1
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Wang G, Su Q, Wang L, Plotkin JB. The evolution of social behaviors and risk preferences in settings with uncertainty. Proc Natl Acad Sci U S A 2024; 121:e2406993121. [PMID: 39018189 PMCID: PMC11287271 DOI: 10.1073/pnas.2406993121] [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: 04/08/2024] [Accepted: 06/13/2024] [Indexed: 07/19/2024] Open
Abstract
Humans update their social behavior in response to past experiences and changing environments. Behavioral decisions are further complicated by uncertainty in the outcome of social interactions. Faced with uncertainty, some individuals exhibit risk aversion while others seek risk. Attitudes toward risk may depend on socioeconomic status; and individuals may update their risk preferences over time, which will feedback on their social behavior. Here, we study how uncertainty and risk preferences shape the evolution of social behaviors. We extend the game-theoretic framework for behavioral evolution to incorporate uncertainty about payoffs and variation in how individuals respond to this uncertainty. We find that different attitudes toward risk can substantially alter behavior and long-term outcomes, as individuals seek to optimize their rewards from social interactions. In a standard setting without risk, for example, defection always overtakes a well-mixed population engaged in the classic Prisoner's Dilemma, whereas risk aversion can reverse the direction of evolution, promoting cooperation over defection. When individuals update their risk preferences along with their strategic behaviors, a population can oscillate between periods dominated by risk-averse cooperators and periods of risk-seeking defectors. Our analysis provides a systematic account of how risk preferences modulate, and even coevolve with, behavior in an uncertain social world.
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Affiliation(s)
- Guocheng Wang
- Center for Systems and Control, College of Engineering, Peking University, Beijing100871, China
- Department of Biology, University of Pennsylvania, Philadelphia, PA19104
| | - Qi Su
- Department of Automation, Shanghai Jiao Tong University, Shanghai200240, China
- Ministry of Education of China, Key Laboratory of System Control and Information Processing, Shanghai200240, China
- Shanghai Engineering Research Center of Intelligent Control and Management, Shanghai200240, China
| | - Long Wang
- Center for Systems and Control, College of Engineering, Peking University, Beijing100871, China
- Center for Multi-Agent Research, Institute for Artificial Intelligence, Peking University, Beijing100871, China
| | - Joshua B. Plotkin
- Department of Biology, University of Pennsylvania, Philadelphia, PA19104
- Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA19014
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2
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Wang C, Perc M, Szolnoki A. Evolutionary dynamics of any multiplayer game on regular graphs. Nat Commun 2024; 15:5349. [PMID: 38914550 PMCID: PMC11196707 DOI: 10.1038/s41467-024-49505-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/23/2023] [Accepted: 06/05/2024] [Indexed: 06/26/2024] Open
Abstract
Multiplayer games on graphs are at the heart of theoretical descriptions of key evolutionary processes that govern vital social and natural systems. However, a comprehensive theoretical framework for solving multiplayer games with an arbitrary number of strategies on graphs is still missing. Here, we solve this by drawing an analogy with the Balls-and-Boxes problem, based on which we show that the local configuration of multiplayer games on graphs is equivalent to distributing k identical co-players among n distinct strategies. We use this to derive the replicator equation for any n-strategy multiplayer game under weak selection, which can be solved in polynomial time. As an example, we revisit the second-order free-riding problem, where costly punishment cannot truly resolve social dilemmas in a well-mixed population. Yet, in structured populations, we derive an accurate threshold for the punishment strength, beyond which punishment can either lead to the extinction of defection or transform the system into a rock-paper-scissors-like cycle. The analytical solution also qualitatively agrees with the phase diagrams that were previously obtained for non-marginal selection strengths. Our framework thus allows an exploration of any multi-strategy multiplayer game on regular graphs.
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Affiliation(s)
- Chaoqian Wang
- Department of Computational and Data Sciences, George Mason University, Fairfax, VA, 22030, USA.
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000, Maribor, Slovenia
- Community Healthcare Center Dr. Adolf Drolc Maribor, Vošnjakova ulica 2, 2000, Maribor, Slovenia
- Complexity Science Hub Vienna, Josefstädterstraße 39, 1080, Vienna, Austria
- Department of Physics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, Republic of Korea
| | - Attila Szolnoki
- Institute of Technical Physics and Materials Science, Centre for Energy Research, P.O. Box 49, H-1525, Budapest, Hungary
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3
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Kuo YP, Carja O. Evolutionary graph theory beyond single mutation dynamics: on how network-structured populations cross fitness landscapes. Genetics 2024; 227:iyae055. [PMID: 38639307 PMCID: PMC11151934 DOI: 10.1093/genetics/iyae055] [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: 02/07/2024] [Revised: 03/28/2024] [Accepted: 04/01/2024] [Indexed: 04/20/2024] Open
Abstract
Spatially resolved datasets are revolutionizing knowledge in molecular biology, yet are under-utilized for questions in evolutionary biology. To gain insight from these large-scale datasets of spatial organization, we need mathematical representations and modeling techniques that can both capture their complexity, but also allow for mathematical tractability. Evolutionary graph theory utilizes the mathematical representation of networks as a proxy for heterogeneous population structure and has started to reshape our understanding of how spatial structure can direct evolutionary dynamics. However, previous results are derived for the case of a single new mutation appearing in the population and the role of network structure in shaping fitness landscape crossing is still poorly understood. Here we study how network-structured populations cross fitness landscapes and show that even a simple extension to a two-mutational landscape can exhibit complex evolutionary dynamics that cannot be predicted using previous single-mutation results. We show how our results can be intuitively understood through the lens of how the two main evolutionary properties of a network, the amplification and acceleration factors, change the expected fate of the intermediate mutant in the population and further discuss how to link these models to spatially resolved datasets of cellular organization.
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Affiliation(s)
- Yang Ping Kuo
- Computational Biology Department, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15232, USA
| | - Oana Carja
- Computational Biology Department, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15232, USA
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4
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Johnson T. Empirically testing a relationship between cooperation and the prime numbers. ROYAL SOCIETY OPEN SCIENCE 2024; 11:231425. [PMID: 39100144 PMCID: PMC11295909 DOI: 10.1098/rsos.231425] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/20/2023] [Revised: 03/03/2024] [Accepted: 04/08/2024] [Indexed: 08/06/2024]
Abstract
Theoretical models suggest a relationship between cooperation and the prime numbers. In environments where agents play multiple one-shot prisoner's dilemma games per generation, cooperators evolve to fixation more frequently when cooperating on a cyclical schedule with a prime-number period length. This finding parrots classic predator-prey models showing selection for prime-number prey life cycles. Here, I report an empirical test of the former models using previously published data concerning humans playing one-shot public goods games across multiple time points-i.e. an analogue to multiple one-shot prisoner's dilemma games. I find very modest evidence of cyclicality at prime-numbered time intervals, though results indicate rough agreement between theoretical predictions and observed rates of full cooperation across time points. Analyses of individual decisions find increased contributions to the public good at prime-number time points and separate placebo tests indicate a 4-in-1000 chance of spuriously estimating this effect. However, when exploratory analyses exclude low-value prime-numbered time points, the magnitude of the estimated effect decreases and the hypothesis of no effect cannot be rejected, implying that low-value, prime-number time points drive estimates, contrary to theoretical model predictions. These findings cast doubt on the hypothesis of increased cooperation at prime-number time points-at least among humans playing public goods games.
