Liang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, People's Republic of China
| | - Qinqin Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, People's Republic of China
| | - Jiqiang Zhang
- School of Physics and Electronic-Electrical Engineering, Ningxia University, Yinchuan 750021, People's Republic of China
- Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100191, People's Republic of China
| | - Guozhong Zheng
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, People's Republic of China
| | - Lin Ma
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, People's Republic of China
| | - Li Chen
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, People's Republic of China
- Robert Koch-Institute, Nordufer 20, 13353 Berlin, Germany
| |
Collapse
|
13
|
A Mutation Threshold for Cooperative Takeover. Life (Basel) 2022; 12:life12020254. [PMID: 35207541 PMCID: PMC8874834 DOI: 10.3390/life12020254] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2021] [Revised: 01/27/2022] [Accepted: 01/29/2022] [Indexed: 11/17/2022] Open
Abstract
One of the leading theories for the origin of life includes the hypothesis according to which life would have evolved as cooperative networks of molecules. Explaining cooperation—and particularly, its emergence in favoring the evolution of life-bearing molecules—is thus a key element in describing the transition from nonlife to life. Using agent-based modeling of the iterated prisoner’s dilemma, we investigate the emergence of cooperative behavior in a stochastic and spatially extended setting and characterize the effects of inheritance and variability. We demonstrate that there is a mutation threshold above which cooperation is—counterintuitively—selected, which drives a dramatic and robust cooperative takeover of the whole system sustained consistently up to the error catastrophe, in a manner reminiscent of typical phase transition phenomena in statistical physics. Moreover, our results also imply that one of the simplest conditional cooperative strategies, “Tit-for-Tat”, plays a key role in the emergence of cooperative behavior required for the origin of life.
Collapse
|
14
|
Takács K, Gross J, Testori M, Letina S, Kenny AR, Power EA, Wittek RPM. Networks of reliable reputations and cooperation: a review. Philos Trans R Soc Lond B Biol Sci 2021; 376:20200297. [PMID: 34601917 PMCID: PMC8487750 DOI: 10.1098/rstb.2020.0297] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022] Open
Abstract
Reputation has been shown to provide an informal solution to the problem of cooperation in human societies. After reviewing models that connect reputations and cooperation, we address how reputation results from information exchange embedded in a social network that changes endogenously itself. Theoretical studies highlight that network topologies have different effects on the extent of cooperation, since they can foster or hinder the flow of reputational information. Subsequently, we review models and empirical studies that intend to grasp the coevolution of reputations, cooperation and social networks. We identify open questions in the literature concerning how networks affect the accuracy of reputations, the honesty of shared information and the spread of reputational information. Certain network topologies may facilitate biased beliefs and intergroup competition or in-group identity formation that could lead to high cooperation within but conflicts between different subgroups of a network. Our review covers theoretical, experimental and field studies across various disciplines that target these questions and could explain how the dynamics of interactions and reputations help or prevent the establishment and sustainability of cooperation in small- and large-scale societies. This article is part of the theme issue ‘The language of cooperation: reputation and honest signalling’.
Collapse
Affiliation(s)
- Károly Takács
- The Institute for Analytical Sociology, Linköping University, 601 74 Norrköping, Sweden.,Computational Social Science-Research Center for Educational and Network Studies (CSS-RECENS), Centre for Social Sciences, Tóth Kálmán u. 4., 1097 Budapest, Hungary
| | - Jörg Gross
- Institute of Psychology, Leiden University, Wassenaarseweg 52, 2333 AK, Leiden, The Netherlands
| | - Martina Testori
- Organization Sciences, Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands
| | - Srebrenka Letina
- The Institute for Analytical Sociology, Linköping University, 601 74 Norrköping, Sweden.,Institute of Health and Wellbeing, MRC/CSO Social and Public Health Sciences Unit, University of Glasgow, Berkeley Square, 99 Berkeley Street, Glasgow G3 7HR, UK
| | - Adam R Kenny
- Institute of Cognitive and Evolutionary Anthropology, University of Oxford, 64 Banbury Road, Oxford OX2 6PN, UK.,Calleva Research Centre for Evolution and Human Sciences, Magdalen College, High Street, Oxford OX1 4AU, UK
| | - Eleanor A Power
- Department of Methodology, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK
| | - Rafael P M Wittek
- Department of Sociology, University of Groningen, Grote Rozenstraat 31, 9712 TG Groningen, The Netherlands
| |
Collapse
|
15
|
Jeong W, Yu U. Critical phenomena and strategy ordering with hub centrality approach in the aspiration-based coordination game. CHAOS (WOODBURY, N.Y.) 2021; 31:093114. [PMID: 34598449 DOI: 10.1063/5.0064406] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2021] [Accepted: 08/24/2021] [Indexed: 06/13/2023]
Abstract
We study the coordination game with an aspiration-driven update rule in regular graphs and scale-free networks. We prove that the model coincides exactly with the Ising model and shows a phase transition at the critical selection noise when the aspiration level is zero. It is found that the critical selection noise decreases with clustering in random regular graphs. With a non-zero aspiration level, the model also exhibits a phase transition as long as the aspiration level is smaller than the degree of graphs. We also show that the critical exponents are independent of clustering and aspiration level to confirm that the coordination game belongs to the Ising universality class. As for scale-free networks, the effect of aspiration level on the order parameter at a low selection noise is examined. In model networks (the Barabási-Albert network and the Holme-Kim network), the order parameter abruptly decreases when the aspiration level is the same as the average degree of the network. In contrast, in real-world networks, the order parameter decreases gradually. We explain this difference by proposing the concepts of hub centrality and local hub. The histogram of hub centrality of real-world networks separates into two parts unlike model networks, and local hubs exist only in real-world networks. We conclude that the difference of network structures in model and real-world networks induces qualitatively different behavior in the coordination game.
Collapse
Affiliation(s)
- Wonhee Jeong
- Department of Physics and Photon Science, Gwangju Institute of Science and Technology, Gwangju 61005, South Korea
| | - Unjong Yu
- Department of Physics and Photon Science, Gwangju Institute of Science and Technology, Gwangju 61005, South Korea
| |
Collapse
|
16
|
Wang SY, Liu YP, Li ML, Li C, Wang RW. Super-rational aspiration induced strategy updating helps resolve the tragedy of the commons in a cooperation system with exit rights. Biosystems 2021; 208:104496. [PMID: 34332036 DOI: 10.1016/j.biosystems.2021.104496] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2021] [Revised: 06/30/2021] [Accepted: 07/23/2021] [Indexed: 10/20/2022]
Abstract
Avoiding the tragedy of the commons requires altruists to incur some losses to benefit the group. Although specific rules and self-enforcing agreements could help maintain the cooperation system stable, the costly recognition and free-rider problem are still questioned these two cooperation maintenance mechanisms. We here considered the situation of both exit costs and exit benefits in the asymmetric prisoner's dilemma game and introduced a super-rational aspiration induced strategy updating, where players adjust strategies in line with their payoffs and aspirations. If their payoffs reach or exceed the aspiration levels, which may be rational or super-rational, they keep their strategies. Otherwise, they imitate a local neighbor's strategy. We explored this rule in the structured and well-mixed population. The results show that super-rationality and asymmetry could together promote cooperation when exit costs exist. With exit benefit, super-rationality promotes cooperation in both structures and asymmetry only works in the well-mixed population. This suggests that the introduced strategy updating rule could sustain cooperation among egoists with exit rights.
