1
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Hansen M, Chalub FACC. Population dynamics and games of variable size. J Theor Biol 2024; 589:111842. [PMID: 38685528 DOI: 10.1016/j.jtbi.2024.111842] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2023] [Revised: 04/12/2024] [Accepted: 04/22/2024] [Indexed: 05/02/2024]
Abstract
This work introduces the concept of Variable Size Game Theory (VSGT), in which the number of players in a game is a strategic decision made by the players themselves. We start by discussing the main examples in game theory: dominance, coexistence, and coordination. We show that the same set of pay-offs can result in coordination-like or coexistence-like games depending on the strategic decision of each player type. We also solve an inverse problem to find a d-player game that reproduces the same fixation pattern of the VSGT. In the sequel, we consider a game involving prosocial and antisocial players, i.e., individuals who tend to play with large groups and small groups, respectively. In this game, a certain task should be performed, that will benefit one of the participants at the expense of the other players. We show that individuals able to gather large groups to perform the task may prevail, even if this task is costly, providing a possible scenario for the evolution of eusociality. The next example shows that different strategies regarding game size may lead to spontaneous separation of different types, a possible scenario for speciation without physical separation (sympatric speciation). In the last example, we generalize to three types of populations from the previous analysis and study compartmental epidemic models: in particular, we recast the SIRS model into the VSGT framework: Susceptibles play 2-player games, while Infectious and Removed play a 1-player game. The SIRS epidemic model is then obtained as the replicator equation of the VSGT. We finish with possible applications of VSGT to be addressed in the future.
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Affiliation(s)
- Matheus Hansen
- Center for Mathematics and Applications (NOVA Math), NOVA FCT, Universidade NOVA de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal.
| | - Fabio A C C Chalub
- Center for Mathematics and Applications (NOVA Math), NOVA FCT, Universidade NOVA de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal; Department of Mathematics, NOVA FCT, Universidade NOVA de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal.
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2
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Bera RK, Rana S, Bhattacharya S. Interaction intensity in strategic fitness: A quantifying yardstick of selection optimization for evolutionary game. Math Biosci 2024; 375:109241. [PMID: 38936543 DOI: 10.1016/j.mbs.2024.109241] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2023] [Revised: 05/22/2024] [Accepted: 06/21/2024] [Indexed: 06/29/2024]
Abstract
The notion of the fitness of a strategy has been assimilated as the reproductive success in the evolutionary game. Initially, this fitness was tied to the game's pay-off and the strategy's relative frequency. However, density dependence becomes exigent in order to make ecologically reliable fitness. However, the contributions of each different type of interaction to the species's overall growth process were surprisingly under-explored. This oversight has occasionally led to either more or less prediction of strategy selection compared to the actual possibility. Moreover, density regulation of the population has always been analysed in a general way compared to strategy selection. In this context, our study introduces the concept of mean relative death payoff, which helps in assessing interaction intensity coefficients and integrates them into strategic fitness. Based on this fitness function, we develop the frequency-density replicator dynamics, which eventually provides distinguishing criteria for directional and balancing selection. Our optimized, evolutionarily stable strategy emerges as a superior alternative to the conventional trade-off between selection forces and ecological processes. More significantly, mean relative death pay-off has both conditional and quantitative roles in getting a stable population size. As a case study, we have extensively analysed the evolution of aggression using the Hawk-Dove game. We have shown that pure Dove selection is always beneficial for species growth rather than pure Hawk selection, and the condition of selection is dependent on external mortality pressure. However, the condition of coexistence is independent of external mortality pressure, representing a strong evolutionary selection that optimizes population density governed by interaction intensity.
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Affiliation(s)
- Ritesh Kumar Bera
- Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, BT Road, Kolkata, 700108, West Bengal, India.
| | - Sourav Rana
- Department of Statistics, Visva-Bharati University, Santiniketan, 700035, West Bengal, India.
| | - Sabyasachi Bhattacharya
- Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, BT Road, Kolkata, 700108, West Bengal, India.
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3
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Király B, Varga T, Szabó G, Garay J. Evolutionarily stable payoff matrix in hawk-dove games. BMC Ecol Evol 2024; 24:65. [PMID: 38769504 PMCID: PMC11107024 DOI: 10.1186/s12862-024-02257-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2024] [Accepted: 05/15/2024] [Indexed: 05/22/2024] Open
Abstract
BACKGROUND Classical matrix game models aim to find the endpoint of behavioural evolution for a set of fixed possible interaction outcomes. Here, we introduce an evolutionary model in which not only the players' strategies but also the payoff matrix evolves according to natural selection. RESULTS We start out from the hawk-dove matrix game and, in a way that is consistent with the monomorphic model setup of Maynard Smith and Price, introduce an evolving phenotypic trait that quantifies fighting ability and determines the probability of winning and the cost of losing escalated hawk-hawk fights. We define evolutionarily stable phenotypes as consisting of an evolutionarily stable strategy and an evolutionarily stable trait, which in turn describes a corresponding evolutionarily stable payoff matrix. CONCLUSIONS We find that the maximal possible cost of escalating fights remains constant during evolution assuming a separation in the time scales of fast behavioural and slow trait selection, despite the fact that the final evolutionarily stable phenotype maximizes the payoff of hawk-hawk fights. Our results mirror the dual nature of Darwinian evolution whereby the criteria of evolutionary success, as well as the successful phenotypes themselves, are a product of natural selection.
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Affiliation(s)
- Balázs Király
- Institute of Technical Physics and Materials Science, HUN-REN Centre for Energy Research, Konkoly-Thege Miklós út 29-33., Budapest, H-1121, Hungary.
| | - Tamás Varga
- Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., Szeged, H-6720, Hungary
| | - György Szabó
- Institute of Technical Physics and Materials Science, HUN-REN Centre for Energy Research, Konkoly-Thege Miklós út 29-33., Budapest, H-1121, Hungary
- Institute of Evolution, HUN-REN Centre for Ecological Research, Konkoly-Thege Miklós út 29-33., Budapest, H-1121, Hungary
| | - József Garay
- Institute of Evolution, HUN-REN Centre for Ecological Research, Konkoly-Thege Miklós út 29-33., Budapest, H-1121, Hungary
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4
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Koskela J, Łatuszyński K, Spanò D. Bernoulli factories and duality in Wright-Fisher and Allen-Cahn models of population genetics. Theor Popul Biol 2024; 156:40-45. [PMID: 38301934 DOI: 10.1016/j.tpb.2024.01.002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2023] [Revised: 12/12/2023] [Accepted: 01/29/2024] [Indexed: 02/03/2024]
Abstract
Mathematical models of genetic evolution often come in pairs, connected by a so-called duality relation. The most seminal example are the Wright-Fisher diffusion and the Kingman coalescent, where the former describes the stochastic evolution of neutral allele frequencies in a large population forwards in time, and the latter describes the genetic ancestry of randomly sampled individuals from the population backwards in time. As well as providing a richer description than either model in isolation, duality often yields equations satisfied by quantities of interest. We employ the so-called Bernoulli factory - a celebrated tool in simulation-based computing - to derive duality relations for broad classes of genetics models. As concrete examples, we present Wright-Fisher diffusions with general drift functions, and Allen-Cahn equations with general, nonlinear forcing terms. The drift and forcing functions can be interpreted as the action of frequency-dependent selection. To our knowledge, this work is the first time a connection has been drawn between Bernoulli factories and duality in models of population genetics.