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Affiliation(s)
- Tim Johnson
- Atkinson School of Management, Willamette University, Salem, OR, 97301, USA
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5
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Dragicevic AZ. The Unification of Evolutionary Dynamics through the Bayesian Decay Factor in a Game on a Graph. Bull Math Biol 2024; 86:69. [PMID: 38714590 DOI: 10.1007/s11538-024-01299-9] [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: 03/07/2024] [Accepted: 04/18/2024] [Indexed: 05/10/2024]
Abstract
We unify evolutionary dynamics on graphs in strategic uncertainty through a decaying Bayesian update. Our analysis focuses on the Price theorem of selection, which governs replicator(-mutator) dynamics, based on a stratified interaction mechanism and a composite strategy update rule. Our findings suggest that the replication of a certain mutation in a strategy, leading to a shift from competition to cooperation in a well-mixed population, is equivalent to the replication of a strategy in a Bayesian-structured population without any mutation. Likewise, the replication of a strategy in a Bayesian-structured population with a certain mutation, resulting in a move from competition to cooperation, is equivalent to the replication of a strategy in a well-mixed population without any mutation. This equivalence holds when the transition rate from competition to cooperation is equal to the relative strength of selection acting on either competition or cooperation in relation to the selection differential between cooperators and competitors. Our research allows us to identify situations where cooperation is more likely, irrespective of the specific payoff levels. This approach provides new perspectives into the intended purpose of Price's equation, which was initially not designed for this type of analysis.
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Affiliation(s)
- Arnaud Zlatko Dragicevic
- Faculty of Economics, Chulalongkorn University, Bangkok, Thailand.
- Sustainable Development, CIRANO, Montréal, Canada.
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6
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Wang X, Fu F, Wang L. Deterministic theory of evolutionary games on temporal networks. J R Soc Interface 2024; 21:20240055. [PMID: 38807526 DOI: 10.1098/rsif.2024.0055] [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: 01/24/2024] [Accepted: 03/28/2024] [Indexed: 05/30/2024] Open
Abstract
Recent empirical studies have revealed that social interactions among agents in realistic networks merely exist intermittently and occur in a particular sequential order. However, it remains unexplored how to theoretically describe evolutionary dynamics of multiple strategies on temporal networks. Herein, we develop a deterministic theory for studying evolutionary dynamics of any [Formula: see text] pairwise games in structured populations where individuals are connected and organized by temporally activated edges. In the limit of weak selection, we derive replicator-like equations with a transformed payoff matrix characterizing how the mean frequency of each strategy varies over time, and then obtain critical conditions for any strategy to be evolutionarily stable on temporal networks. Interestingly, the re-scaled payoff matrix is a linear combination of the original payoff matrix with an additional one describing local competitions between any pair of different strategies, whose weights are solely determined by network topology and selection intensity. As a particular example, we apply the deterministic theory to analysing the impacts of temporal networks in the mini-ultimatum game, and find that temporally networked population structures result in the emergence of fairness. Our work offers theoretical insights into the subtle effects of network temporality on evolutionary game dynamics.
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Affiliation(s)
- Xiaofeng Wang
- Department of Automation, School of Information Science and Technology, Donghua University , Shanghai 201620, People's Republic of China
- Engineering Research Center of Digitized Textile and Apparel Technology (Ministry of Education), Donghua University , Shanghai 201620, People's Republic of China
| | - Feng Fu
- Department of Mathematics, Dartmouth College , Hanover, NH 03755, USA
- Department of Biomedical Data Science, Geisel School of Medicine at Dartmouth , Lebanon, NH 03756, USA
| | - Long Wang
- Center for Systems and Control, College of Engineering, Peking University , Beijing 100871, People's Republic of China
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7
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Chiba-Okabe H, Plotkin JB. Can institutions foster cooperation by wealth redistribution? J R Soc Interface 2024; 21:20230698. [PMID: 38471530 PMCID: PMC10932717 DOI: 10.1098/rsif.2023.0698] [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: 11/27/2023] [Accepted: 02/06/2024] [Indexed: 03/14/2024] Open
Abstract
Theoretical models prescribe how institutions can promote cooperation in a population by imposing appropriate punishments or rewards on individuals. However, many real-world institutions are not sophisticated or responsive enough to ensure cooperation by calibrating their policies. Or, worse yet, an institution might selfishly exploit the population it governs for its own benefit. Here, we study the evolution of cooperation in the presence of an institution that is autonomous, in the sense that it has its own interests that may or may not align with those of the population. The institution imposes a tax on the population and redistributes a portion of the tax revenue to cooperators, withholding the remaining revenue for itself. The institution adjusts its rates of taxation and redistribution to optimize its own long-term, discounted utility. We consider three types of institutions with different goals, embodied in their utility functions. We show that a prosocial institution, whose goal is to maximize the average payoff of the population, can indeed promote cooperation-but only if it is sufficiently forward-looking. On the other hand, an institution that seeks to maximize welfare among cooperators alone will successfully promote collective cooperation even if it is myopic. Remarkably, even a selfish institution, which seeks to maximize the revenue it withholds for itself, can nonetheless promote cooperation. The average payoff of the population increases when a selfish institution is more forward-looking, so that a population under a selfish regime can sometimes fare better than under anarchy. Our analysis highlights the potential benefits of institutional wealth redistribution, even when an institution does not share the interests of the population it governs.
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Affiliation(s)
- Hiroaki Chiba-Okabe
- Program in Applied Mathematics and Computational Science, University of Pennsylvania, Philadelphia, PA, USA
- Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA, USA
| | - Joshua B. Plotkin
- Program in Applied Mathematics and Computational Science, University of Pennsylvania, Philadelphia, PA, USA
- Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA, USA
- Department of Biology, University of Pennsylvania, Philadelphia, PA, USA
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8
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Aguilar-Janita M, Khalil N, Leyva I, Sendiña-Nadal I. Cooperation transitions in social games induced by aspiration-driven players. Phys Rev E 2024; 109:024107. [PMID: 38491644 DOI: 10.1103/physreve.109.024107] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2023] [Accepted: 01/16/2024] [Indexed: 03/18/2024]
Abstract
Cooperation and defection are social traits whose evolutionary origin is still unresolved. Recent behavioral experiments with humans suggested that strategy changes are driven mainly by the individuals' expectations and not by imitation. This work theoretically analyzes and numerically explores an aspiration-driven strategy updating in a well-mixed population playing games. The payoffs of the game matrix and the aspiration are condensed into just two parameters that allow a comprehensive description of the dynamics. We find continuous and abrupt transitions in the cooperation density with excellent agreement between theory and the Gillespie simulations. Under strong selection, the system can display several levels of steady cooperation or get trapped into absorbing states. These states are still relevant for experiments even when irrational choices are made due to their prolonged relaxation times. Finally, we show that for the particular case of the prisoner dilemma, where defection is the dominant strategy under imitation mechanisms, the self-evaluation update instead favors cooperation nonlinearly with the level of aspiration. Thus, our work provides insights into the distinct role between imitation and self-evaluation with no learning dynamics.