Collapse
Affiliation(s)
- Si-Yi Wang
- School of Ecology and Environment, Northwestern Polytechnical University, Xi'an, 710129, China; Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710129, China
| | - Yan-Ping Liu
- School of Ecology and Environment, Northwestern Polytechnical University, Xi'an, 710129, China; Center for Quantitative Biology, College of Science, Gansu Agricultural University, Lanzhou, 730070, China
| | - Min-Lan Li
- School of Ecology and Environment, Northwestern Polytechnical University, Xi'an, 710129, China; Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710129, China
| | - Cong Li
- School of Ecology and Environment, Northwestern Polytechnical University, Xi'an, 710129, China.
| | - Rui-Wu Wang
- School of Ecology and Environment, Northwestern Polytechnical University, Xi'an, 710129, China.
| |
Collapse
|
17
|
Szolnoki A, Chen X. Cooperation and competition between pair and multi-player social games in spatial populations. Sci Rep 2021; 11:12101. [PMID: 34103617 PMCID: PMC8187490 DOI: 10.1038/s41598-021-91532-5] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2021] [Accepted: 05/21/2021] [Indexed: 11/25/2022] Open
Abstract
The conflict between individual and collective interests is in the heart of every social dilemmas established by evolutionary game theory. We cannot avoid these conflicts but sometimes we may choose which interaction framework to use as a battlefield. For instance some people like to be part of a larger group while other persons prefer to interact in a more personalized, individual way. Both attitudes can be formulated via appropriately chosen traditional games. In particular, the prisoner’s dilemma game is based on pair interaction while the public goods game represents multi-point interactions of group members. To reveal the possible advantage of a certain attitude we extend these models by allowing players not simply to change their strategies but also let them to vary their attitudes for a higher individual income. We show that both attitudes could be the winner at a specific parameter value. Interestingly, however, the subtle interplay between different states may result in a counterintuitive evolutionary outcome where the increase of the multiplication factor of public goods game drives the population to a fully defector state. We point out that the accompanying pattern formation can only be understood via the multipoint or multi-player interactions of different microscopic states where the vicinity of a particular state may influence the relation of two other competitors.
Collapse
Affiliation(s)
- Attila Szolnoki
- Centre for Energy Research, Institute of Technical Physics and Materials Science, P.O. Box 49, 1525, Budapest, Hungary.
| | - Xiaojie Chen
- School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China
| |
Collapse
|
18
|
Random choices facilitate solutions to collective network coloring problems by artificial agents. iScience 2021; 24:102340. [PMID: 33870136 PMCID: PMC8047171 DOI: 10.1016/j.isci.2021.102340] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2021] [Revised: 03/09/2021] [Accepted: 03/17/2021] [Indexed: 11/22/2022] Open
Abstract
Global coordination is required to solve a wide variety of challenging collective action problems from network colorings to the tragedy of the commons. Recent empirical study shows that the presence of a few noisy autonomous agents can greatly improve collective performance of humans in solving networked color coordination games. To provide analytical insights into the role of behavioral randomness, here we study myopic artificial agents attempting to solve similar network coloring problems using decision update rules that are only based on local information but allow random choices at various stages of their heuristic reasonings. We show that the resulting efficacy of resolving color conflicts is dependent on the implementation of random behavior of agents and specific population characteristics. Our work demonstrates that distributed greedy optimization algorithms exploiting local information should be deployed in combination with occasional exploration via random choices in order to overcome local minima and achieve global coordination. Local information makes solving distributed network coloring problems difficult Greedy agents can become gridlocked, making it difficult to find a global solution Agents making random choices can facilitate the finding of a global coloring Randomness can be finely tuned to a specific underlying population structure
Collapse
|
19
|
Self-questioning dynamical evolutionary game with altruistic behavior and sharing mechanism in scale-free network. INT J MACH LEARN CYB 2021. [DOI: 10.1007/s13042-021-01311-x] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
|
20
|
Chae S, Lee N, Baek SK, Jeong HC. Assortative clustering in a one-dimensional population with replication strategies. Phys Rev E 2021; 103:032114. [PMID: 33862713 DOI: 10.1103/physreve.103.032114] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2020] [Accepted: 02/22/2021] [Indexed: 11/07/2022]
Abstract
In a geographically distributed population, assortative clustering plays an important role in evolution by modifying local environments. To examine its effects in a linear habitat, we consider a one-dimensional grid of cells, where each cell is either empty or occupied by an organism whose replication strategy is genetically inherited to offspring. The strategy determines whether to have offspring in surrounding cells, as a function of the neighborhood configuration. If more than one offspring compete for a cell, then they can be all exterminated due to the cost of conflict depending on environmental conditions. We find that the system is more densely populated in an unfavorable environment than in a favorable one because only the latter has to pay the cost of conflict. This observation agrees reasonably well with a mean-field analysis which takes assortative clustering of strategies into consideration. Our finding suggests a possibility of intrinsic nonlinearity between environmental conditions and population density when an evolutionary process is involved.
Collapse
Affiliation(s)
- Sunhee Chae
- Department of Physics and Astronomy, Sejong University, Seoul 05006, Korea
| | - Nahyeon Lee
- Department of Physics and Astronomy, Sejong University, Seoul 05006, Korea
| | - Seung Ki Baek
- Department of Physics, Pukyong National University, Busan 48513, Korea
| | - Hyeong-Chai Jeong
- Department of Physics and Astronomy, Sejong University, Seoul 05006, Korea
| |
Collapse
|
21
|
Shi Z, Wei W, Feng X, Li X, Zheng Z. Dynamic aspiration based on Win-Stay-Lose-Learn rule in spatial prisoner's dilemma game. PLoS One 2021; 16:e0244814. [PMID: 33395443 PMCID: PMC7781394 DOI: 10.1371/journal.pone.0244814] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2020] [Accepted: 12/16/2020] [Indexed: 11/18/2022] Open
Abstract
Prisoner’s dilemma game is the most commonly used model of spatial evolutionary game which is considered as a paradigm to portray competition among selfish individuals. In recent years, Win-Stay-Lose-Learn, a strategy updating rule base on aspiration, has been proved to be an effective model to promote cooperation in spatial prisoner’s dilemma game, which leads aspiration to receive lots of attention. In this paper, according to Expected Value Theory and Achievement Motivation Theory, we propose a dynamic aspiration model based on Win-Stay-Lose-Learn rule in which individual’s aspiration is inspired by its payoff. It is found that dynamic aspiration has a significant impact on the evolution process, and different initial aspirations lead to different results, which are called Stable Coexistence under Low Aspiration, Dependent Coexistence under Moderate aspiration and Defection Explosion under High Aspiration respectively. Furthermore, a deep analysis is performed on the local structures which cause defectors’ re-expansion, the concept of END- and EXP-periods are used to justify the mechanism of network reciprocity in view of time-evolution, typical feature nodes for defectors’ re-expansion called Infectors, Infected nodes and High-risk cooperators respectively are found. Compared to fixed aspiration model, dynamic aspiration introduces a more satisfactory explanation on population evolution laws and can promote deeper comprehension for the principle of prisoner’s dilemma.