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Affiliation(s)
- Jere Koskela
- School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK; Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
| | | | - Dario Spanò
- Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
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5
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Roitershtein A, Rastegar R, Chapkin RS, Ivanov I. Extinction scenarios in evolutionary processes: a multinomial Wright-Fisher approach. J Math Biol 2023; 87:63. [PMID: 37751048 PMCID: PMC10586398 DOI: 10.1007/s00285-023-01993-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2019] [Revised: 08/16/2023] [Accepted: 08/31/2023] [Indexed: 09/27/2023]
Abstract
We study a discrete-time multi-type Wright-Fisher population process. The mean-field dynamics of the stochastic process is induced by a general replicator difference equation. We prove several results regarding the asymptotic behavior of the model, focusing on the impact of the mean-field dynamics on it. One of the results is a limit theorem that describes sufficient conditions for an almost certain path to extinction, first eliminating the type which is the least fit at the mean-field equilibrium. The effect is explained by the metastability of the stochastic system, which under the conditions of the theorem spends almost all time before the extinction event in a neighborhood of the equilibrium. In addition to the limit theorems, we propose a maximization principle for a general deterministic replicator dynamics and study its implications for the stochastic model.
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Affiliation(s)
| | - Reza Rastegar
- Occidental Petroleum Corporation, Houston, TX, 77046, USA
| | - Robert S Chapkin
- Department of Nutrition - Program in Integrative Nutrition & Complex Diseases, Texas A &M University, College Station, TX, 77843, USA
| | - Ivan Ivanov
- Department of Veterinary Physiology and Pharmacology, Texas A &M University, College Station, TX, 77843, USA.
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6
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Fan L, Song Z, Wang L, Liu Y, Wang Z. Incorporating social payoff into reinforcement learning promotes cooperation. CHAOS (WOODBURY, N.Y.) 2022; 32:123140. [PMID: 36587319 DOI: 10.1063/5.0093996] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/31/2022] [Accepted: 12/01/2022] [Indexed: 06/17/2023]
Abstract
Reinforcement learning has been demonstrated to be an effective approach to investigate the dynamic of strategy updating and the learning process of agents in game theory. Most studies have shown that Q-learning failed to resolve the problem of cooperation in well-mixed populations or homogeneous networks. To this aim, we investigate the self-regarding Q-learning's effect on cooperation in spatial prisoner's dilemma games by incorporating the social payoff. Here, we redefine the reward term of self-regarding Q-learning by involving the social payoff; that is, the reward is defined as a monotonic function of the individual payoff and the social payoff represented by its neighbors' payoff. Numerical simulations reveal that such a framework can facilitate cooperation remarkably because the social payoff ensures agents learn to cooperate toward socially optimal outcomes. Moreover, we find that self-regarding Q-learning is an innovative rule that ensures cooperators coexist with defectors even at high temptations to defection. The investigation of the emergence and stability of the sublattice-ordered structure shows that such a mechanism tends to generate a checkerboard pattern to increase agents' payoff. Finally, the effects of Q-learning parameters are also analyzed, and the robustness of this mechanism is verified on different networks.
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Affiliation(s)
- Litong Fan
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China
| | - Zhao Song
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China
| | - Lu Wang
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China
| | - Yang Liu
- School of Artificial Intelligence, Optics and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China
| | - Zhen Wang
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China
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7
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Abstract
In order to accommodate the empirical fact that population structures are rarely simple, modern studies of evolutionary dynamics allow for complicated and highly heterogeneous spatial structures. As a result, one of the most difficult obstacles lies in making analytical deductions, either qualitative or quantitative, about the long-term outcomes of evolution. The "structure-coefficient" theorem is a well-known approach to this problem for mutation-selection processes under weak selection, but a general method of evaluating the terms it comprises is lacking. Here, we provide such a method for populations of fixed (but arbitrary) size and structure, using easily interpretable demographic measures. This method encompasses a large family of evolutionary update mechanisms and extends the theorem to allow for asymmetric contests to provide a better understanding of the mutation-selection balance under more realistic circumstances. We apply the method to study social goods produced and distributed among individuals in spatially heterogeneous populations, where asymmetric interactions emerge naturally and the outcome of selection varies dramatically, depending on the nature of the social good, the spatial topology, and the frequency with which mutations arise.
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8
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Evolutionary game dynamics with non-uniform interaction rates in finite population. J Theor Biol 2022; 540:111086. [PMID: 35271866 DOI: 10.1016/j.jtbi.2022.111086] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2021] [Revised: 02/11/2022] [Accepted: 03/03/2022] [Indexed: 11/22/2022]
Abstract
In this study, we extend evolutionary game dynamics with non-uniform interaction rates to the situation with finite population. Our main goal is to show how the fixation probability is influenced by the non-uniform interaction rates under weak selection. Based on the diffusion approximation of the Moran process and assumption of weak selection, the stochastic dynamic properties of a two-phenotype game with non-uniform interaction rates in a finite population are investigated. By the analysis of some cases, we show that the non-uniform interaction rates may result in the potential evolutionary complexity of game dynamics in finite population.
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9
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Wang X, Perc M. Bilateral costly expulsions resolve the public goods dilemma. Proc Math Phys Eng Sci 2021; 477:20210627. [PMID: 35601085 PMCID: PMC9097491 DOI: 10.1098/rspa.2021.0627] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/02/2021] [Accepted: 08/24/2021] [Indexed: 11/12/2022] Open
Abstract
Expulsion has been found to promote cooperation in social dilemmas, but only if it does not incur costs or is applied unilaterally. Here, we show that removing both conditions leads to a spontaneous resolution of the costly expulsion problem. Namely, by studying the public goods game where cooperators and defectors can expel others at a personal cost, we find that public cooperation thrives as expulsion costs increase. This is counterintuitive, as the cost of other-regarding behaviour typically places an additional burden on cooperation, which is in itself costly. Such scenarios are referred to as second-order free-rider problems, and they typically require an additional mechanism, such as network reciprocity, to be resolved. We perform a mean field analysis of the public goods game with bilateral costly expulsion, showing analytically that the expected payoff difference between cooperators and defectors increases with expulsion costs as long as players with the same strategy have, on average, a higher frequency to interact with each other. As the latter condition is often satisfied in social networks, our results thus reveal a fascinating new path to public cooperation, and they show that the costs of well-intended actions need not be low for them to be effective.
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Affiliation(s)
- Xiaofeng Wang
- Department of Automation, School of Information Science and Technology, Donghua University, Shanghai 201620, China
- Engineering Research Center of Digitized Textile and Apparel Technology, Donghua University, Ministry of Education, Shanghai 201620, China
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, Maribor 2000, Slovenia
- Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 404332, Taiwan
- Alma Mater Europaea ECM, Slovenska ulica 17, Maribor 2000, Slovenia
- Complexity Science Hub Vienna, Josefstädterstraße 39, Vienna 1080, Austria
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10
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Salagnac O, Wakeley J. The consequences of switching strategies in a two-player iterated survival game. J Math Biol 2021; 82:17. [PMID: 33547962 PMCID: PMC7867574 DOI: 10.1007/s00285-021-01569-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2020] [Revised: 01/08/2021] [Accepted: 01/17/2021] [Indexed: 11/25/2022]
Abstract
We consider two-player iterated survival games in which players are able to switch from a more cooperative behavior to a less cooperative one at some step of an n-step game. Payoffs are survival probabilities and lone individuals have to finish the game on their own. We explore the potential of these games to support cooperation, focusing on the case in which each single step is a Prisoner’s Dilemma. We find that incentives for or against cooperation depend on the number of defections at the end of the game, as opposed to the number of steps in the game. Broadly, cooperation is supported when the survival prospects of lone individuals are relatively bleak. Specifically, we find three critical values or cutoffs for the loner survival probability which, in concert with other survival parameters, determine the incentives for or against cooperation. One cutoff determines the existence of an optimal number of defections against a fully cooperative partner, one determines whether additional defections eventually become disfavored as the number of defections by the partner increases, and one determines whether additional cooperations eventually become favored as the number of defections by the partner increases. We obtain expressions for these switch-points and for optimal numbers of defections against partners with various strategies. These typically involve small numbers of defections even in very long games. We show that potentially long stretches of equilibria may exist, in which there is no incentive to defect more or cooperate more. We describe how individuals find equilibria in best-response walks among n-step strategies.