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Affiliation(s)
- M Aguilar-Janita
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Spain
| | - N Khalil
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Spain
| | - I Leyva
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, Pozuelo de Alarcón, 28223 Madrid, Spain
| | - I Sendiña-Nadal
- Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, Pozuelo de Alarcón, 28223 Madrid, Spain
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9
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Zhao C, Zhu Y. Heterogeneous decision-making dynamics of threshold-switching agents on complex networks. CHAOS (WOODBURY, N.Y.) 2023; 33:123133. [PMID: 38149990 DOI: 10.1063/5.0172442] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2023] [Accepted: 11/21/2023] [Indexed: 12/28/2023]
Abstract
In the classical two-player decision-making scenario, individuals may have different tendencies to take a certain action, given that there exists a sufficient number of neighbors adopting a particular option. This is ubiquitous in many real-life contexts including traffic congestion, crowd evacuation, and minimal vertex cover problem. Under best-response dynamics, we investigate the decision-making behaviors of heterogeneous agents on complex networks. Results of the networked games are twofold: for networks of uniform degree distribution (e.g., the lattice) and fraction of the strategy is of a linear function of the threshold setting. Moreover, the equilibrium analysis is provided and the relationship between the equilibrium dynamics and the change of the threshold value is given quantitatively. Next, if the games are played on networks with non-uniform degree distribution (e.g., random regular and scale-free networks), influence of the threshold-switching will be weakened. Robust experiments indicate that it is not the value of the average degree, but the degree distribution that influences how the strategy evolves affected by the threshold settings. Our result shows that the decision-making behaviors can be effectively manipulated by tuning the parameters in the utility function (i.e., thresholds) of some agents for more regular network structures.
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Affiliation(s)
- Chengli Zhao
- College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China
| | - Yuying Zhu
- School of Artificial Intelligence, Tiangong University, Tianjin 300387, China
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10
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Wang X, Zhou L, McAvoy A, Li A. Imitation dynamics on networks with incomplete information. Nat Commun 2023; 14:7453. [PMID: 37978181 PMCID: PMC10656501 DOI: 10.1038/s41467-023-43048-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2023] [Accepted: 10/30/2023] [Indexed: 11/19/2023] Open
Abstract
Imitation is an important learning heuristic in animal and human societies. Previous explorations report that the fate of individuals with cooperative strategies is sensitive to the protocol of imitation, leading to a conundrum about how different styles of imitation quantitatively impact the evolution of cooperation. Here, we take a different perspective on the personal and external social information required by imitation. We develop a general model of imitation dynamics with incomplete information in networked systems, which unifies classical update rules including the death-birth and pairwise-comparison rule on complex networks. Under pairwise interactions, we find that collective cooperation is most promoted if individuals neglect personal information. If personal information is considered, cooperators evolve more readily with more external information. Intriguingly, when interactions take place in groups on networks with low degrees of clustering, using more personal and less external information better facilitates cooperation. Our unifying perspective uncovers intuition by examining the rate and range of competition induced by different information situations.
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Affiliation(s)
- Xiaochen Wang
- Center for Systems and Control, College of Engineering, Peking University, Beijing, 100871, China
| | - Lei Zhou
- School of Automation, Beijing Institute of Technology, Beijing, 100081, China
| | - Alex McAvoy
- School of Data Science and Society, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA
- Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA
| | - Aming Li
- Center for Systems and Control, College of Engineering, Peking University, Beijing, 100871, China.
- Center for Multi-Agent Research, Institute for Artificial Intelligence, Peking University, Beijing, 100871, China.
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11
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Raducha T, San Miguel M. Evolutionary games on multilayer networks: coordination and equilibrium selection. Sci Rep 2023; 13:11818. [PMID: 37479729 PMCID: PMC10362047 DOI: 10.1038/s41598-023-38589-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/21/2023] [Accepted: 07/11/2023] [Indexed: 07/23/2023] Open
Abstract
We study mechanisms of synchronisation, coordination, and equilibrium selection in two-player coordination games on multilayer networks. We investigate three possible update rules: the replicator dynamics (RD), the best response (BR), and the unconditional imitation (UI). Players interact on a two-layer random regular network. The population on each layer plays a different game, with layer I preferring the opposite strategy to layer II. We measure the difference between the two games played on the layers by a difference in payoffs, and the inter-connectedness by a node overlap parameter. We discover a critical value of the overlap below which layers do not synchronise, i.e. they display different levels of coordination. Above this threshold both layers typically coordinate on the same strategy. Surprisingly, there is a symmetry breaking in the selection of equilibrium-for RD and UI there is a phase where only the payoff-dominant equilibrium is selected. It is not observed, however, for BR update rule. Our work is an example of previously observed differences between the update rules. Nonetheless, we took a novel approach with the game being played on two inter-connected layers. As we show, the multilayer structure enhances the abundance of the Pareto-optimal equilibrium in coordination games with imitative update rules.
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Affiliation(s)
- Tomasz Raducha
- Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Spain.
- Institute for Cross-Disciplinary Physics and Complex Systems, IFISC (CSIC-UIB), Palma, Spain.
| | - Maxi San Miguel
- Institute for Cross-Disciplinary Physics and Complex Systems, IFISC (CSIC-UIB), Palma, Spain
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12
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Kessinger TA, Tarnita CE, Plotkin JB. Evolution of norms for judging social behavior. Proc Natl Acad Sci U S A 2023; 120:e2219480120. [PMID: 37276388 PMCID: PMC10268218 DOI: 10.1073/pnas.2219480120] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2022] [Accepted: 05/03/2023] [Indexed: 06/07/2023] Open
Abstract
Reputations provide a powerful mechanism to sustain cooperation, as individuals cooperate with those of good social standing. But how should someone's reputation be updated as we observe their social behavior, and when will a population converge on a shared norm for judging behavior? Here, we develop a mathematical model of cooperation conditioned on reputations, for a population that is stratified into groups. Each group may subscribe to a different social norm for assessing reputations and so norms compete as individuals choose to move from one group to another. We show that a group initially comprising a minority of the population may nonetheless overtake the entire population-especially if it adopts the Stern Judging norm, which assigns a bad reputation to individuals who cooperate with those of bad standing. When individuals do not change group membership, stratifying reputation information into groups tends to destabilize cooperation, unless individuals are strongly insular and favor in-group social interactions. We discuss the implications of our results for the structure of information flow in a population and for the evolution of social norms of judgment.