Collapse
Affiliation(s)
- Zhenyu Shi
- School of Mathematical Sciences, Beihang University, Beijing, China
- Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education, Beijing, China
- Peng Cheng Laboratory, Shenzhen, Guangdong, China
- Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, China
| | - Wei Wei
- School of Mathematical Sciences, Beihang University, Beijing, China
- Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education, Beijing, China
- Peng Cheng Laboratory, Shenzhen, Guangdong, China
- Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, China
- * E-mail:
| | - Xiangnan Feng
- School of Mathematical Sciences, Beihang University, Beijing, China
- Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education, Beijing, China
- Peng Cheng Laboratory, Shenzhen, Guangdong, China
- Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, China
| | - Xing Li
- School of Mathematical Sciences, Beihang University, Beijing, China
- Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education, Beijing, China
- Peng Cheng Laboratory, Shenzhen, Guangdong, China
- Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, China
| | - Zhiming Zheng
- School of Mathematical Sciences, Beihang University, Beijing, China
- Key Laboratory of Mathematics, Informatics and Behavioral Semantics, Ministry of Education, Beijing, China
- Peng Cheng Laboratory, Shenzhen, Guangdong, China
- Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing, China
| |
Collapse
|
22
|
Fu Y, Zhang Y, Guo Y, Xie Y. Evolutionary dynamics of cooperation with the celebrity effect in complex networks. CHAOS (WOODBURY, N.Y.) 2021; 31:013130. [PMID: 33754779 DOI: 10.1063/5.0033335] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/15/2020] [Accepted: 12/28/2020] [Indexed: 06/12/2023]
Abstract
How long-term cooperation is maintained in a society is an important and interesting question. The evolutionary game theory is often used as the basic framework to study this topic. The social status of game participants has an important influence on individual decision-making. Enlightened by this thought, we present a classification imitation model where the mechanisms of the celebrity effect and incomplete egoism are presented. The celebrity effect is reflected in each strategy update process to probe how individual decision-making is dynamically adjusted by comparing the social status of both parties in the game. The incomplete egoism refers to the irrational imitation of celebrities while self-interest is ignored. With this model, the group cooperation decision-making mechanism led by celebrities is revealed. Large-scale Monte Carlo simulations show that the incomplete egoism of individuals cannot stimulate cooperation but guarantee the stable existence of cooperation. Furthermore, the scale-free and community structure of the network enables cooperation to spread widely and maintains long-term survival. Our conclusion might provide practically new insight into the understanding and controlling of cooperation in the complex social systems.
Collapse
Affiliation(s)
- Yanyu Fu
- School of Business, Tianjin University of Finance and Economics, Tianjin 300222, China
| | - Yan Zhang
- School of Systems Science, Beijing Normal University, Beijing 100875, China
| | - Yu Guo
- Software Institute, Nanjing University, Nanjing 210093, China
| | - Yunya Xie
- Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China
| |
Collapse
|
23
|
Jia D, Wang X, Song Z, Romić I, Li X, Jusup M, Wang Z. Evolutionary dynamics drives role specialization in a community of players. J R Soc Interface 2020; 17:20200174. [PMID: 32693747 DOI: 10.1098/rsif.2020.0174] [Citation(s) in RCA: 32] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023] Open
Abstract
The progression of game theory from classical to evolutionary and spatial games provided a powerful means to study cooperation, and enabled a better understanding of general cooperation-promoting mechanisms. However, current standard models assume that at any given point players must choose either cooperation or defection, meaning that regardless of the spatial structure in which they exist, they cannot differentiate between their neighbours and adjust their behaviour accordingly. This is at odds with interactions among organisms in nature who are well capable of behaving differently towards different members of their communities. We account for this natural fact by introducing a new type of player-dubbed link players-who can adjust their behaviour to each individual neighbour. This is in contrast to more common node players whose behaviour affects all neighbours in the same way. We proceed to study cooperation in pure and mixed populations, showing that cooperation peaks at moderately low densities of link players. In such conditions, players naturally specialize in different roles. Node players tend to be either cooperators or defectors, while link players form social insulation between cooperative and defecting clusters by acting both as cooperators and defectors. Such fairly complex processes emerging from a simple model reflect some of the complexities observed in experimental studies on social behaviour in microbes and pave a way for the development of richer game models.
Collapse
Affiliation(s)
- Danyang Jia
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China.,Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi'an 710072, People's Republic of China
| | - Xinyu Wang
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China.,Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi'an 710072, People's Republic of China
| | - Zhao Song
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China.,Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi'an 710072, People's Republic of China
| | - Ivan Romić
- Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi'an 710072, People's Republic of China.,Statistics and Mathematics College, Yunnan University of Finance and Economics, Kunming 650221, People's Republic of China.,Graduate School of Economics, Osaka City University, Osaka 558-8585, Japan
| | - Xuelong Li
- Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi'an 710072, People's Republic of China.,School of Computer Science, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China
| | - Marko Jusup
- Tokyo Tech World Research Hub Initiative (WRHI), Institute of Innovative Research, Tokyo Institute of Technology, Tokyo 152-8550, Japan
| | - Zhen Wang
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China.,Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi'an 710072, People's Republic of China
| |
Collapse
|
24
|
Evolution of Cooperation in the Presence of Higher-Order Interactions: From Networks to Hypergraphs. ENTROPY 2020; 22:e22070744. [PMID: 33286516 PMCID: PMC7517288 DOI: 10.3390/e22070744] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/08/2020] [Revised: 07/02/2020] [Accepted: 07/04/2020] [Indexed: 01/04/2023]
Abstract
Many real systems are strongly characterized by collective cooperative phenomena whose existence and properties still need a satisfactory explanation. Coherently with their collective nature, they call for new and more accurate descriptions going beyond pairwise models, such as graphs, in which all the interactions are considered as involving only two individuals at a time. Hypergraphs respond to this need, providing a mathematical representation of a system allowing from pairs to larger groups. In this work, through the use of different hypergraphs, we study how group interactions influence the evolution of cooperation in a structured population, by analyzing the evolutionary dynamics of the public goods game. Here we show that, likewise to network reciprocity, group interactions also promote cooperation. More importantly, by means of an invasion analysis in which the conditions for a strategy to survive are studied, we show how, in heterogeneously-structured populations, reciprocity among players is expected to grow with the increasing of the order of the interactions. This is due to the heterogeneity of connections and, particularly, to the presence of individuals standing out as hubs in the population. Our analysis represents a first step towards the study of evolutionary dynamics through higher-order interactions, and gives insights into why cooperation in heterogeneous higher-order structures is enhanced. Lastly, it also gives clues about the co-existence of cooperative and non-cooperative behaviors related to the structural properties of the interaction patterns.
Collapse
|
25
|
Amaral MA, Javarone MA. Strategy equilibrium in dilemma games with off-diagonal payoff perturbations. Phys Rev E 2020; 101:062309. [PMID: 32688499 DOI: 10.1103/physreve.101.062309] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2020] [Accepted: 06/03/2020] [Indexed: 06/11/2023]
Abstract
We analyze the strategy equilibrium of dilemma games considering a payoff matrix affected by small and random perturbations on the off-diagonal. Notably, a recent work [Proc. R. Soc. A 476, 20200116 (2020)1364-502110.1098/rspa.2020.0116] reported that while cooperation is sustained by perturbations acting on the main diagonal, a less clear scenario emerges when perturbations act on the off-diagonal. Thus, the second case represents the core of this investigation, aimed at completing the description of the effects that payoff perturbations have on the dynamics of evolutionary games. Our results, achieved by analyzing the proposed model under a variety of configurations as different update rules, suggest that off-diagonal perturbations actually constitute a nontrivial form of noise. In particular, the most interesting effects are detected near the phase transition, as perturbations tend to move the strategy distribution towards nonordered states of equilibrium, supporting cooperation when defection is pervading the population, and supporting defection in the opposite case. To conclude, we identified a form of noise that, under controlled conditions, could be used to enhance cooperation and greatly delay its extinction.