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Affiliation(s)
| | - John Wakeley
- Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138 USA
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11
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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.
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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
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12
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Broom M, Křivan V. Two-strategy games with time constraints on regular graphs. J Theor Biol 2020; 506:110426. [PMID: 32777217 DOI: 10.1016/j.jtbi.2020.110426] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2019] [Revised: 07/24/2020] [Accepted: 07/27/2020] [Indexed: 11/19/2022]
Abstract
Evolutionary game theory is a powerful method for modelling animal conflicts. The original evolutionary game models were used to explain specific biological features of interest, such as the existence of ritualised contests, and were necessarily simple models that ignored many properties of real populations, including the duration of events and spatial and related structural effects. Both of these areas have subsequently received much attention. Spatial and structural effects have been considered in evolutionary graph theory, and a significant body of literature has been built up to deal with situations where the population is not homogeneous. More recently a theory of time constraints has been developed to take account of the fact that different events can take different times, and that interaction times can explicitly depend upon selected strategies, which can, in turn, influence the distribution of different opponent types within the population. Here, for the first time, we build a model of time constraint games which explicitly considers a spatial population, by considering a population evolving on an underlying graph, using two graph dynamics, birth-death and death-birth. We consider one short time scale along which frequencies of pairs and singles change as individuals interact with their neighbours, and another, evolutionary time scale, along which frequencies of strategies change in the population. We show that for graphs with large degree, both dynamics reproduce recent results from well-mixed time constraint models, including two ESSs being common in Hawk-Dove and Prisoner's Dilemma games, but for low degree there can be marked differences. For birth-death processes the effect of the graph degree is small, whereas for death-birth dynamics there is a large effect. The general prediction for both Hawk-Dove and Prisoner's dilemma games is that as the graph degree decreases, i.e., as the number of neighbours decreases, mixed ESS do appear. In particular, for the Prisoner's dilemma game this means that cooperation is easier to establish in situations where individuals have low number of neighbours. We thus see that solutions depend non-trivially on the combination of graph degree, dynamics and game.
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Affiliation(s)
- Mark Broom
- Department of Mathematics, City, University of London, London, UK.
| | - Vlastimil Křivan
- Centre for Mathematical Biology, Department of Mathematics, Faculty of Science, University of South Bohemia, Branišovská 1760, 370 05 České Budějovice, Czech Republic; Czech Academy of Sciences, Biology Centre, Institute of Entomology, Branišovská 31, 370 05 České Budějovice, Czech Republic.
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13
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Abstract
Cooperation in social dilemmas plays a pivotal role in the formation of systems at all levels of complexity, from replicating molecules to multi-cellular organisms to human and animal societies. In spite of its ubiquity, the origin and stability of cooperation pose an evolutionary conundrum, since cooperation, though beneficial to others, is costly to the individual cooperator. Thus natural selection would be expected to favor selfish behavior in which individuals reap the benefits of cooperation without bearing the costs of cooperating themselves. Many proximate mechanisms have been proposed to account for the origin and maintenance of cooperation, including kin selection, direct reciprocity, indirect reciprocity, and evolution in structured populations. Despite the apparent diversity of these approaches they all share a unified underlying logic: namely, each mechanism results in assortative interactions in which individuals using the same strategy interact with a higher probability than they would at random. Here we study the evolution of cooperation in both discrete strategy and continuous strategy social dilemmas with assortative interactions. For the sake of tractability, assortativity is modeled by an individual interacting with another of the same type with probability r and interacting with a random individual in the population with probability 1−r, where r is a parameter that characterizes the degree of assortativity in the system. For discrete strategy social dilemmas we use both a generalization of replicator dynamics and individual-based simulations to elucidate the donation, snowdrift, and sculling games with assortative interactions, and determine the analogs of Hamilton’s rule, which govern the evolution of cooperation in these games. For continuous strategy social dilemmas we employ both a generalization of deterministic adaptive dynamics and individual-based simulations to study the donation, snowdrift, and tragedy of the commons games, and determine the effect of assortativity on the emergence and stability of cooperation.
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14
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Wu B, Park HJ, Wu L, Zhou D. Evolution of cooperation driven by self-recommendation. Phys Rev E 2019; 100:042303. [PMID: 31770974 DOI: 10.1103/physreve.100.042303] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2019] [Indexed: 11/07/2022]
Abstract
Cooperators increase the fitness of others at a cost to themselves. Thus cooperation should not be favored by natural selection in a well-mixed population. It challenges the evolutionists since cooperation is widespread. Information spreading has been revealed to play a key role in the emergence of cooperation. Individuals, however, are typically assumed to be passive in the information spreading. Here we assume that individuals self-recommend themselves to those that are about to have new neighbors. Individuals with higher tendencies of self-recommendation are likely to have more neighbors. In this way, individuals are active to spread the information. We analytically obtain a critical cost-to-benefit ratio, below which cooperation emerges. It reveals quantitatively how eloquent cooperators have to be compared with defectors to ensure that cooperation takes over the population. It also indicates that individuals need to be open enough to the self-recommendation to enhance cooperation level. In addition, the critical cost-to-benefit ratio represents the viscosity of the population, measuring how close cooperators are to each other. Our results highlight the role self-recommendation plays in cooperation.
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Affiliation(s)
- Bin Wu
- School of Sciences, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
| | - Hye Jin Park
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, August-Thienemann-Strasse 2, 24306 Plön, Germany
| | - Lingshan Wu
- School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China
| | - Da Zhou
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, August-Thienemann-Strasse 2, 24306 Plön, Germany.,School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China
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15
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Xiao Y, Wu B. Close spatial arrangement of mutants favors and disfavors fixation. PLoS Comput Biol 2019; 15:e1007212. [PMID: 31525178 PMCID: PMC6746358 DOI: 10.1371/journal.pcbi.1007212] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2019] [Accepted: 06/25/2019] [Indexed: 11/26/2022] Open
Abstract
Cooperation is ubiquitous across all levels of biological systems ranging from microbial communities to human societies. It, however, seemingly contradicts the evolutionary theory, since cooperators are exploited by free-riders and thus are disfavored by natural selection. Many studies based on evolutionary game theory have tried to solve the puzzle and figure out the reason why cooperation exists and how it emerges. Network reciprocity is one of the mechanisms to promote cooperation, where nodes refer to individuals and links refer to social relationships. The spatial arrangement of mutant individuals, which refers to the clustering of mutants, plays a key role in network reciprocity. Besides, many other mechanisms supporting cooperation suggest that the clustering of mutants plays an important role in the expansion of mutants. However, the clustering of mutants and the game dynamics are typically coupled. It is still unclear how the clustering of mutants alone alters the evolutionary dynamics. To this end, we employ a minimal model with frequency independent fitness on a circle. It disentangles the clustering of mutants from game dynamics. The distance between two mutants on the circle is adopted as a natural indicator for the clustering of mutants or assortment. We find that the assortment is an amplifier of the selection for the connected mutants compared with the separated ones. Nevertheless, as mutants are separated, the more dispersed mutants are, the greater the chance of invasion is. It gives rise to the non-monotonic effect of clustering, which is counterintuitive. On the other hand, we find that less assortative mutants speed up fixation. Our model shows that the clustering of mutants plays a non-trivial role in fixation, which has emerged even if the game interaction is absent. Evolutionary dynamics on networks are key for biological and social evolution. Typically, the clustering mutants on networks can dramatically alter the direction of selection. Previous studies on the assortment of mutants assume that individuals interact in a frequency-dependent way. It is hard to tell how assortment alone alters the evolutionary fate. We establish a minimal network model to disentangle the assortment from the game interaction. We find that for weak selection limit, the assortment of mutants plays little role in fixation probability. For strong selection limit, connected mutants, i.e., the maximum assortment, are best for fixation. When the mutants are separated by only one wild-type individual, it is worse off than that separated by more than one wild-type individual in fixation probability. Our results show the nontrivial yet fundamental effect of the clustering on fixation. Noteworthily, it has already arisen, even if the game interaction is absent.