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Affiliation(s)
| | - Corina E. Tarnita
- Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ08544
| | - Joshua B. Plotkin
- Department of Biology, University of Pennsylvania, Philadelphia, PA19104
- Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA19104
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13
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Bhattacharjee S, Dubey VK, Mukhopadhyay A, Chakraborty S. Periodic orbits in deterministic discrete-time evolutionary game dynamics: An information-theoretic perspective. Phys Rev E 2023; 107:064405. [PMID: 37464624 DOI: 10.1103/physreve.107.064405] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2022] [Accepted: 05/30/2023] [Indexed: 07/20/2023]
Abstract
Even though the existence of nonconvergent evolution of the states of populations in ecological and evolutionary contexts is an undeniable fact, insightful game-theoretic interpretations of such outcomes are scarce in the literature of evolutionary game theory. As a proof-of-concept, we tap into the information-theoretic concept of relative entropy in order to construct a game-theoretic interpretation for periodic orbits in a wide class of deterministic discrete-time evolutionary game dynamics, primarily investigating the two-player two-strategy case. Effectively, we present a consistent generalization of the evolutionarily stable strategy-the cornerstone of the evolutionary game theory-and aptly term the generalized concept "information stable orbit." The information stable orbit captures the essence of the evolutionarily stable strategy in that it compares the total payoff obtained against an evolving mutant with the total payoff that the mutant gets while playing against itself. Furthermore, we discuss the connection of the information stable orbit with the dynamical stability of the corresponding periodic orbit.
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Affiliation(s)
- Sayak Bhattacharjee
- Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India
| | - Vikash Kumar Dubey
- Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India
| | - Archan Mukhopadhyay
- School of Computer Science, University of Birmingham, Birmingham B15 2TT, United Kingdom
| | - Sagar Chakraborty
- Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India
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14
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Cooney DB, Levin SA, Mori Y, Plotkin JB. Evolutionary dynamics within and among competing groups. Proc Natl Acad Sci U S A 2023; 120:e2216186120. [PMID: 37155901 PMCID: PMC10193939 DOI: 10.1073/pnas.2216186120] [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: 09/22/2022] [Accepted: 03/22/2023] [Indexed: 05/10/2023] Open
Abstract
Biological and social systems are structured at multiple scales, and the incentives of individuals who interact in a group may diverge from the collective incentive of the group as a whole. Mechanisms to resolve this tension are responsible for profound transitions in evolutionary history, including the origin of cellular life, multicellular life, and even societies. Here, we synthesize a growing literature that extends evolutionary game theory to describe multilevel evolutionary dynamics, using nested birth-death processes and partial differential equations to model natural selection acting on competition within and among groups of individuals. We analyze how mechanisms known to promote cooperation within a single group-including assortment, reciprocity, and population structure-alter evolutionary outcomes in the presence of competition among groups. We find that population structures most conducive to cooperation in multiscale systems can differ from those most conducive within a single group. Likewise, for competitive interactions with a continuous range of strategies we find that among-group selection may fail to produce socially optimal outcomes, but it can nonetheless produce second-best solutions that balance individual incentives to defect with the collective incentives for cooperation. We conclude by describing the broad applicability of multiscale evolutionary models to problems ranging from the production of diffusible metabolites in microbes to the management of common-pool resources in human societies.
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Affiliation(s)
- Daniel B Cooney
- Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104
- Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA 19104
| | - Simon A Levin
- Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544
| | - Yoichiro Mori
- Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104
- Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA 19104
- Department of Biology, University of Pennsylvania, Philadelphia, PA 19104
| | - Joshua B Plotkin
- Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104
- Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA 19104
- Department of Biology, University of Pennsylvania, Philadelphia, PA 19104
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15
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Li JY, Wu WH, Li ZZ, Wang WX, Zhang B. Data-driven evolutionary game models for the spread of fairness and cooperation in heterogeneous networks. Front Psychiatry 2023; 14:1131769. [PMID: 37229392 PMCID: PMC10204145 DOI: 10.3389/fpsyt.2023.1131769] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 12/26/2022] [Accepted: 03/10/2023] [Indexed: 05/27/2023] Open
Abstract
Unique large-scale cooperation and fairness norms are essential to human society, but the emergence of prosocial behaviors is elusive. The fact that heterogeneous social networks prevail raised a hypothesis that heterogeneous networks facilitate fairness and cooperation. However, the hypothesis has not been validated experimentally, and little is known about the evolutionary psychological basis of cooperation and fairness in human networks. Fortunately, research about oxytocin, a neuropeptide, may provide novel ideas for confirming the hypothesis. Recent oxytocin-modulated network game experiments observed that intranasal administration of oxytocin to a few central individuals significantly increases global fairness and cooperation. Here, based on the experimental phenomena and data, we show a joint effect of social preference and network heterogeneity on promoting prosocial behaviors by building evolutionary game models. In the network ultimatum game and the prisoner's dilemma game with punishment, inequality aversion can lead to the spread of costly punishment for selfish and unfair behaviors. This effect is initiated by oxytocin, then amplified via influential nodes, and finally promotes global cooperation and fairness. In contrast, in the network trust game, oxytocin increases trust and altruism, but these effects are confined locally. These results uncover general oxytocin-initiated mechanisms underpinning fairness and cooperation in human networks.
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Affiliation(s)
- Jing-Yi Li
- School of Systems Science, Beijing Normal University, Beijing, China
- CSSC Intelligent Innovation Research Institute, Beijing, China
- CSSC System Engineering Research Institute, Beijing, China
| | - Wen-Hao Wu
- School of Systems Science, Beijing Normal University, Beijing, China
| | - Ze-Zheng Li
- School of Systems Science, Beijing Normal University, Beijing, China
| | - Wen-Xu Wang
- School of Systems Science, Beijing Normal University, Beijing, China
- Chinese Institute for Brain Research, Beijing, China
| | - Boyu Zhang
- Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, China
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16
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Yagoobi S, Sharma N, Traulsen A. Categorizing update mechanisms for graph-structured metapopulations. J R Soc Interface 2023; 20:20220769. [PMID: 36919418 PMCID: PMC10015335 DOI: 10.1098/rsif.2022.0769] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/16/2023] Open
Abstract
The structure of a population strongly influences its evolutionary dynamics. In various settings ranging from biology to social systems, individuals tend to interact more often with those present in their proximity and rarely with those far away. A common approach to model the structure of a population is evolutionary graph theory. In this framework, each graph node is occupied by a reproducing individual. The links connect these individuals to their neighbours. The offspring can be placed on neighbouring nodes, replacing the neighbours-or the progeny of its neighbours can replace a node during the course of ongoing evolutionary dynamics. Extending this theory by replacing single individuals with subpopulations at nodes yields a graph-structured metapopulation. The dynamics between the different local subpopulations is set by an update mechanism. There are many such update mechanisms. Here, we classify update mechanisms for structured metapopulations, which allows to find commonalities between past work and illustrate directions for further research and current gaps of investigation.