Collapse
Affiliation(s)
- Marco A Amaral
- Instituto de Humanidades, Artes e Ciências, Universidade Federal do Sul da Bahia-BA, 45996-108, Brazil
| | - Marco A Javarone
- Department of Mathematics, University College London, London WC1E 6BT, United Kingdom
| |
Collapse
|
26
|
Amaral MA, Javarone MA. Heterogeneity in evolutionary games: an analysis of the risk perception. Proc Math Phys Eng Sci 2020; 476:20200116. [PMID: 32523420 DOI: 10.1098/rspa.2020.0116] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/21/2020] [Accepted: 03/24/2020] [Indexed: 11/12/2022] Open
Abstract
In this study, we analyse the relationship between heterogeneity and cooperation. Previous investigations suggest that this relation is non-trivial, as some authors found that heterogeneity sustains cooperation, while others obtained different results. Among the possible forms of heterogeneity, we focus on the individual perception of risks and rewards related to a generic event, which can appear in a number of social and biological systems. The modelling approach is based on the framework of evolutionary game theory. To represent this kind of heterogeneity, we implement small and local perturbations on the pay-off matrix of simple two-strategy games, such as the Prisoner's Dilemma. So, while usually the pay-off is considered to be a global and time-invariant structure, i.e. it is the same for all individuals of a population at any time, in our model its value is continuously affected by small variations, in both time and space (i.e. position on a lattice). We found that such perturbations can be beneficial or detrimental to cooperation, depending on their setting. Notably, cooperation is strongly supported when perturbations act on the main diagonal of the pay-off matrix, whereas when they act on the off-diagonal the resulting effect is more difficult to quantify. To conclude, the proposed model shows a rich spectrum of possible equilibria, whose interpretation might offer insights and enrich the description of several systems.
Collapse
Affiliation(s)
- Marco A Amaral
- Instituto de Humanidades, Artes e Ciências, Universidade Federal do Sul da Bahia, Teixeira de Freitas, Bahia 45988, Brazil
| | - Marco A Javarone
- Department of Mathematics, University College London, London, UK
| |
Collapse
|
27
|
Zhu P, Song Z, Guo H, Wang Z, Zhao T. Adaptive willingness resolves social dilemma in network populations. CHAOS (WOODBURY, N.Y.) 2019; 29:113114. [PMID: 31779363 DOI: 10.1063/1.5093046] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/23/2019] [Accepted: 10/15/2019] [Indexed: 06/10/2023]
Abstract
Cooperation is an effective manner to enable different elements of complex networks to work well. In this work, we propose a coevolution mechanism of learning willingness in the network population: an agent will be more likely to imitate a given neighbor's strategy if her payoff is not less than the average performance of all her neighbors. Interestingly, increase of learning willingness will greatly promote cooperation even under the environment of extremely beneficial temptation to defectors. Through a microscopic analysis, it is unveiled that cooperators are protected due to the appearance of large-size clusters. Pair approximation theory also validates all these findings. Such an adaptive mechanism thus provides a feasible solution to relieve social dilemmas and will inspire further studies.
Collapse
Affiliation(s)
- Peican Zhu
- School of Computer Science and Engineering, Northwestern Polytechnical University (NWPU), Xi'an, Shaanxi 710072, China
| | - Zhao Song
- School of Mechanical Engineering, NWPU, Xi'an, Shaanxi 710072, China
| | - Hao Guo
- School of Mechanical Engineering, NWPU, Xi'an, Shaanxi 710072, China
| | - Zhen Wang
- School of Mechanical Engineering, NWPU, Xi'an, Shaanxi 710072, China
| | - Tianyun Zhao
- School of Automation, NWPU, Xi'an, Shaanxi 710072, China
| |
Collapse
|
28
|
Rong Z, Wu ZX, Li X, Holme P, Chen G. Heterogeneous cooperative leadership structure emerging from random regular graphs. CHAOS (WOODBURY, N.Y.) 2019; 29:103103. [PMID: 31675848 DOI: 10.1063/1.5120349] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2019] [Accepted: 09/10/2019] [Indexed: 06/10/2023]
Abstract
This paper investigates the evolution of cooperation and the emergence of hierarchical leadership structure in random regular graphs. It is found that there exist different learning patterns between cooperators and defectors, and cooperators are able to attract more followers and hence more likely to become leaders. Hence, the heterogeneous distributions of reputation and leadership can emerge from homogeneous random graphs. The important directed game-learning skeleton is then studied, revealing some important structural properties, such as the heavy-tailed degree distribution and the positive in-in degree correlation.
Collapse
Affiliation(s)
- Zhihai Rong
- School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China
| | - Zhi-Xi Wu
- Institute of Computational Physics and Complex Systems, Lanzhou University, Lanzhou 730000, China
| | - Xiang Li
- Adaptive Networks and Control Lab, Department of Electronic Engineering, Fudan University, Shanghai 200433, China
| | - Petter Holme
- Tokyo Tech World Research Hub Initiative (WRHI), Institute of Innovative Research, Tokyo Institute of Technology, Nagatsuta-cho 4259, Midori-ku, Yokohama, Kanagawa 226-8503, Japan
| | - Guanrong Chen
- Department of Electrical Engineering, City University of Hong Kong, Hong Kong, China
| |
Collapse
|
29
|
Xu H, Tian C, Fan S, Li J. Information flows in the market: An evolutionary game approach. CHAOS (WOODBURY, N.Y.) 2019; 29:023126. [PMID: 30823723 DOI: 10.1063/1.5084070] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/03/2018] [Accepted: 01/29/2019] [Indexed: 06/09/2023]
Abstract
Information influences the decisions that investors make in the markets. Whether this information is true or false can be quantified and distinguished by markets. To study how information propagates through markets, we propose an information flow game based on an evolutionary game approach. In reality, investors transmit profits or losses when they transmit information, because there are values associated with information in the market. In the information flow game, information is represented by its value. Investors in the game can choose to be sharers or silencers. Sharers share their information with their neighbors according to a sharing rate α, which is a key quantity in the model. In the evolutionary process, we show that more sharers emerge when the market is full of rumors, especially as the sharing rate increases. Higher values of the sharing rate reduce the standard deviation of the information value in such markets, whereas the opposite occurs in markets that largely consist of true information. The reactions of the investors are asymmetric, which indicates that investors are more sensitive to losses than to profits. Furthermore, as the network becomes more random, a higher sharing rate becomes more beneficial for the stability of the emergence of sharers if information is generally false, whereas a lower sharing rate is helpful for the stability of the emergence of sharers if information is generally true.
Collapse
Affiliation(s)
- Hedong Xu
- Institute of Finance, Jinan University, Guangzhou 510632, China
| | - Cunzhi Tian
- Institute of Finance, Jinan University, Guangzhou 510632, China
| | - Suohai Fan
- School of Information Science and Technology, Jinan University, Guangzhou 510632, China
| | - Jiajia Li
- China Merchants Bank Guangzhou Branch, Guangzhou 510632, China
| |
Collapse
|
30
|
Chang S, Zhang Z, Li Y, Wu YE, Xie Y. Investment preference promotes cooperation in spatial public goods game. PLoS One 2018; 13:e0206486. [PMID: 30427895 PMCID: PMC6235307 DOI: 10.1371/journal.pone.0206486] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/20/2017] [Accepted: 10/02/2018] [Indexed: 11/18/2022] Open
Abstract
It is usually assumed that each cooperator contributes equally to different public pools in spatial public goods game. However, it is more reasonable to invest differently according to individual investment preference. In this paper, an extended public goods game, in which cooperators contribute to the groups according to the investment preference, is developed. The investment preference of a cooperator is characterized by the fraction of the cooperator from his/her own memory about a group and the intensity of investment preference is represented by a tunable parameter α. The well-mixed population and the structured population are analyzed under this mechanism. It is shown that the investment preference can give rise to coordination. Moreover, the extensive numerical simulation results show that with the increasing of investment preference density or memory length, the proportion of cooperation can increase monotonously. This is because the investment preference could help cooperators resist the invasion from defectors. Compared with the basic version, the new mechanism is able to promote cooperation effectively. Our research may provide a valuable insight for further exploring the nature of cooperation in the real world.