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Affiliation(s)
- Yunming Xiao
- School of Sciences, Beijing University of Posts and Telecommunications, Beijing, China
| | - Bin Wu
- School of Sciences, Beijing University of Posts and Telecommunications, Beijing, China
- * E-mail:
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16
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Park J, Jang B. Robust coexistence with alternative competition strategy in the spatial cyclic game of five species. CHAOS (WOODBURY, N.Y.) 2019; 29:051105. [PMID: 31154778 DOI: 10.1063/1.5097003] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2019] [Accepted: 04/22/2019] [Indexed: 06/09/2023]
Abstract
Alternative strategy is common in animal populations to promote reproductive fitness by obtaining resources. In spatial dynamics of cyclic competition, reproduction can occur when individuals obtain vacant rooms and, in this regard, empty sites should be resources for reproduction which can be induced by interspecific competition. In this paper, we study the role of alternative competition in the spatial system of cyclically competing five species by utilizing rock-paper-scissors-lizard-spock game. From Monte-Carlo simulations, we found that strong alternative competition can lead to the reemergence of coexistence of five species regardless of mobility, which is never reported in previous works under the symmetric competition structure. By investigating the coexistence probability, we also found that coexistence alternates by passing certain degrees of alternative competition in combination with mobility. In addition, we provided evidences in the opposite scenario by strengthening spontaneous competition, which exhibits the reemergence of coexistence similarly. Our findings may suggest more comprehensive perspectives to interpret mechanisms for biodiversity by alternative strategies in spatially extended systems than previously reported.
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Affiliation(s)
- Junpyo Park
- Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 44919, South Korea
| | - Bongsoo Jang
- Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 44919, South Korea
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17
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Effects of Relatedness on the Evolution of Cooperation in Nonlinear Public Goods Games. GAMES 2018. [DOI: 10.3390/g9040087] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
Evolution of cooperation has traditionally been studied by assuming that individuals adopt either of two pure strategies, to cooperate or defect. Recent work has considered continuous cooperative investments, turning full cooperation and full defection into two opposing ends of a spectrum and sometimes allowing for the emergence of the traditionally-studied pure strategies through evolutionary diversification. These studies have typically assumed a well-mixed population in which individuals are encountered with equal probability. Here, we allow for the possibility of assortative interactions by assuming that, with specified probabilities, an individual interacts with one or more other individuals of the same strategy. A closely related assumption has previously been made in evolutionary game theory and has been interpreted in terms of relatedness. We systematically study the effect of relatedness and find, among other conclusions, that the scope for evolutionary branching is reduced by either higher average degree of, or higher uncertainty in, relatedness with interaction partners. We also determine how different types of non-linear dependencies of benefits and costs constrain the types of evolutionary outcomes that can occur. While our results overall corroborate the conclusions of earlier studies, i.e. higher relatedness promotes the evolution of cooperation, our investigation gives a comprehensive picture of how relatedness affects the evolution of cooperation with continuous investments.
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18
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He QQ, Feng TJ, Tao Y, Zhang B, Ji T. Asymmetric evolutionary game dynamics based on individuals' own volition. J Theor Biol 2018; 454:118-125. [PMID: 29775681 DOI: 10.1016/j.jtbi.2018.05.012] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/29/2018] [Revised: 05/10/2018] [Accepted: 05/14/2018] [Indexed: 11/16/2022]
Abstract
For the pairwise interactions, the evolution of individual behavior should involve two major factors: one is what you will do in an interaction with a given opponent, and another is what type of opponents you prefer to interact with. In this study, we developed a two-phenotype iterated bimatrix replicator dynamics model based on individuals' own volition, where, different from the classic iterated game model, we assume that (i) for all interaction pairs, the maximum expected interaction time is same and it is limited even if two individuals in an interaction pair would like to keep their interaction; and (ii) all individuals are able to unilaterally break off the interactions with their opponents according to their own volition. Therefore, we define that, at any time t, an interaction pair will be disbanded with a given probability and the new interaction pairs will be randomly formed. The main results show that: (i) the existence of locally asymmetrically stable interior equilibrium is possible; and (ii) the evolutionary stability of the system is similar to the classic asymmetric evolutionary game. These results may provide a new insight for revealing the evolutionary significance of asymmetric game dynamics.
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Affiliation(s)
- Qiao-Qiao He
- Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing, PR China
| | - Tian-Jiao Feng
- Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, PR China
| | - Yi Tao
- Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing, PR China; University of Chinese Academy of Sciences, PR China
| | - Boyu Zhang
- Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing, PR China.
| | - Ting Ji
- Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing, PR China.
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19
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Křivan V, Galanthay TE, Cressman R. Beyond replicator dynamics: From frequency to density dependent models of evolutionary games. J Theor Biol 2018; 455:232-248. [PMID: 29990466 DOI: 10.1016/j.jtbi.2018.07.003] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Game theoretic models of evolution such as the Hawk-Dove game assume that individuals gain fitness (which is a proxy of the per capita population growth rate) in pair-wise contests only. These models assume that the equilibrium distribution of phenotypes involved (e.g., Hawks and Doves) in the population is given by the Hardy-Weinberg law, which is based on instantaneous, random pair formation. On the other hand, models of population dynamics do not consider pairs, newborns are produced by singles, and interactions between phenotypes or species are described by the mass action principle. This article links game theoretic and population approaches. It shows that combining distribution dynamics with population dynamics can lead to stable coexistence of Hawk and Dove population numbers in models that do not assume a priori that fitness is negative density dependent. Our analysis shows clearly that the interior Nash equilibrium of the Hawk and Dove model depends both on population size and on interaction times between different phenotypes in the population. This raises the question of the applicability of classic evolutionary game theory that requires all interactions take the same amount of time and that all single individuals have the same payoff per unit of time, to real populations. Furthermore, by separating individual fitness into birth and death effects on singles and pairs, it is shown that stable coexistence in these models depends on the time-scale of the distribution dynamics relative to the population dynamics. When explicit density-dependent fitness is included through competition over a limited resource, the combined dynamics of the Hawk-Dove model often lead to Dove extinction no matter how costly fighting is for Hawk pairs.
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Affiliation(s)
- Vlastimil Křivan
- Czech Academy of Sciences, Biology Centre, Institute of Entomology, Branišovská 31, České Budějovice 370 05, Czech Republic; Department of Mathematics, Faculty of Sciences, University of South Bohemia, Branišovská 1760, České Budějovice 370 05, Czech Republic.
| | | | - Ross Cressman
- Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada.
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20
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Argasinski K, Broom M. Interaction rates, vital rates, background fitness and replicator dynamics: how to embed evolutionary game structure into realistic population dynamics. Theory Biosci 2018; 137:33-50. [PMID: 29159683 PMCID: PMC5893772 DOI: 10.1007/s12064-017-0257-y] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/03/2016] [Accepted: 10/16/2017] [Indexed: 11/13/2022]
Abstract
In this paper we are concerned with how aggregated outcomes of individual behaviours, during interactions with other individuals (games) or with environmental factors, determine the vital rates constituting the growth rate of the population. This approach needs additional elements, namely the rates of event occurrence (interaction rates). Interaction rates describe the distribution of the interaction events in time, which seriously affects the population dynamics, as is shown in this paper. This leads to the model of a population of individuals playing different games, where focal game affected by the considered trait can be extracted from the general model, and the impact on the dynamics of other events (which is not neutral) can be described by an average background fertility and mortality. This leads to a distinction between two types of background fitness, strategically neutral elements of the focal games (correlated with the focal game events) and the aggregated outcomes of other interactions (independent of the focal game). The new approach is useful for clarification of the biological meaning of concepts such as weak selection. Results are illustrated by a Hawk-Dove example.