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Affiliation(s)
- Sedigheh Yagoobi
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, August-Thienemann Strasse 2, Plön 24306, Germany
| | - Nikhil Sharma
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, August-Thienemann Strasse 2, Plön 24306, Germany
| | - Arne Traulsen
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, August-Thienemann Strasse 2, Plön 24306, Germany
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17
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Wang S, Chen X, Xiao Z, Szolnoki A, Vasconcelos VV. Optimization of institutional incentives for cooperation in structured populations. J R Soc Interface 2023; 20:20220653. [PMID: 36722070 PMCID: PMC9890111 DOI: 10.1098/rsif.2022.0653] [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: 09/05/2022] [Accepted: 01/03/2023] [Indexed: 02/02/2023] Open
Abstract
The application of incentives, such as reward and punishment, is a frequently applied way for promoting cooperation among interacting individuals in structured populations. However, how to properly use the incentives is still a challenging problem for incentive-providing institutions. In particular, since the implementation of incentive is costly, to explore the optimal incentive protocol, which ensures the desired collective goal at a minimal cost, is worthy of study. In this work, we consider the positive and negative incentives for a structured population of individuals whose conflicting interactions are characterized by a Prisoner's Dilemma game. We establish an index function for quantifying the cumulative cost during the process of incentive implementation, and theoretically derive the optimal positive and negative incentive protocols for cooperation on regular networks. We find that both types of optimal incentive protocols are identical and time-invariant. Moreover, we compare the optimal rewarding and punishing schemes concerning implementation cost and provide a rigorous basis for the usage of incentives in the game-theoretical framework. We further perform computer simulations to support our theoretical results and explore their robustness for different types of population structures, including regular, random, small-world and scale-free networks.
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Affiliation(s)
- Shengxian Wang
- School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, People’s Republic of China
- Faculty of Science and Engineering, University of Groningen, Groningen 9747 AG, The Netherlands
| | - Xiaojie Chen
- School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, People’s Republic of China
| | - Zhilong Xiao
- School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, People’s Republic of China
- School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou 510006, People’s Republic of China
| | - Attila Szolnoki
- Institute of Technical Physics and Materials Science, Centre for Energy Research, P.O. Box 49, Budapest 1525, Hungary
| | - Vítor V. Vasconcelos
- Computational Science Lab, Informatics Institute, University of Amsterdam, Amsterdam 1098XH, The Netherlands
- Institute for Advanced Study, University of Amsterdam, Amsterdam 1012 GC, The Netherlands
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18
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Wang C, Szolnoki A. Evolution of cooperation under a generalized death-birth process. Phys Rev E 2023; 107:024303. [PMID: 36932485 DOI: 10.1103/physreve.107.024303] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/12/2022] [Accepted: 01/24/2023] [Indexed: 02/09/2023]
Abstract
According to the evolutionary death-birth protocol, a player is chosen randomly to die and neighbors compete for the available position proportional to their fitness. Hence, the status of the focal player is completely ignored and has no impact on the strategy update. In this paper, we revisit and generalize this rule by introducing a weight factor to compare the payoff values of the focal and invading neighbors. By means of evolutionary graph theory, we analyze the model on joint transitive graphs to explore the possible consequences of the presence of a weight factor. We find that focal weight always hinders cooperation under weak selection strength. Surprisingly, the results show a nontrivial tipping point of the weight factor where the threshold of cooperation success shifts from positive to negative infinity. Once focal weight exceeds this tipping point, cooperation becomes unreachable. Our theoretical predictions are confirmed by Monte Carlo simulations on a square lattice of different sizes. We also verify the robustness of the conclusions to arbitrary two-player prisoner's dilemmas, to dispersal graphs with arbitrary edge weights, and to interaction and dispersal graphs overlapping arbitrarily.
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Affiliation(s)
- Chaoqian Wang
- Department of Computational and Data Sciences, George Mason University, Fairfax, Virginia 22030, USA
| | - Attila Szolnoki
- Institute of Technical Physics and Materials Science, Centre for Energy Research, P.O. Box 49, H-1525 Budapest, Hungary
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19
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Universal scaling of extinction time in stochastic evolutionary dynamics. Sci Rep 2022; 12:22403. [PMID: 36575301 PMCID: PMC9794815 DOI: 10.1038/s41598-022-27102-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/18/2022] [Accepted: 12/26/2022] [Indexed: 12/28/2022] Open
Abstract
Evolutionary dynamics is well captured by the replicator equations when the population is infinite and well-mixed. However, the extinction dynamics is modified with finite and structured populations. Experiments on the non-transitive ecosystem containing three populations of bacteria found that the ecological stability sensitively depends on the spatial structure of the populations. Based on the Reference-Gamble-Birth algorithm, we use agent-based Monte Carlo simulations to investigate the extinction dynamics in the rock-paper-scissors ecosystem with finite and structured populations. On the fully-connected network, the extinction time in stable and unstable regimes falls into two universal functions when plotted with the rescaled variables. On the two dimensional grid, the spatial structure changes the transition boundary between stable and unstable regimes but doesn't change its extinction trend. The finding of universal scaling in extinction dynamics is unexpected, and may provide a powerful method to classify different evolutionary dynamics into universal classes.
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20
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Chen YT. The replicator equation in stochastic spatial evolutionary games. Stoch Process Their Appl 2022. [DOI: 10.1016/j.spa.2022.11.013] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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21
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Cooney DB. Assortment and Reciprocity Mechanisms for Promotion of Cooperation in a Model of Multilevel Selection. Bull Math Biol 2022; 84:126. [PMID: 36136162 DOI: 10.1007/s11538-022-01082-8] [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: 05/04/2022] [Accepted: 09/05/2022] [Indexed: 11/26/2022]
Abstract
In the study of the evolution of cooperation, many mechanisms have been proposed to help overcome the self-interested cheating that is individually optimal in the Prisoners' Dilemma game. These mechanisms include assortative or networked social interactions, other-regarding preferences considering the payoffs of others, reciprocity rules to establish cooperation as a social norm, and multilevel selection involving simultaneous competition between individuals favoring cheaters and competition between groups favoring cooperators. In this paper, we build on recent work studying PDE replicator equations for multilevel selection to understand how within-group mechanisms of assortment, other-regarding preferences, and both direct and indirect reciprocity can help to facilitate cooperation in concert with evolutionary competition between groups. We consider a group-structured population in which interactions between individuals consist of Prisoners' Dilemma games and study the dynamics of multilevel competition determined by the payoffs individuals receive when interacting according to these within-group mechanisms. We find that the presence of each of these mechanisms acts synergistically with multilevel selection for the promotion of cooperation, decreasing the strength of between-group competition required to sustain long-time cooperation and increasing the collective payoff achieved by the population. However, we find that only other-regarding preferences allow for the achievement of socially optimal collective payoffs for Prisoners' Dilemma games in which average payoff is maximized by an intermediate mix of cooperators and defectors. For the other three mechanisms, the multilevel dynamics remain susceptible to a shadow of lower-level selection, as the collective outcome fails to exceed the payoff of the all-cooperator group.