Collapse
Affiliation(s)
- Shuhua Chang
- Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, China
| | - Zhipeng Zhang
- Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, China
| | - Yu Li
- Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, China
| | - Yu E Wu
- Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, China
| | - Yunya Xie
- Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, China
| |
Collapse
|
31
|
He X, Du H, Cai M, Feldman MW. The evolution of cooperation in signed networks under the impact of structural balance. PLoS One 2018; 13:e0205084. [PMID: 30296278 PMCID: PMC6175270 DOI: 10.1371/journal.pone.0205084] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/07/2017] [Accepted: 09/18/2018] [Indexed: 11/27/2022] Open
Abstract
Structural balance plays an important role in the dynamics of signed networks. Based on structural balance, we generalize the evolution of cooperation in signed networks. Here we develop a new simulation model to study the impact of structural balance on the evolution of cooperation in signed networks. The simulation shows that cooperation prevails when an individual has a higher probability of adjusting the signs of its relations. We also find that structural balance forces the coexistence of cooperators and defectors, while the initial attributes of networks have little impact on the evolution of cooperation in the presence of structural balance, although they have a strong effect on the evolution of structural balance.
Collapse
Affiliation(s)
- Xiaochen He
- Center for Administration and Complexity Science of Xi’an Jiaotong University, Xi’an, Shanxi Province
| | - Haifeng Du
- Center for Administration and Complexity Science of Xi’an Jiaotong University, Xi’an, Shanxi Province
| | - Meng Cai
- Center for Administration and Complexity Science of Xi’an Jiaotong University, Xi’an, Shanxi Province
- School of Economics and Management, Xidian University, Xi’an, Shanxi Province
- Department of Physics, Boston University, Silber Way, Boston, United States of America
| | - Marcus W. Feldman
- Center for Administration and Complexity Science of Xi’an Jiaotong University, Xi’an, Shanxi Province
- Morrison Institute for Population and Resource Studies, Stanford University, Stanford, United States of America
| |
Collapse
|
32
|
Mao Y, Xu X, Rong Z, Wu ZX. The emergence of cooperation-extortion alliance on scale-free networks with normalized payoff. ACTA ACUST UNITED AC 2018. [DOI: 10.1209/0295-5075/122/50005] [Citation(s) in RCA: 38] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
|
33
|
Santos M, Ferreira AL, Figueiredo W. Phase diagram and criticality of the two-dimensional prisoner's dilemma model. Phys Rev E 2018; 96:012120. [PMID: 29347229 DOI: 10.1103/physreve.96.012120] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2017] [Indexed: 11/07/2022]
Abstract
The stationary states of the prisoner's dilemma model are studied on a square lattice taking into account the role of a noise parameter in the decision-making process. Only first neighboring players-defectors and cooperators-are considered in each step of the game. Through Monte Carlo simulations we determined the phase diagrams of the model in the plane noise versus the temptation to defect for a large range of values of the noise parameter. We observed three phases: cooperators and defectors absorbing phases, and a coexistence phase between them. The phase transitions as well as the critical exponents associated with them were determined using both static and dynamical scaling laws.
Collapse
Affiliation(s)
- M Santos
- Physics Department, I3N, Aveiro University, Aveiro 3810-193, Portugal
| | - A L Ferreira
- Physics Department, I3N, Aveiro University, Aveiro 3810-193, Portugal
| | - W Figueiredo
- Physics Department, Federal University of Santa Catarina, Florianópolis, SC 88040-900, Brazil
| |
Collapse
|
34
|
Szabó G, Bunth G. Social dilemmas in multistrategy evolutionary potential games. Phys Rev E 2018; 97:012305. [PMID: 29448430 DOI: 10.1103/physreve.97.012305] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/30/2017] [Indexed: 06/08/2023]
Abstract
The nature of social dilemmas is studied in n-strategy evolutionary potential games on a square lattice with nearest-neighbor interactions and the logit rule. For symmetric games with symmetric payoff matrices there are no dilemmas because of the coincidence of individual and common interests. The dilemmas are caused by the antisymmetric parts of the self- and cross-dependent payoff components if it modifies the preferred Nash equilibrium. The contentment of players and the emergence of dilemmas in the preferred Nash equilibria are illustrated on some two-dimensional cross sections of the parameter space.
Collapse
Affiliation(s)
- György Szabó
- Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
| | - Gergely Bunth
- Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary
| |
Collapse
|
35
|
Pinheiro FL, Hartmann D. Intermediate Levels of Network Heterogeneity Provide the Best Evolutionary Outcomes. Sci Rep 2017; 7:15242. [PMID: 29127336 PMCID: PMC5681591 DOI: 10.1038/s41598-017-15555-7] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2017] [Accepted: 10/30/2017] [Indexed: 11/24/2022] Open
Abstract
Complex networks impact the diffusion of ideas and innovations, the formation of opinions, and the evolution of cooperative behavior. In this context, heterogeneous structures have been shown to generate a coordination-like dynamics that drives a population towards a monomorphic state. In contrast, homogeneous networks tend to result in a stable co-existence of multiple traits in the population. These conclusions have been reached through the analysis of networks with either very high or very low levels of degree heterogeneity. In this paper, we use methods from Evolutionary Game Theory to explore how different levels of degree heterogeneity impact the fate of cooperation in structured populations whose individuals face the Prisoner’s Dilemma. Our results suggest that in large networks a minimum level of heterogeneity is necessary for a society to become evolutionary viable. Moreover, there is an optimal range of heterogeneity levels that maximize the resilience of the society facing an increasing number of social dilemmas. Finally, as the level of degree heterogeneity increases, the evolutionary dominance of either cooperators or defectors in a society increasingly depends on the initial state of a few influential individuals. Our findings imply that neither very unequal nor very equal societies offer the best evolutionary outcome.
Collapse
Affiliation(s)
- Flávio L Pinheiro
- Collective Learning Group, The MIT Media Lab, Massachusetts Institute of Technology, Cambridge, MA, USA.
| | - Dominik Hartmann
- Chair for Innovation Management and Innovation Economics, University of Leipzig, Leipzig, Germany. .,Fraunhofer Center for International Management and Knowledge Economy, Leipzig, Germany.
| |
Collapse
|
36
|
Xie F, Shi J, Lin J. Impact of interaction style and degree on the evolution of cooperation on Barabási-Albert scale-free network. PLoS One 2017; 12:e0182523. [PMID: 28806757 PMCID: PMC5555699 DOI: 10.1371/journal.pone.0182523] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2017] [Accepted: 07/19/2017] [Indexed: 11/19/2022] Open
Abstract
In this work, we study an evolutionary prisoner's dilemma game (PDG) on Barabási-Albert scale-free networks with limited player interactions, and explore the effect of interaction style and degree on cooperation. The results show that high-degree preference interaction, namely the most applicable interaction in the real world, is less beneficial for emergence of cooperation on scale-free networks than random interaction. Besides, cooperation on scale-free networks is enhanced with the increase of interaction degree regardless whether the interaction is high-degree preference or random. If the interaction degree is very low, the cooperation level on scale-free networks is much lower than that on regular ring networks, which is against the common belief that scale-free networks must be more beneficial for cooperation. Our analysis indicates that the interaction relations, the strategy and the game payoff of high-connectivity players play important roles in the evolution of cooperation on scale-free networks. A certain number of interactions are necessary for scale-free networks to exhibit strong capability of facilitating cooperation. Our work provides important insight for members on how to interact with others in a social organization.