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Affiliation(s)
- K. Argasinski
- Institute of Mathematics of Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa 10, Poland
| | - M. Broom
- Department of Mathematics, City, University of London, Northampton Square, London, EC1V 0HB UK
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21
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Emergence of unusual coexistence states in cyclic game systems. Sci Rep 2017; 7:7465. [PMID: 28785001 PMCID: PMC5547111 DOI: 10.1038/s41598-017-07911-4] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/04/2017] [Accepted: 07/04/2017] [Indexed: 11/08/2022] Open
Abstract
Evolutionary games of cyclic competitions have been extensively studied to gain insights into one of the most fundamental phenomena in nature: biodiversity that seems to be excluded by the principle of natural selection. The Rock-Paper-Scissors (RPS) game of three species and its extensions [e.g., the Rock-Paper-Scissors-Lizard-Spock (RPSLS) game] are paradigmatic models in this field. In all previous studies, the intrinsic symmetry associated with cyclic competitions imposes a limitation on the resulting coexistence states, leading to only selective types of such states. We investigate the effect of nonuniform intraspecific competitions on coexistence and find that a wider spectrum of coexistence states can emerge and persist. This surprising finding is substantiated using three classes of cyclic game models through stability analysis, Monte Carlo simulations and continuous spatiotemporal dynamical evolution from partial differential equations. Our finding indicates that intraspecific competitions or alternative symmetry-breaking mechanisms can promote biodiversity to a broader extent than previously thought.
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22
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Khadem H, Kebriaei H, Veisi Z. Inactivation of tumor suppressor genes and cancer therapy: An evolutionary game theory approach. Math Biosci 2017; 288:84-93. [DOI: 10.1016/j.mbs.2017.03.001] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2016] [Revised: 02/27/2017] [Accepted: 03/01/2017] [Indexed: 01/02/2023]
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23
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Interaction times change evolutionary outcomes: Two-player matrix games. J Theor Biol 2017; 416:199-207. [DOI: 10.1016/j.jtbi.2017.01.010] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/26/2016] [Revised: 12/20/2016] [Accepted: 01/06/2017] [Indexed: 11/23/2022]
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24
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Zheng XD, Li C, Yu JR, Wang SC, Fan SJ, Zhang BY, Tao Y. A simple rule of direct reciprocity leads to the stable coexistence of cooperation and defection in the Prisoner's Dilemma game. J Theor Biol 2017; 420:12-17. [PMID: 28259660 DOI: 10.1016/j.jtbi.2017.02.036] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2016] [Revised: 02/23/2017] [Accepted: 02/27/2017] [Indexed: 11/25/2022]
Abstract
The long-term coexistence of cooperation and defection is a common phenomenon in nature and human society. However, none of the theoretical models based on the Prisoner's Dilemma (PD) game can provide a concise theoretical model to explain what leads to the stable coexistence of cooperation and defection in the long-term even though some rules for promoting cooperation have been summarized (Nowak, 2006, Science 314, 1560-1563). Here, based on the concept of direct reciprocity, we develop an elementary model to show why stable coexistence of cooperation and defection in the PD game is possible. The basic idea behind our theoretical model is that all players in a PD game prefer a cooperator as an opponent, and our results show that considering strategies allowing opting out against defection provide a general and concise way of understanding the fundamental importance of direct reciprocity in driving the evolution of cooperation.
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Affiliation(s)
- Xiu-Deng Zheng
- Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing, PR China
| | - Cong Li
- Department of Mathematics and Statistics, University of Montreal, Montreal, Canada
| | - Jie-Ru Yu
- College of Resources and Environmental Sciences, Gansu Agricultural University, Lanzhou, PR China
| | - Shi-Chang Wang
- Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing, PR China
| | - Song-Jia Fan
- Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing, PR China
| | - Bo-Yu Zhang
- School of Mathematical Science, Beijing Normal University, Beijing, PR China.
| | - Yi Tao
- Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing, PR China.
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25
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Wu B, Mao S, Wang J, Zhou D. Control of epidemics via social partnership adjustment. Phys Rev E 2017; 94:062314. [PMID: 28085324 PMCID: PMC7217516 DOI: 10.1103/physreve.94.062314] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2016] [Indexed: 11/07/2022]
Abstract
Epidemic control is of great importance for human society. Adjusting interacting partners is an effective individualized control strategy. Intuitively, it is done either by shortening the interaction time between susceptible and infected individuals or by increasing the opportunities for contact between susceptible individuals. Here, we provide a comparative study on these two control strategies by establishing an epidemic model with nonuniform stochastic interactions. It seems that the two strategies should be similar, since shortening the interaction time between susceptible and infected individuals somehow increases the chances for contact between susceptible individuals. However, analytical results indicate that the effectiveness of the former strategy sensitively depends on the infectious intensity and the combinations of different interaction rates, whereas the latter one is quite robust and efficient. Simulations are shown to verify our analytical predictions. Our work may shed light on the strategic choice of disease control.
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Affiliation(s)
- Bin Wu
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
| | - Shanjun Mao
- School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China
| | - Jiazeng Wang
- Department of Mathematics, Beijing Technology and Business University, Beijing 100048, People's Republic of China
| | - Da Zhou
- School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, Xiamen University, Xiamen 361005, People's Republic of China
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26
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Zhang BY, Fan SJ, Li C, Zheng XD, Bao JZ, Cressman R, Tao Y. Opting out against defection leads to stable coexistence with cooperation. Sci Rep 2016; 6:35902. [PMID: 27775099 PMCID: PMC5075917 DOI: 10.1038/srep35902] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2016] [Accepted: 10/05/2016] [Indexed: 11/09/2022] Open
Abstract
Cooperation coexisting with defection is a common phenomenon in nature and human society. Previous studies for promoting cooperation based on kin selection, direct and indirect reciprocity, graph selection and group selection have provided conditions that cooperators outcompete defectors. However, a simple mechanism of the long-term stable coexistence of cooperation and defection is still lacking. To reveal the effect of direct reciprocity on the coexistence of cooperation and defection, we conducted a simple experiment based on the Prisoner’s Dilemma (PD) game, where the basic idea behind our experiment is that all players in a PD game should prefer a cooperator as an opponent. Our experimental and theoretical results show clearly that the strategies allowing opting out against defection are able to maintain this stable coexistence.
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Affiliation(s)
- Bo-Yu Zhang
- Key Lab of Animal Ecology and Conservation Biology, Chinese Academy of Science, Beijing, China.,Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, China
| | - Song-Jia Fan
- Key Lab of Animal Ecology and Conservation Biology, Chinese Academy of Science, Beijing, China
| | - Cong Li
- Key Lab of Animal Ecology and Conservation Biology, Chinese Academy of Science, Beijing, China.,Department of Mathematics and Statistics, University of Montreal, Montreal, Canada
| | - Xiu-Deng Zheng
- Key Lab of Animal Ecology and Conservation Biology, Chinese Academy of Science, Beijing, China
| | - Jian-Zhang Bao
- School of Complex Systems, Beijing Normal University, Beijing, China
| | - Ross Cressman
- Department of Mathematics, Wilfrid Laurier University, Waterloo, Canada
| | - Yi Tao
- Key Lab of Animal Ecology and Conservation Biology, Chinese Academy of Science, Beijing, China
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27
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Ghachem M. The institution as a blunt instrument: Cooperation through imperfect observability. J Theor Biol 2016; 396:182-90. [DOI: 10.1016/j.jtbi.2016.02.013] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2014] [Revised: 01/20/2016] [Accepted: 02/10/2016] [Indexed: 11/29/2022]
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28
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Wardil L, Hauert C. Targeted Cooperative Actions Shape Social Networks. PLoS One 2016; 11:e0147850. [PMID: 26824240 PMCID: PMC4733057 DOI: 10.1371/journal.pone.0147850] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2015] [Accepted: 01/08/2016] [Indexed: 11/23/2022] Open
Abstract
Individual acts of cooperation give rise to dynamic social networks. Traditionally, models for cooperation in structured populations are based on a separation of individual strategies and of population structure. Individuals adopt a strategy—typically cooperation or defection, which determines their behaviour toward their neighbours as defined by an interaction network. Here, we report a behavioural experiment that amalgamates strategies and structure to empirically investigate the dynamics of social networks. The action of paying a cost c to provide a benefit b is represented as a directed link point from the donor to the recipient. Participants can add and/or remove links to up to two recipients in each round. First, we show that dense networks emerge, where individuals are characterized by fairness: they receive to the same extent they provide. More specifically, we investigate how participants use information about the generosity and payoff of others to update their links. It turns out that aversion to payoff inequity was the most consistent update rule: adding links to individuals that are worse off and removing links to individuals that are better off. We then investigate the effect of direct reciprocation, showing that the possibility of direct reciprocation does not increase cooperation as compared to the treatment where participants are totally unaware of who is providing benefits to them.