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Affiliation(s)
- Daniel B Cooney
- Department of Mathematics and Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA, USA.
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22
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McAvoy A, Kates-Harbeck J, Chatterjee K, Hilbe C. Evolutionary instability of selfish learning in repeated games. PNAS NEXUS 2022; 1:pgac141. [PMID: 36714856 PMCID: PMC9802390 DOI: 10.1093/pnasnexus/pgac141] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/06/2021] [Accepted: 07/22/2022] [Indexed: 02/01/2023]
Abstract
Across many domains of interaction, both natural and artificial, individuals use past experience to shape future behaviors. The results of such learning processes depend on what individuals wish to maximize. A natural objective is one's own success. However, when two such "selfish" learners interact with each other, the outcome can be detrimental to both, especially when there are conflicts of interest. Here, we explore how a learner can align incentives with a selfish opponent. Moreover, we consider the dynamics that arise when learning rules themselves are subject to evolutionary pressure. By combining extensive simulations and analytical techniques, we demonstrate that selfish learning is unstable in most classical two-player repeated games. If evolution operates on the level of long-run payoffs, selection instead favors learning rules that incorporate social (other-regarding) preferences. To further corroborate these results, we analyze data from a repeated prisoner's dilemma experiment. We find that selfish learning is insufficient to explain human behavior when there is a trade-off between payoff maximization and fairness.
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Affiliation(s)
| | | | | | - Christian Hilbe
- Max Planck Research Group: Dynamics of Social Behavior, Max Planck Institute for Evolutionary Biology, Plön, Germany
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23
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Li Z, Chen X, Yang HX, Szolnoki A. Game-theoretical approach for opinion dynamics on social networks. CHAOS (WOODBURY, N.Y.) 2022; 32:073117. [PMID: 35907745 DOI: 10.1063/5.0084178] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/04/2022] [Accepted: 06/21/2022] [Indexed: 06/15/2023]
Abstract
Opinion dynamics on social networks have received considerable attentions in recent years. Nevertheless, just a few works have theoretically analyzed the condition in which a certain opinion can spread in the whole structured population. In this article, we propose an evolutionary game approach for a binary opinion model to explore the conditions for an opinion's spreading. Inspired by real-life observations, we assume that an agent's choice to select an opinion is not random but is based on a score rooted from both public knowledge and the interactions with neighbors. By means of coalescing random walks, we obtain a condition in which opinion A can be favored to spread on social networks in the weak selection limit. We find that the successfully spreading condition of opinion A is closely related to the basic scores of binary opinions, the feedback scores on opinion interactions, and the structural parameters including the edge weights, the weighted degrees of vertices, and the average degree of the network. In particular, when individuals adjust their opinions based solely on the public information, the vitality of opinion A depends exclusively on the difference of basic scores of A and B. When there are no negative (positive) feedback interactions between connected individuals, we find that the success of opinion A depends on the ratio of the obtained positive (negative) feedback scores of competing opinions. To complete our study, we perform computer simulations on fully connected, small-world, and scale-free networks, respectively, which support and confirm our theoretical findings.
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Affiliation(s)
- Zhifang Li
- School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
| | - Xiaojie Chen
- School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
| | - Han-Xin Yang
- Department of Physics, Fuzhou University, Fuzhou 350108, China
| | - Attila Szolnoki
- Institute of Technical Physics and Materials Science, Centre for Energy Research, P.O. Box 49, H-1525 Budapest, Hungary
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24
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Coordination and equilibrium selection in games: the role of local effects. Sci Rep 2022; 12:3373. [PMID: 35233046 PMCID: PMC8888577 DOI: 10.1038/s41598-022-07195-3] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2021] [Accepted: 02/11/2022] [Indexed: 01/28/2023] Open
Abstract
We study the role of local effects and finite size effects in reaching coordination and in equilibrium selection in two-player coordination games. We investigate three update rules - the replicator dynamics (RD), the best response (BR), and the unconditional imitation (UI). For the pure coordination game with two equivalent strategies we find a transition from a disordered state to coordination for a critical value of connectivity. The transition is system-size-independent for the BR and RD update rules. For the IU it is system-size-dependent, but coordination can always be reached below the connectivity of a complete graph. We also consider the general coordination game which covers a range of games, such as the stag hunt. For these games there is a payoff-dominant strategy and a risk-dominant strategy with associated states of equilibrium coordination. We analyse equilibrium selection analytically and numerically. For the RD and BR update rules mean-field predictions agree with simulations and the risk-dominant strategy is evolutionary favoured independently of local effects. When players use the unconditional imitation, however, we observe coordination in the payoff-dominant strategy. Surprisingly, the selection of pay-off dominant equilibrium only occurs below a critical value of the network connectivity and disappears in complete graphs. As we show, it is a combination of local effects and update rule that allows for coordination on the payoff-dominant strategy.
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25
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Hajihashemi M, Aghababaei Samani K. Multi-strategy evolutionary games: A Markov chain approach. PLoS One 2022; 17:e0263979. [PMID: 35176094 PMCID: PMC8853582 DOI: 10.1371/journal.pone.0263979] [Citation(s) in RCA: 2] [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: 12/27/2021] [Accepted: 02/01/2022] [Indexed: 11/18/2022] Open
Abstract
Interacting strategies in evolutionary games is studied analytically in a well-mixed population using a Markov chain method. By establishing a correspondence between an evolutionary game and Markov chain dynamics, we show that results obtained from the fundamental matrix method in Markov chain dynamics are equivalent to corresponding ones in the evolutionary game. In the conventional fundamental matrix method, quantities like fixation probability and fixation time are calculable. Using a theorem in the fundamental matrix method, conditional fixation time in the absorbing Markov chain is calculable. Also, in the ergodic Markov chain, the stationary probability distribution that describes the Markov chain’s stationary state is calculable analytically. Finally, the Rock, scissor, paper evolutionary game are evaluated as an example, and the results of the analytical method and simulations are compared. Using this analytical method saves time and computational facility compared to prevalent simulation methods.