Collapse
Affiliation(s)
- Fengjie Xie
- Department of Information Management, College of Economics and Management, Xi’an University of Posts and Telecommunications, Xi’an, Shaan Xi, China
| | - Jing Shi
- Department of Mechanical and Materials Engineering, College of Engineering & Applied Science, University of Cincinnati, Cincinnati, Ohio, United States of America
- * E-mail: (JS); (JL)
| | - Jun Lin
- Department of Management Science, School of Management, Xi’an Jiaotong University, Xi’an, Shaan Xi, China
- * E-mail: (JS); (JL)
| |
Collapse
|
37
|
Xu X, Rong Z, Wu ZX, Zhou T, Tse CK. Extortion provides alternative routes to the evolution of cooperation in structured populations. Phys Rev E 2017; 95:052302. [PMID: 28618489 DOI: 10.1103/physreve.95.052302] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/19/2016] [Indexed: 06/07/2023]
Abstract
In this paper, we study the evolution of cooperation in structured populations (individuals are located on either a regular lattice or a scale-free network) in the context of repeated games by involving three types of strategies, namely, unconditional cooperation, unconditional defection, and extortion. The strategy updating of the players is ruled by the replicator-like dynamics. We find that extortion strategies can act as catalysts to promote the emergence of cooperation in structured populations via different mechanisms. Specifically, on regular lattice, extortioners behave as both a shield, which can enwrap cooperators inside and keep them away from defectors, and a spear, which can defeat those surrounding defectors with the help of the neighboring cooperators. Particularly, the enhancement of cooperation displays a resonance-like behavior, suggesting the existence of optimal extortion strength mostly favoring the evolution of cooperation, which is in good agreement with the predictions from the generalized mean-field approximation theory. On scale-free network, the hubs, who are likely occupied by extortioners or defectors at the very beginning, are then prone to be conquered by cooperators on small-degree nodes as time elapses, thus establishing a bottom-up mechanism for the emergence and maintenance of cooperation.
Collapse
Affiliation(s)
- Xiongrui Xu
- CompleX Lab, Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 611731, China
| | - Zhihai Rong
- CompleX Lab, Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 611731, China
- Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, China
- Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
| | - Zhi-Xi Wu
- Institute of Computational Physics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, People's Republic of China
| | - Tao Zhou
- Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, China
| | - Chi Kong Tse
- Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
| |
Collapse
|
38
|
Amaral MA, Perc M, Wardil L, Szolnoki A, da Silva Júnior EJ, da Silva JKL. Role-separating ordering in social dilemmas controlled by topological frustration. Phys Rev E 2017; 95:032307. [PMID: 28415219 DOI: 10.1103/physreve.95.032307] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2016] [Indexed: 06/07/2023]
Abstract
''Three is a crowd" is an old proverb that applies as much to social interactions as it does to frustrated configurations in statistical physics models. Accordingly, social relations within a triangle deserve special attention. With this motivation, we explore the impact of topological frustration on the evolutionary dynamics of the snowdrift game on a triangular lattice. This topology provides an irreconcilable frustration, which prevents anticoordination of competing strategies that would be needed for an optimal outcome of the game. By using different strategy updating protocols, we observe complex spatial patterns in dependence on payoff values that are reminiscent to a honeycomb-like organization, which helps to minimize the negative consequence of the topological frustration. We relate the emergence of these patterns to the microscopic dynamics of the evolutionary process, both by means of mean-field approximations and Monte Carlo simulations. For comparison, we also consider the same evolutionary dynamics on the square lattice, where of course the topological frustration is absent. However, with the deletion of diagonal links of the triangular lattice, we can gradually bridge the gap to the square lattice. Interestingly, in this case the level of cooperation in the system is a direct indicator of the level of topological frustration, thus providing a method to determine frustration levels in an arbitrary interaction network.
Collapse
Affiliation(s)
- Marco A Amaral
- Departamento de Física, Universidade Federal de Minas Gerais, Caixa Postal 702, CEP 30161-970, Belo Horizonte-MG, Brazil
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia
- CAMTP-Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia
| | - Lucas Wardil
- Departamento de Fisica, Universidade Federal de Ouro Preto, Ouro Preto, MG, Brazil
| | - Attila Szolnoki
- Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, Post Office Box 49, H-1525 Budapest, Hungary
| | - Elton J da Silva Júnior
- Departamento de Física, Universidade Federal de Minas Gerais, Caixa Postal 702, CEP 30161-970, Belo Horizonte-MG, Brazil
| | - Jafferson K L da Silva
- Departamento de Física, Universidade Federal de Minas Gerais, Caixa Postal 702, CEP 30161-970, Belo Horizonte-MG, Brazil
| |
Collapse
|
39
|
Chu C, Liu J, Shen C, Jin J, Shi L. Win-stay-lose-learn promotes cooperation in the prisoner's dilemma game with voluntary participation. PLoS One 2017; 12:e0171680. [PMID: 28182707 PMCID: PMC5300200 DOI: 10.1371/journal.pone.0171680] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/01/2016] [Accepted: 01/24/2017] [Indexed: 11/19/2022] Open
Abstract
Voluntary participation, demonstrated to be a simple yet effective mechanism to promote persistent cooperative behavior, has been extensively studied. It has also been verified that the aspiration-based win-stay-lose-learn strategy updating rule promotes the evolution of cooperation. Inspired by this well-known fact, we combine the Win-Stay-Lose-Learn updating rule with voluntary participation: Players maintain their strategies when they are satisfied, or players attempt to imitate the strategy of one randomly chosen neighbor. We find that this mechanism maintains persistent cooperative behavior, even further promotes the evolution of cooperation under certain conditions.
Collapse
Affiliation(s)
- Chen Chu
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, China
| | - Jinzhuo Liu
- School of Software, Yunnan University, Kunming, Yunnan, China
| | - Chen Shen
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, China
| | - Jiahua Jin
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, China
- Library of Yunnan Normal University, Kunming, Yunnan, China
| | - Lei Shi
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, China
- * E-mail:
| |
Collapse
|
40
|
Evolutionary stability for matrix games under time constraints. J Theor Biol 2017; 415:1-12. [DOI: 10.1016/j.jtbi.2016.11.029] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2016] [Revised: 11/25/2016] [Accepted: 11/29/2016] [Indexed: 11/19/2022]
|
41
|
Wu T, Wang L, Fu F. Coevolutionary dynamics of phenotypic diversity and contingent cooperation. PLoS Comput Biol 2017; 13:e1005363. [PMID: 28141806 PMCID: PMC5308777 DOI: 10.1371/journal.pcbi.1005363] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/23/2016] [Revised: 02/14/2017] [Accepted: 01/14/2017] [Indexed: 01/03/2023] Open
Abstract
Phenotypic diversity is considered beneficial to the evolution of contingent cooperation, in which cooperators channel their help preferentially towards others of similar phenotypes. However, it remains largely unclear how phenotypic variation arises in the first place and thus leads to the construction of phenotypic complexity. Here we propose a mathematical model to study the coevolutionary dynamics of phenotypic diversity and contingent cooperation. Unlike previous models, our model does not assume any prescribed level of phenotypic diversity, but rather lets it be an evolvable trait. Each individual expresses one phenotype at a time and only the phenotypes expressed are visible to others. Moreover, individuals can differ in their potential of phenotypic variation, which is characterized by the number of distinct phenotypes they can randomly switch to. Each individual incurs a cost proportional to the number of potentially expressible phenotypes so as to retain phenotypic variation and expression. Our results show that phenotypic diversity coevolves with contingent cooperation under a wide range of conditions and that there exists an optimal level of phenotypic diversity best promoting contingent cooperation. It pays for contingent cooperators to elevate their potential of phenotypic variation, thereby increasing their opportunities of establishing cooperation via novel phenotypes, as these new phenotypes serve as secret tags that are difficult for defector to discover and chase after. We also find that evolved high levels of phenotypic diversity can occasionally collapse due to the invasion of defector mutants, suggesting that cooperation and phenotypic diversity can mutually reinforce each other. Thus, our results provide new insights into better understanding the coevolution of cooperation and phenotypic diversity. Phenotypic variation is commonly observed from human cells to the intestinal pathogen Salmonella enterica serovar Typhimurium to the wrinkly-spreader morphs. Such phenotypic diversity proves effective in promoting cooperation, or confers survival advantage against unfavorable environmental changes. Prior studies show that interactions based on phenotypic similarity can promote cooperation. Yet in these models, the level of phenotypic diversity is prescribed such that individuals each possess the same number of available phenotypes, and thereby no evolution of phenotypic diversity per se. We here take into consideration important aspects of the diversity of phenotype and contingent cooperation and show that phenotypic diversity coevolves with cooperation under a variety of conditions. Our work provides a potential mechanism for the evolution of cooperation, and individuals, especially cooperators, endowed with diverse phenotypes constitute the backbone in inducing the coevolution.