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Affiliation(s)
- Lucas Wardil
- Department of Mathematics, University of British Columbia, Vancouver, BC, Canada
- * E-mail:
| | - Christoph Hauert
- Department of Mathematics, University of British Columbia, Vancouver, BC, Canada
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29
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Sarkar B. Random and non-random mating populations: Evolutionary dynamics in meiotic drive. Math Biosci 2015; 271:29-41. [PMID: 26524140 DOI: 10.1016/j.mbs.2015.10.013] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/08/2015] [Revised: 10/11/2015] [Accepted: 10/16/2015] [Indexed: 10/22/2022]
Abstract
Game theoretic tools are utilized to analyze a one-locus continuous selection model of sex-specific meiotic drive by considering nonequivalence of the viabilities of reciprocal heterozygotes that might be noticed at an imprinted locus. The model draws attention to the role of viability selections of different types to examine the stable nature of polymorphic equilibrium. A bridge between population genetics and evolutionary game theory has been built up by applying the concept of the Fundamental Theorem of Natural Selection. In addition to pointing out the influences of male and female segregation ratios on selection, configuration structure reveals some noted results, e.g., Hardy-Weinberg frequencies hold in replicator dynamics, occurrence of faster evolution at the maximized variance fitness, existence of mixed Evolutionarily Stable Strategy (ESS) in asymmetric games, the tending evolution to follow not only a 1:1 sex ratio but also a 1:1 different alleles ratio at particular gene locus. Through construction of replicator dynamics in the group selection framework, our selection model introduces a redefining bases of game theory to incorporate non-random mating where a mating parameter associated with population structure is dependent on the social structure. Also, the model exposes the fact that the number of polymorphic equilibria will depend on the algebraic expression of population structure.
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Affiliation(s)
- Bijan Sarkar
- Department of Mathematics, Neotia Institute of Technology, Management and Science, Diamond Harbour Road, 24 Parganas (South), 743368, West Bengal, India.
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30
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Allen B, Nowak MA. Games among relatives revisited. J Theor Biol 2015; 378:103-16. [PMID: 25953388 DOI: 10.1016/j.jtbi.2015.04.031] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2014] [Revised: 03/09/2015] [Accepted: 04/22/2015] [Indexed: 11/28/2022]
Abstract
We present a simple model for the evolution of social behavior in family-structured, finite sized populations. Interactions are represented as evolutionary games describing frequency-dependent selection. Individuals interact more frequently with siblings than with members of the general population, as quantified by an assortment parameter r, which can be interpreted as "relatedness". Other models, mostly of spatially structured populations, have shown that assortment can promote the evolution of cooperation by facilitating interaction between cooperators, but this effect depends on the details of the evolutionary process. For our model, we find that sibling assortment promotes cooperation in stringent social dilemmas such as the Prisoner's Dilemma, but not necessarily in other situations. These results are obtained through straightforward calculations of changes in gene frequency. We also analyze our model using inclusive fitness. We find that the quantity of inclusive fitness does not exist for general games. For special games, where inclusive fitness exists, it provides less information than the straightforward analysis.
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Affiliation(s)
- Benjamin Allen
- Department of Mathematics, Emmanuel College, Boston, MA 02115, United States; Center for Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, United States; Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, United States.
| | - Martin A Nowak
- Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, United States; Department of Mathematics, Harvard University, Cambridge, MA 02138, United States; Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, United States
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31
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Wardil L, Hauert C. Origin and structure of dynamic cooperative networks. Sci Rep 2014; 4:5725. [PMID: 25030202 PMCID: PMC4101522 DOI: 10.1038/srep05725] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2014] [Accepted: 06/26/2014] [Indexed: 11/09/2022] Open
Abstract
Societies are built on social interactions among individuals. Cooperation represents the simplest form of a social interaction: one individual provides a benefit to another one at a cost to itself. Social networks represent a dynamical abstraction of social interactions in a society. The behaviour of an individual towards others and of others towards the individual shape the individual's neighbourhood and hence the local structure of the social network. Here we propose a simple theoretical framework to model dynamic social networks by focussing on each individual's actions instead of interactions between individuals. This eliminates the traditional dichotomy between the strategy of individuals and the structure of the population and easily complements empirical studies. As a consequence, altruists, egoists and fair types are naturally determined by the local social structures, while globally egalitarian networks or stratified structures arise. Cooperative interactions drive the emergence and shape the structure of social networks.
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Affiliation(s)
- Lucas Wardil
- Department of Mathematics, The University of British Columbia 1984 Mathematics Road, Vancouver B.C., Canada V6T 1Z2
| | - Christoph Hauert
- Department of Mathematics, The University of British Columbia 1984 Mathematics Road, Vancouver B.C., Canada V6T 1Z2
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32
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Li A, Wu B, Wang L. Cooperation with both synergistic and local interactions can be worse than each alone. Sci Rep 2014; 4:5536. [PMID: 24985887 PMCID: PMC4078301 DOI: 10.1038/srep05536] [Citation(s) in RCA: 33] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/27/2014] [Accepted: 06/16/2014] [Indexed: 11/25/2022] Open
Abstract
Cooperation is ubiquitous ranging from multicellular organisms to human societies. Population structures indicating individuals' limited interaction ranges are crucial to understand this issue. But it remains unknown to what extend multiple interactions involving nonlinearity in payoff influence the cooperation in structured populations. Here we show a rule, which determines the emergence and stabilization of cooperation, under multiple discounted, linear, and synergistic interactions. The rule is validated by simulations in homogenous and heterogenous structured populations. We find that the more neighbours there are the harder for cooperation to evolve for multiple interactions with linearity and discounting. For synergistic scenario, however, distinct from its pairwise counterpart, moderate number of neighbours can be the worst, indicating that synergistic interactions work with strangers but not with neighbours. Our results suggest that the combination of different factors which promotes cooperation alone can be worse than that with every single factor.
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Affiliation(s)
- Aming Li
- Center for Systems and Control, State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, 100871 Beijing, China
| | - Bin Wu
- Department of Evolutionary Theory, Max-Planck-Institute for Evolutionary Biology, August-Thienemann-Str. 2, 24306 Plön, Germany
| | - Long Wang
- Center for Systems and Control, State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, 100871 Beijing, China
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Masuda N. Evolution via imitation among like-minded individuals. J Theor Biol 2014; 349:100-8. [PMID: 24530826 DOI: 10.1016/j.jtbi.2014.02.003] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/05/2013] [Revised: 02/02/2014] [Accepted: 02/03/2014] [Indexed: 11/16/2022]
Abstract
In social situations with which evolutionary game is concerned, individuals are considered to be heterogeneous in various aspects. In particular, they may differently perceive the same outcome of the game owing to heterogeneity in idiosyncratic preferences, fighting abilities, and positions in a social network. In such a population, an individual may imitate successful and similar others, where similarity refers to that in the idiosyncratic fitness function. I propose an evolutionary game model with two subpopulations on the basis of multipopulation replicator dynamics to describe such a situation. In the proposed model, pairs of players are involved in a two-person game as a well-mixed population, and imitation occurs within subpopulations in each of which players have the same payoff matrix. It is shown that the model does not allow any internal equilibrium such that the dynamics differs from that of other related models such as the bimatrix game. In particular, even a slight difference in the payoff matrix in the two subpopulations can make the opposite strategies to be stably selected in the two subpopulations in the snowdrift and coordination games.