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Affiliation(s)
- Mahdi Hajihashemi
- Department of Physics, Isfahan University of Technology, Isfahan, Iran
- * E-mail:
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26
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Bahk J, Jeong HC. Emergence of cooperation through chain-reaction death. Phys Rev E 2022; 105:024116. [PMID: 35291063 DOI: 10.1103/physreve.105.024116] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2020] [Accepted: 01/25/2022] [Indexed: 06/14/2023]
Abstract
We generalize the Bak-Sneppen model of coevolution to a game model for evolutionary dynamics which provides a natural way for the emergence of cooperation. Interaction between members is mimicked by a prisoner's dilemma game with a memoryless stochastic strategy. The fitness of each member is determined by the payoffs π of the games with its neighbors. We investigate the evolutionary dynamics using a mean-field calculation and Monte Carlo method with two types of death processes, fitness-dependent death and chain-reaction death. In the former, the death probability is proportional to e^{-βπ} where β is the "selection intensity." The neighbors of the death site also die with a probability R through the chain-reaction process invoked by the abrupt change of the interaction environment. When a cooperator interacts with defectors, the cooperator is likely to die due to its low payoff, but the neighboring defectors also tend to disappear through the chain-reaction death, giving rise to an assortment of cooperators. Owing to this assortment, cooperation can emerge for a wider range of R values than the mean-field theory predicts. We present the detailed evolutionary dynamics of our model and the conditions for the emergence of cooperation.
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Affiliation(s)
- Jiwon Bahk
- Department of Physics and Astronomy, Sejong University, Seoul 05006, Korea
| | - Hyeong-Chai Jeong
- Department of Physics and Astronomy, Sejong University, Seoul 05006, Korea
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
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27
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Identification and Control of Game-Based Epidemic Models. GAMES 2022. [DOI: 10.3390/g13010010] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
The effectiveness of control measures against the diffusion of the COVID-19 pandemic is grounded on the assumption that people are prepared and disposed to cooperate. From a strategic decision point of view, cooperation is the unreachable strategy of the Prisoner’s Dilemma game, where the temptation to exploit the others and the fear of being betrayed by them drives the people’s behavior, which eventually results in a fully defective outcome. In this work, we integrate a standard epidemic model with the replicator equation of evolutionary games in order to study the interplay between the infection spreading and the propensity of people to be cooperative under the pressure of the epidemic. The developed model shows high performance in fitting real measurements of infected, recovered and dead people during the whole period of COVID-19 epidemic spread, from March 2020 to September 2021 in Italy. The estimated parameters related to cooperation result to be significantly correlated with vaccination and screening data, thus validating the model. The stability analysis of the multiple steady states present in the proposed model highlights the possibility to tune fundamental control parameters to dramatically reduce the number of potential dead people with respect to the non-controlled case.
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28
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Podobnik B, Jusup M, Korošak D, Holme P, Lipić T. The microdynamics shaping the relationship between democracy and corruption. Proc Math Phys Eng Sci 2022; 478:20210567. [PMID: 35153611 PMCID: PMC8753146 DOI: 10.1098/rspa.2021.0567] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/13/2021] [Accepted: 12/01/2021] [Indexed: 11/12/2022] Open
Abstract
Physics has a long tradition of laying rigorous quantitative foundations for social phenomena. Here, we up the ante for physics' forays into the territory of social sciences by (i) empirically documenting a tipping point in the relationship between democratic norms and corruption suppression, and then (ii) demonstrating how such a tipping point emerges from a micro-scale mechanistic model of spin dynamics in a complex network. Specifically, the tipping point in the relationship between democratic norms and corruption suppression is such that democratization has little effect on suppressing corruption below a critical threshold, but a large effect above the threshold. The micro-scale model of spin dynamics underpins this phenomenon by reinterpreting spins in terms of unbiased (i.e. altruistic) and biased (i.e. parochial) other-regarding behaviour, as well as the corresponding voting preferences. Under weak democratic norms, dense social connections of parochialists enable coercing enough opportunist voters to vote in favour of perpetuating parochial in-group bias. Society may, however, strengthen democratic norms in a rapid turn of events during which opportunists adopt altruism and vote to subdue bias. The emerging model outcome at the societal scale thus mirrors the data, implying that democracy either perpetuates or suppresses corruption depending on the prevailing democratic norms.
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Affiliation(s)
- Boris Podobnik
- Faculty of Civil Engineering, University of Rijeka, 51000 Rijeka, Croatia.,Faculty of Information Studies in Novo Mesto, 8000 Novo Mesto, Slovenia.,Luxembourg School of Business, 2453 Luxembourg, Luxembourg.,Zagreb School of Economics and Management, 10000 Zagreb, Croatia
| | - Marko Jusup
- Tokyo Tech World Research Hub Initiative (WRHI), Institute of Innovative Research, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Dean Korošak
- Institute of Physiology, Faculty of Medicine, University of Maribor, 2000 Maribor, Slovenia.,Faculty of Civil Engineering, Transportation Engineering and Architecture, University of Maribor, 2000 Maribor, Slovenia
| | - Petter Holme
- Tokyo Tech World Research Hub Initiative (WRHI), Institute of Innovative Research, Tokyo Institute of Technology, Tokyo 152-8552, Japan
| | - Tomislav Lipić
- Division of Electronics, Rudjer Boskovic Institute, 10000 Zagreb, Croatia
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29
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Czárán T, Scheuring I. Weak selection helps cheap but harms expensive cooperation in spatial threshold dilemmas. J Theor Biol 2021; 536:110995. [PMID: 34979105 DOI: 10.1016/j.jtbi.2021.110995] [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: 09/30/2021] [Revised: 12/17/2021] [Accepted: 12/21/2021] [Indexed: 11/30/2022]
Abstract
Public Goods Games (PGGs) are n-person games with dependence of individual fitness benefits on the collective investment by the players. We have studied a simple PGG scenario played out by cooperating (C) and defecting (D) agents, applying the highly nonlinear threshold benefit function in an individual-based lattice model. A semi-analytical approximation of the lattice model has been developed and shown to describe the dynamics fairly well in the vicinity of the steady state. Besides the expected outcomes (i.e., the negative effect on cooperator persistence of higher cooperation costs and/or more intensive mixing of the population) we have found a surprising, counter-intuitive effect of the strength of selection on the steady state of the model. The effect is different at low and high cooperation costs, and it shows up only in the lattice model, suggesting that stochastic effects and higher order spatial correlations due to the emergent spatial clustering of cooperators (not taken into account in the semi-analytical approximation) must be responsible for the unexpected results for which we propose an intuitive explanation, present a tentative demonstration, and shortly discuss their biological relevance.