Collapse
Affiliation(s)
- Te Wu
- Center for Complex Systems, Xidian University, Xi’an, China
- Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
| | - Long Wang
- Center for Systems and Control, College of Engineering, Peking University, Beijing, China
- * E-mail: (LW); (FF)
| | - Feng Fu
- Department of Mathematics, Dartmouth College, Hanover, New Hampshire, United States of America
- Department of Biomedical Data Science, Geisel School of Medicine, Dartmouth College, Hanover, New Hampshire, United States of America
- * E-mail: (LW); (FF)
| |
Collapse
|
42
|
Wu Y, Chang S, Zhang Z, Deng Z. Impact of Social Reward on the Evolution of the Cooperation Behavior in Complex Networks. Sci Rep 2017; 7:41076. [PMID: 28112276 PMCID: PMC5253654 DOI: 10.1038/srep41076] [Citation(s) in RCA: 38] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2016] [Accepted: 12/12/2016] [Indexed: 11/17/2022] Open
Abstract
Social reward, as a significant mechanism explaining the evolution of cooperation, has attracted great attention both theoretically and experimentally. In this paper, we study the evolution of cooperation by proposing a reward model in network population, where a third strategy, reward, as an independent yet particular type of cooperation is introduced in 2-person evolutionary games. Specifically, a new kind of role corresponding to reward strategy, reward agents, is defined, which is aimed at increasing the income of cooperators by applying to them a social reward. Results from numerical simulations show that consideration of social reward greatly promotes the evolution of cooperation, which is confirmed for different network topologies and two evolutionary games. Moreover, we explore the microscopic mechanisms for the promotion of cooperation in the three-strategy model. As expected, the reward agents play a vital role in the formation of cooperative clusters, thus resisting the aggression of defectors. Our research might provide valuable insights into further exploring the nature of cooperation in the real world.
Collapse
Affiliation(s)
- Yu'e Wu
- Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China
| | - Shuhua Chang
- Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China
| | - Zhipeng Zhang
- Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China
| | - Zhenghong Deng
- Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China
| |
Collapse
|
43
|
Szabó G, Varga L, Szabó M. Anisotropic invasion and its consequences in two-strategy evolutionary games on a square lattice. Phys Rev E 2016; 94:052314. [PMID: 27967092 DOI: 10.1103/physreve.94.052314] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2016] [Indexed: 11/07/2022]
Abstract
We have studied invasion processes in two-strategy evolutionary games on a square lattice for imitation rule when the players interact with their nearest neighbors. Monte Carlo simulations are performed for systems where the pair interactions are composed of a unit strength coordination game when varying the strengths of the self-dependent and cross-dependent components at a fixed noise level. The visualization of strategy distributions has clearly indicated that circular homogeneous domains evolve into squares with an orientation dependent on the composition. This phenomenon is related to the anisotropy of invasion velocities along the interfaces separating the two homogeneous regions. The quantified invasion velocities indicate the existence of a parameter region in which the invasions are opposite for the horizontal (or vertical) and the tilted interfaces. In this parameter region faceted islands of both strategies shrink and the system evolves from a random initial state into the homogeneous state that first percolated.
Collapse
Affiliation(s)
- György Szabó
- Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
| | - Levente Varga
- Babeş-Bolyai University, Faculty of Physics, RO-400084 Cluj-Napoca, Romania
| | - Mátyás Szabó
- Reed College, Department of Physics, 97202 Portland, Oregon, USA
| |
Collapse
|
44
|
Amaral MA, Wardil L, Perc M, da Silva JKL. Stochastic win-stay-lose-shift strategy with dynamic aspirations in evolutionary social dilemmas. Phys Rev E 2016; 94:032317. [PMID: 27739792 DOI: 10.1103/physreve.94.032317] [Citation(s) in RCA: 66] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2016] [Indexed: 11/07/2022]
Abstract
In times of plenty expectations rise, just as in times of crisis they fall. This can be mathematically described as a win-stay-lose-shift strategy with dynamic aspiration levels, where individuals aspire to be as wealthy as their average neighbor. Here we investigate this model in the realm of evolutionary social dilemmas on the square lattice and scale-free networks. By using the master equation and Monte Carlo simulations, we find that cooperators coexist with defectors in the whole phase diagram, even at high temptations to defect. We study the microscopic mechanism that is responsible for the striking persistence of cooperative behavior and find that cooperation spreads through second-order neighbors, rather than by means of network reciprocity that dominates in imitation-based models. For the square lattice the master equation can be solved analytically in the large temperature limit of the Fermi function, while for other cases the resulting differential equations must be solved numerically. Either way, we find good qualitative agreement with the Monte Carlo simulation results. Our analysis also reveals that the evolutionary outcomes are to a large degree independent of the network topology, including the number of neighbors that are considered for payoff determination on lattices, which further corroborates the local character of the microscopic dynamics. Unlike large-scale spatial patterns that typically emerge due to network reciprocity, here local checkerboard-like patterns remain virtually unaffected by differences in the macroscopic properties of the interaction network.
Collapse
Affiliation(s)
- Marco A Amaral
- Departamento de Física, Universidade Federal de Minas Gerais, Caixa Postal 702, CEP 30161-970, Belo Horizonte-MG, Brazil
| | - Lucas Wardil
- Departamento de Fisica, Universidade Federal de Ouro Preto, Ouro Preto, CEP 35400-000 MG, Brazil
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia.,CAMTP-Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia
| | - Jafferson K L da Silva
- Departamento de Física, Universidade Federal de Minas Gerais, Caixa Postal 702, CEP 30161-970, Belo Horizonte-MG, Brazil
| |
Collapse
|
45
|
Kim J, Yook SH, Kim Y. Reciprocity in spatial evolutionary public goods game on double-layered network. Sci Rep 2016; 6:31299. [PMID: 27503801 PMCID: PMC4977568 DOI: 10.1038/srep31299] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2016] [Accepted: 07/15/2016] [Indexed: 11/25/2022] Open
Abstract
Spatial evolutionary games have mainly been studied on a single, isolated network. However, in real world systems, many interaction topologies are not isolated but many different types of networks are inter-connected to each other. In this study, we investigate the spatial evolutionary public goods game (SEPGG) on double-layered random networks (DRN). Based on the mean-field type arguments and numerical simulations, we find that SEPGG on DRN shows very rich interesting phenomena, especially, depending on the size of each layer, intra-connectivity, and inter-connected couplings, the network reciprocity of SEPGG on DRN can be drastically enhanced through the inter-connected coupling. Furthermore, SEPGG on DRN can provide a more general framework which includes the evolutionary dynamics on multiplex networks and inter-connected networks at the same time.