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Affiliation(s)
- Naoki Masuda
- Department of Mathematical Informatics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-8656, Japan.
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Abou Chakra M, Hilbe C, Traulsen A. Plastic behaviors in hosts promote the emergence of retaliatory parasites. Sci Rep 2014; 4:4251. [PMID: 24589512 PMCID: PMC3940972 DOI: 10.1038/srep04251] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2014] [Accepted: 02/13/2014] [Indexed: 11/09/2022] Open
Abstract
Mafia like behavior, where individuals cooperate under the threat of punishment, occurs not only in humans, but is also observed in several animal species. Observations suggest that avian hosts tend to accept a certain degree of parasitism in order to avoid retaliating punishment from the brood parasite. To understand under which conditions it will be beneficial for a host to cooperate, we model the interaction between hosts and parasites as an evolutionary game. In our model, the host's behavior is plastic, and thus, its response depends on the previous interactions with the parasite. We find that such learned behavior in turn is crucial for the evolution of retaliating parasites. The abundance of this kind of mafia behavior oscillates in time and does not settle to an equilibrium. Our results suggest that retaliation is a mechanism for the parasite to evade specialization and to induce acceptance by the host.
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Affiliation(s)
- Maria Abou Chakra
- Evolutionary Theory Group, Max Planck Institute, D-24306 Plön, Germany
| | - Christian Hilbe
- 1] Evolutionary Theory Group, Max Planck Institute, D-24306 Plön, Germany [2] Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, USA
| | - Arne Traulsen
- Evolutionary Theory Group, Max Planck Institute, D-24306 Plön, Germany
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35
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Garcia T, Brunnet LG, De Monte S. Differential adhesion between moving particles as a mechanism for the evolution of social groups. PLoS Comput Biol 2014; 10:e1003482. [PMID: 24586133 PMCID: PMC3937110 DOI: 10.1371/journal.pcbi.1003482] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2013] [Accepted: 12/13/2013] [Indexed: 11/18/2022] Open
Abstract
The evolutionary stability of cooperative traits, that are beneficial to other individuals but costly to their carrier, is considered possible only through the establishment of a sufficient degree of assortment between cooperators. Chimeric microbial populations, characterized by simple interactions between unrelated individuals, restrain the applicability of standard mechanisms generating such assortment, in particular when cells disperse between successive reproductive events such as happens in Dicyostelids and Myxobacteria. In this paper, we address the evolutionary dynamics of a costly trait that enhances attachment to others as well as group cohesion. By modeling cells as self-propelled particles moving on a plane according to local interaction forces and undergoing cycles of aggregation, reproduction and dispersal, we show that blind differential adhesion provides a basis for assortment in the process of group formation. When reproductive performance depends on the social context of players, evolution by natural selection can lead to the success of the social trait, and to the concomitant emergence of sizeable groups. We point out the conditions on the microscopic properties of motion and interaction that make such evolutionary outcome possible, stressing that the advent of sociality by differential adhesion is restricted to specific ecological contexts. Moreover, we show that the aggregation process naturally implies the existence of non-aggregated particles, and highlight their crucial evolutionary role despite being largely neglected in theoretical models for the evolution of sociality. Although pervasive in the living world, collective behavior is a puzzle for evolutionary biology. The genetic traits that sustain it are costly for their carriers and make them vulnerable to the exploitation of asocial “free-riders” that benefit from the group without contributing to its cohesion. This paradox has spawned an extensive literature mainly concerned with elaborate cooperative behaviors that might be inoperant for simple biological entities such as microbes. We model successive life cycles of aggregation, reproduction and dispersal in a biological population combining a statistical physics approach to mimic the group formation process and an evolutionary game theory approach to account for the conflict between individual competition and collective success. Our results show a parsimonious way to the advent of sociality based on differential physical adhesion in organisms deprived of complex cognitive abilities. We also stress the key role of ungrouped individuals and specify the conditions on motion properties that make sociality possible. In detailing a mechanism akin to promote social behavior in microbes in the absence of genealogical relatedness, our work might shed light on both the maintenance of facultative multicellular lifestyles and the evolutionary origins of multicellularity.
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Affiliation(s)
- Thomas Garcia
- CNRS UMR 7625 Écologie et Évolution, École Normale Supérieure, Paris, France
- CNRS UMR 7625 Écologie et Évolution, Université Pierre et Marie Curie-Paris 6, Paris, France
- * E-mail:
| | | | - Silvia De Monte
- CNRS UMR 8197 INSERM U1024, Institut de Biologie de l'École Normale Supérieure (IBENS), Paris, France
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Hauser OP, Traulsen A, Nowak MA. Heterogeneity in background fitness acts as a suppressor of selection. J Theor Biol 2013; 343:178-85. [PMID: 24211522 DOI: 10.1016/j.jtbi.2013.10.013] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2013] [Revised: 09/25/2013] [Accepted: 10/23/2013] [Indexed: 10/26/2022]
Abstract
We introduce the concept of heterogeneity in background fitness to evolutionary dynamics in finite populations. Background fitness is specific to an individual but not linked to its strategy. It can be thought of as a property that is related to the physical or societal position of an individual, but is not dependent on the strategy that is adopted in the evolutionary process under consideration. In our model, an individual's total fitness is the sum of its background fitness and the fitness derived from using a specific strategy. This approach has important implications for the imitation of behavioural strategies: if we imitate others for their success, but can only adopt their behaviour and not their social and economic ties, we may imitate in vain. We study the effect of heterogeneity in background fitness on the fixation of a mutant strategy with constant fitness. We find that heterogeneity suppresses selection, but also decreases the time until a novel strategy either takes over the population or is lost again. We derive analytical solutions of the fixation probability in small populations. In the case of large total background fitness in a population with maximum inequality, we find a particularly simple approximation of the fixation probability. Numerical simulations suggest that this simple approximation also holds for larger population sizes.
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Affiliation(s)
- Oliver P Hauser
- Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, USA
| | - Arne Traulsen
- Evolutionary Theory Group, Max Planck Institute for Evolutionary Biology, August-Thienemann-Str. 2, 24306 Plön, Germany
| | - Martin A Nowak
- Evolutionary Theory Group, Max Planck Institute for Evolutionary Biology, August-Thienemann-Str. 2, 24306 Plön, Germany.
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37
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Tang C, Li X, Cao L, Zhan J. The law of evolutionary dynamics in community-structured population. J Theor Biol 2012; 306:1-6. [DOI: 10.1016/j.jtbi.2012.04.024] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2011] [Revised: 04/16/2012] [Accepted: 04/17/2012] [Indexed: 11/28/2022]
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38
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Szabó G, Szolnoki A. Selfishness, fraternity, and other-regarding preference in spatial evolutionary games. J Theor Biol 2012; 299:81-7. [DOI: 10.1016/j.jtbi.2011.03.015] [Citation(s) in RCA: 46] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2011] [Revised: 03/07/2011] [Accepted: 03/15/2011] [Indexed: 10/18/2022]
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Peña J, Pestelacci E, Berchtold A, Tomassini M. Participation costs can suppress the evolution of upstream reciprocity. J Theor Biol 2011; 273:197-206. [PMID: 21216253 DOI: 10.1016/j.jtbi.2010.12.043] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/25/2010] [Revised: 12/24/2010] [Accepted: 12/30/2010] [Indexed: 11/25/2022]
Abstract
Indirect reciprocity, one of the many mechanisms proposed to explain the evolution of cooperation, is the idea that altruistic actions can be rewarded by third parties. Upstream or generalized reciprocity is one type of indirect reciprocity in which individuals help someone if they have been helped by somebody else in the past. Although empirically found to be at work in humans, the evolution of upstream reciprocity is difficult to explain from a theoretical point of view. A recent model of upstream reciprocity, first proposed by Nowak and Roch (2007) and further analyzed by Iwagami and Masuda (2010), shows that while upstream reciprocity alone does not lead to the evolution of cooperation, it can act in tandem with mechanisms such as network reciprocity and increase the total level of cooperativity in the population. We argue, however, that Nowak and Roch's model systematically leads to non-uniform interaction rates, where more cooperative individuals take part in more games than less cooperative ones. As a result, the critical benefit-to-cost ratios derived under this model in previous studies are not invariant with respect to the addition of participation costs. We show that accounting for these costs can hinder and even suppress the evolution of upstream reciprocity, both for populations with non-random encounters and graph-structured populations.
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Affiliation(s)
- Jorge Peña
- Institut de Mathématiques Appliquées, Université de Lausanne, CH-1015 Lausanne, Switzerland.
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40
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Kun Á, Scheuring I. Evolution of cooperation on dynamical graphs. Biosystems 2009; 96:65-8. [DOI: 10.1016/j.biosystems.2008.11.009] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/27/2008] [Revised: 11/19/2008] [Accepted: 11/20/2008] [Indexed: 11/16/2022]
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41
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Masuda N. Oscillatory dynamics in evolutionary games are suppressed by heterogeneous adaptation rates of players. J Theor Biol 2008; 251:181-9. [DOI: 10.1016/j.jtbi.2007.11.010] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/14/2007] [Revised: 11/12/2007] [Accepted: 11/12/2007] [Indexed: 11/16/2022]
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42
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Zhang PA, Nie PY, Hu DQ, Zou FY. The analysis of bi-level evolutionary graphs. Biosystems 2007; 90:897-902. [PMID: 17640797 DOI: 10.1016/j.biosystems.2007.05.008] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2006] [Revised: 05/31/2007] [Accepted: 05/31/2007] [Indexed: 12/01/2022]
Abstract
Evolutionary graphs (EGs), in which evolutionary dynamic is arranged on a graph, were initially proposed by Lieberman et al. [Lieberman, E., Hauert, C., Nowak, M.A., 2005. Evolutionary dynamics on graphs. Nature 433, 312-316] in the biological field and many biological phenomena are successfully explained. EGs on two levels (or bi-level EGs), based on some biological phenomena, are considered in this paper. The bi-level EGs are compared with the one-rooted EGs in two cases. One has the identical numbers of the followers, the other with the same numbers of total individuals. Then, some properties of the bi-level EGs are obtained. It is showed that bi-level EGs are more stable, and the bi-level EGs with just two leaders are the most stable, if they have identical followers respectively. The bi-level EGs theory can successfully explain the phenomena of symbiosis in biology.
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Affiliation(s)
- Pei-ai Zhang
- Department of Mathematics, Jinan University, Guangzhou 510632, PR China
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43
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Masuda N. Participation costs dismiss the advantage of heterogeneous networks in evolution of cooperation. Proc Biol Sci 2007; 274:1815-21. [PMID: 17504741 PMCID: PMC2270926 DOI: 10.1098/rspb.2007.0294] [Citation(s) in RCA: 152] [Impact Index Per Article: 8.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Real social interactions occur on networks in which each individual is connected to some, but not all, of others. In social dilemma games with a fixed population size, heterogeneity in the number of contacts per player is known to promote evolution of cooperation. Under a common assumption of positively biased pay-off structure, well-connected players earn much by playing frequently, and cooperation once adopted by well-connected players is unbeatable and spreads to others. However, maintaining a social contact can be costly, which would prevent local pay-offs from being positively biased. In replicator-type evolutionary dynamics, it is shown that even a relatively small participation cost extinguishes the merit of heterogeneous networks in terms of cooperation. In this situation, more connected players earn less so that they are no longer spreaders of cooperation. Instead, those with fewer contacts win and guide the evolution. The participation cost, or the baseline pay-off, is irrelevant in homogeneous populations, but is essential for evolutionary games on heterogeneous networks.
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Affiliation(s)
- Naoki Masuda
- Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-8656, Japan.
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44
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Lessard S, Ladret V. The probability of fixation of a single mutant in an exchangeable selection model. J Math Biol 2007; 54:721-44. [PMID: 17252282 DOI: 10.1007/s00285-007-0069-7] [Citation(s) in RCA: 74] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/13/2006] [Revised: 11/30/2006] [Indexed: 11/26/2022]
Abstract
The Cannings exchangeable model for a finite population in discrete time is extended to incorporate selection. The probability of fixation of a mutant type is studied under the assumption of weak selection. An exact formula for the derivative of this probability with respect to the intensity of selection is deduced, and developed in the case of a single mutant. This formula is expressed in terms of mean coalescence times under neutrality assuming that the coefficient of selection for the mutant type has a derivative with respect to the intensity of selection that takes a polynomial form with respect to the frequency of the mutant type. An approximation is obtained in the case where this derivative is a continuous function of the mutant frequency and the population size is large. This approximation is consistent with a diffusion approximation under moment conditions on the number of descendants of a single individual in one time step. Applications to evolutionary game theory in finite populations are presented.
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Affiliation(s)
- Sabin Lessard
- Département de mathématiques et de statistique, Université de Montréal, Montréal, QC H3C 3J7, Canada.
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45
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Pacheco JM, Traulsen A, Nowak MA. Coevolution of strategy and structure in complex networks with dynamical linking. PHYSICAL REVIEW LETTERS 2006; 97:258103. [PMID: 17280398 PMCID: PMC2430061 DOI: 10.1103/physrevlett.97.258103] [Citation(s) in RCA: 240] [Impact Index Per Article: 13.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/09/2006] [Indexed: 05/13/2023]
Abstract
We introduce a model in which individuals differ in the rate at which they seek new interactions with others, making rational decisions modeled as general symmetric two-player games. Once a link between two individuals has formed, the productivity of this link is evaluated. Links can be broken off at different rates. We provide analytic results for the limiting cases where linking dynamics is much faster than evolutionary dynamics and vice versa, and show how the individual capacity of forming new links or severing inconvenient ones maps into the problem of strategy evolution in a well-mixed population under a different game. For intermediate ranges, we investigate numerically the detailed interplay determined by these two time scales and show that the scope of validity of the analytical results extends to a much wider ratio of time scales than expected.
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Affiliation(s)
- Jorge M Pacheco
- Program for Evolutionary Dynamics, Harvard University, Cambridge, Massachusetts 02138, USA
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46
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Active linking in evolutionary games. J Theor Biol 2006; 243:437-43. [PMID: 16901509 DOI: 10.1016/j.jtbi.2006.06.027] [Citation(s) in RCA: 114] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2006] [Revised: 06/23/2006] [Accepted: 06/26/2006] [Indexed: 11/15/2022]
Abstract
In the traditional approach to evolutionary game theory, the individuals of a population meet each other at random, and they have no control over the frequency or duration of interactions. Here we remove these simplifying assumptions. We introduce a new model, where individuals differ in the rate at which they seek new interactions. Once a link between two individuals has formed, the productivity of this link is evaluated. Links can be broken off at different rates. In a limiting case, the linking dynamics introduces a simple transformation of the payoff matrix. We outline conditions for evolutionary stability. As a specific example, we study the interaction between cooperators and defectors. We find a simple relationship that characterizes those linking dynamics which allow natural selection to favour cooperation over defection.
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