Collapse
Affiliation(s)
- Jinho Kim
- Department of Social Network Science, Kyung Hee University, Seoul 130-701, Korea
| | - Soon-Hyung Yook
- Department of Social Network Science, Kyung Hee University, Seoul 130-701, Korea
- Department of Physics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, Korea
| | - Yup Kim
- Department of Physics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, Korea
| |
Collapse
|
46
|
Han D, Li D, Sun M. How the initial level of visibility and limited resource affect the evolution of cooperation. Sci Rep 2016; 6:27191. [PMID: 27250335 PMCID: PMC4890014 DOI: 10.1038/srep27191] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/11/2016] [Accepted: 05/16/2016] [Indexed: 11/08/2022] Open
Abstract
This work sheds important light on how the initial level of visibility and limited resource might affect the evolution of the players' strategies under different network structure. We perform the prisoner's dilemma game in the lattice network and the scale-free network, the simulation results indicate that the average density of death in lattice network decreases with the increases of the initial proportion of visibility. However, the contrary phenomenon is observed in the scale-free network. Further results reflect that the individuals' payoff in lattice network is significantly larger than the one in the scale-free network. In the lattice network, the visibility individuals could earn much more than the invisibility one. However, the difference is not apparent in the scale-free network. We also find that a high Successful-Defection-Payoff (SDB) and a rich natural environment have relatively larger deleterious cooperation effects. A high SDB is beneficial to raising the level of visibility in the heterogeneous network, however, that has adverse visibility consequences in homogeneous network. Our result reveals that players are more likely to cooperate voluntarily under homogeneous network structure.
Collapse
Affiliation(s)
- Dun Han
- Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu, 212013, PR China
| | - Dandan Li
- College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 211106, China
| | - Mei Sun
- Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu, 212013, PR China
| |
Collapse
|
47
|
Amaral MA, Wardil L, Perc M, da Silva JKL. Evolutionary mixed games in structured populations: Cooperation and the benefits of heterogeneity. Phys Rev E 2016; 93:042304. [PMID: 27176309 DOI: 10.1103/physreve.93.042304] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/19/2016] [Indexed: 06/05/2023]
Abstract
Evolutionary games on networks traditionally involve the same game at each interaction. Here we depart from this assumption by considering mixed games, where the game played at each interaction is drawn uniformly at random from a set of two different games. While in well-mixed populations the random mixture of the two games is always equivalent to the average single game, in structured populations this is not always the case. We show that the outcome is, in fact, strongly dependent on the distance of separation of the two games in the parameter space. Effectively, this distance introduces payoff heterogeneity, and the average game is returned only if the heterogeneity is small. For higher levels of heterogeneity the distance to the average game grows, which often involves the promotion of cooperation. The presented results support preceding research that highlights the favorable role of heterogeneity regardless of its origin, and they also emphasize the importance of the population structure in amplifying facilitators of cooperation.
Collapse
Affiliation(s)
- Marco A Amaral
- Departamento de Física, Universidade Federal de Minas Gerais, Caixa Postal 702, CEP 30161-970, Belo Horizonte-MG, Brazil
| | - Lucas Wardil
- Departamento de Fisica, Universidade Federal de Ouro Preto, Ouro Preto, 35400-000, MG, Brazil
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia
- CAMTP-Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia
| | - Jafferson K L da Silva
- Departamento de Física, Universidade Federal de Minas Gerais, Caixa Postal 702, CEP 30161-970, Belo Horizonte-MG, Brazil
| |
Collapse
|
48
|
Gianetto DA, Heydari B. Sparse cliques trump scale-free networks in coordination and competition. Sci Rep 2016; 6:21870. [PMID: 26899456 PMCID: PMC4761901 DOI: 10.1038/srep21870] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/15/2015] [Accepted: 02/02/2016] [Indexed: 11/09/2022] Open
Abstract
Cooperative behavior, a natural, pervasive and yet puzzling phenomenon, can be significantly enhanced by networks. Many studies have shown how global network characteristics affect cooperation; however, it is difficult to understand how this occurs based on global factors alone, low-level network building blocks, or motifs are necessary. In this work, we systematically alter the structure of scale-free and clique networks and show, through a stochastic evolutionary game theory model, that cooperation on cliques increases linearly with community motif count. We further show that, for reactive stochastic strategies, network modularity improves cooperation in the anti-coordination Snowdrift game and the Prisoner's Dilemma game but not in the Stag Hunt coordination game. We also confirm the negative effect of the scale-free graph on cooperation when effective payoffs are used. On the flip side, clique graphs are highly cooperative across social environments. Adding cycles to the acyclic scale-free graph increases cooperation when multiple games are considered; however, cycles have the opposite effect on how forgiving agents are when playing the Prisoner's Dilemma game.
Collapse
Affiliation(s)
- David A Gianetto
- School of Systems and Enterprises, Stevens Institute of Technology, Hoboken NJ, USA.,Raytheon Space and Airborne Systems, El Segundo CA, USA
| | - Babak Heydari
- School of Systems and Enterprises, Stevens Institute of Technology, Hoboken NJ, USA
| |
Collapse
|
49
|
Zhang Y, Fu F, Chen X, Xie G, Wang L. Cooperation in group-structured populations with two layers of interactions. Sci Rep 2015; 5:17446. [PMID: 26632251 PMCID: PMC4668372 DOI: 10.1038/srep17446] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2015] [Accepted: 10/29/2015] [Indexed: 11/09/2022] Open
Abstract
Recently there has been a growing interest in studying multiplex networks where individuals are structured in multiple network layers. Previous agent-based simulations of games on multiplex networks reveal rich dynamics arising from interdependency of interactions along each network layer, yet there is little known about analytical conditions for cooperation to evolve thereof. Here we aim to tackle this issue by calculating the evolutionary dynamics of cooperation in group-structured populations with two layers of interactions. In our model, an individual is engaged in two layers of group interactions simultaneously and uses unrelated strategies across layers. Evolutionary competition of individuals is determined by the total payoffs accrued from two layers of interactions. We also consider migration which allows individuals to move to a new group within each layer. An approach combining the coalescence theory with the theory of random walks is established to overcome the analytical difficulty upon local migration. We obtain the exact results for all “isotropic” migration patterns, particularly for migration tuned with varying ranges. When the two layers use one game, the optimal migration ranges are proved identical across layers and become smaller as the migration probability grows.
Collapse
Affiliation(s)
- Yanling Zhang
- School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China.,Center for Systems and Control, State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China
| | - Feng Fu
- Theoretical Biology, Institute of Integrative Biology, ETH Zürich, 8092 Zürich, Switzerland
| | - Xiaojie Chen
- School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
| | - Guangming Xie
- Center for Systems and Control, State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China
| | - Long Wang
- Center for Systems and Control, State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China
| |
Collapse
|
50
|
Choi W, Yook SH, Kim Y. Percolation in spatial evolutionary prisoner's dilemma game on two-dimensional lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:052140. [PMID: 26651679 DOI: 10.1103/physreve.92.052140] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/17/2015] [Indexed: 06/05/2023]
Abstract
We study the spatial evolutionary prisoner's dilemma game with updates of imitation max on triangular, hexagonal, and square lattices. We use the weak prisoner's dilemma game with a single parameter b. Due to the competition between the temptation value b and the coordination number z of the base lattice, a greater variety of percolation properties is expected to occur on the lattice with the larger z. From the numerical analysis, we find six different regimes on the triangular lattice (z=6). Regardless of the initial densities of cooperators and defectors, cooperators always percolate in the steady state in two regimes for small b. In these two regimes, defectors do not percolate. In two regimes for the intermediate value of b, both cooperators and defectors undergo percolation transitions. The defector always percolates in two regimes for large b. On the hexagonal lattice (z=3), there exist two distinctive regimes. For small b, both the cooperators and the defectors undergo percolation transitions while only defectors always percolate for large b. On the square lattice (z=4), there exist three regimes. Combining with the finite-size scaling analyses, we show that all the observed percolation transitions belong to the universality class of the random percolation. We also show how the detailed growth mechanism of cooperator and defector clusters decides each regime.
Collapse
Affiliation(s)
- Woosik Choi
- Department of Physics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, Korea
| | - Soon-Hyung Yook
- Department of Physics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, Korea
| | - Yup Kim
- Department of Physics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, Korea
| |
Collapse
|