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Affiliation(s)
- Tamás Czárán
- Centre for Ecological Research, Institute of Evolution, 1121 Budapest, Konkoly-Thege Miklós út 29-33, Hungary; MTA-ELTE Theoretical Biology and Evolutionary Ecology Research Group, 1117 Budapest, Pázmány P. sétány 1/c, Hungary
| | - István Scheuring
- Centre for Ecological Research, Institute of Evolution, 1121 Budapest, Konkoly-Thege Miklós út 29-33, Hungary; MTA-ELTE Theoretical Biology and Evolutionary Ecology Research Group, 1117 Budapest, Pázmány P. sétány 1/c, Hungary.
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30
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Sadhukhan S, Chattopadhyay R, Chakraborty S. Amplitude death in coupled replicator map lattice: Averting migration dilemma. Phys Rev E 2021; 104:044304. [PMID: 34781425 DOI: 10.1103/physreve.104.044304] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2021] [Accepted: 09/20/2021] [Indexed: 11/07/2022]
Abstract
Populations composed of a collection of subpopulations (demes) with random migration between them are quite common occurrences. The emergence and sustenance of cooperation in such a population is a highly researched topic in the evolutionary game theory. If the individuals in every deme are considered to be either cooperators or defectors, the migration dilemma can be envisaged: The cooperators would not want to migrate to a defector-rich deme as they fear of facing exploitation; but without migration, cooperation cannot be established throughout the network of demes. With a view to studying the aforementioned scenario, in this paper, we set up a theoretical model consisting of a coupled map lattice of replicator maps based on two-player-two-strategy games. The replicator map considered is capable of showing a variety of evolutionary outcomes, like convergent (fixed point) outcomes and nonconvergent (periodic and chaotic) outcomes. Furthermore, this coupled network of the replicator maps undergoes the phenomenon of amplitude death leading to nonoscillatory stable synchronized states. We specifically explore the effect of (i) the nature of coupling that models migration between the maps, (ii) the heterogenous demes (in the sense that not all the demes have the same game being played by the individuals), (iii) the degree of the network, and (iv) the cost associated with the migration. In the course of investigation, we are intrigued by the effectiveness of the random migration in sustaining a uniform cooperator fraction across a population irrespective of the details of the replicator dynamics and the interaction among the demes.
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Affiliation(s)
- Shubhadeep Sadhukhan
- Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India
| | - Rohitashwa Chattopadhyay
- The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India.,Homi Bhabha National Institute, Anushaktinagar, Mumbai 400 094, India
| | - Sagar Chakraborty
- Department of Physics, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India
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31
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Arefin MR, Tanimoto J. Impact of the baseline payoff on evolutionary outcomes. Phys Rev E 2021; 104:044314. [PMID: 34781447 DOI: 10.1103/physreve.104.044314] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2021] [Accepted: 10/06/2021] [Indexed: 11/07/2022]
Abstract
Do individuals enjoying a higher baseline payoff behave similarly in competitive scenarios compared to their counterparts? The classical replicator equation does not answer such a question since it is invariant to the background or baseline payoff of individuals. In reality, however, if one's baseline payoff is higher than the possible payoffs of an interaction (or game), the individual may respond generously or indifferently if s(he) is satisfied with the prevailing benchmark payoff. This work intends to explore such a phenomenon within the realm of pairwise interactions-taking the prisoner's dilemma as a metaphor-in well-mixed finite and infinite populations. In this framework, a player uses the payoff (comprising baseline and game payoffs) -expectation difference to estimate a degree of eagerness and, with that degree of eagerness, revises his or her strategy with a certain probability. We adopt two approaches to explore such a context, naming them as the Fermi and imitation processes, in which the former uses a pairwise Femi function and the latter considers the relative fitness to estimate probabilities for strategy revision. In a finite population, we examine the effect of intensities to payoff-expectation and strategic payoff differences (denoted by k_{1} and k_{2}, respectively) as well as the level of contentment (ω) on the fixation probability and fixation time (for a single defector). We observe that the fixation probability surges with the increase of intensity parameters. Nevertheless, the maximum fixation probability may require a substantially larger time to fixate, especially when the expectation is lower than the baseline payoff. This means that cooperators can persist for a longer period of time. A higher expectation or greed, however, considerably reduces the fixation time. Interestingly, our numerical simulation reveals that both approaches are equivalent under weak k_{2}(≪1) in the Fermi process. We further derive mean-field equations for both approaches in the context of an infinite population, where we observe two possible evolutionary consequences: either full-scale defection or the persistence of the initial frequency of cooperators. The latter scenario indicates players' uninterested or neutral behavior in relation to the interaction due to their satisfaction on the baseline payoff.
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Affiliation(s)
- Md Rajib Arefin
- Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga-koen, Kasuga-shi, Fukuoka 816-8580, Japan.,Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh
| | - Jun Tanimoto
- Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga-koen, Kasuga-shi, Fukuoka 816-8580, Japan.,Faculty of Engineering Sciences, Kyushu University, Kasuga-koen, Kasuga-shi, Fukuoka 816-8580, Japan
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McAvoy A, Rao A, Hauert C. Intriguing effects of selection intensity on the evolution of prosocial behaviors. PLoS Comput Biol 2021; 17:e1009611. [PMID: 34780464 PMCID: PMC8629389 DOI: 10.1371/journal.pcbi.1009611] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/04/2021] [Revised: 11/29/2021] [Accepted: 11/03/2021] [Indexed: 12/05/2022] Open
Abstract
In many models of evolving populations, genetic drift has an outsized role relative to natural selection, or vice versa. While there are many scenarios in which one of these two assumptions is reasonable, intermediate balances between these forces are also biologically relevant. In this study, we consider some natural axioms for modeling intermediate selection intensities, and we explore how to quantify the long-term evolutionary dynamics of such a process. To illustrate the sensitivity of evolutionary dynamics to drift and selection, we show that there can be a “sweet spot” for the balance of these two forces, with sufficient noise for rare mutants to become established and sufficient selection to spread. This balance allows prosocial traits to evolve in evolutionary models that were previously thought to be unconducive to the emergence and spread of altruistic behaviors. Furthermore, the effects of selection intensity on long-run evolutionary outcomes in these settings, such as when there is global competition for reproduction, can be highly non-monotonic. Although intermediate selection intensities (neither weak nor strong) are notoriously difficult to study analytically, they are often biologically relevant; and the results we report suggest that they can elicit novel and rich dynamics in the evolution of prosocial behaviors. Theoretical models of populations have been useful for assessing when and how traits spread, in large part because they are simple. Rather than being used to reproduce empirical data, these idealized models involve relatively few parameters and are utilized to gain a qualitative understanding of what promotes or suppresses a trait. For prosocial traits, which entail a cost to self to help another, one thing that mathematical models often suggest is that competition to reproduce must be localized, meaning an individual must be fitter than just a small subset of the population in order to produce an offspring. We show here that this finding is not robust. Such traits can indeed proliferate when there is global competition for reproduction, which we demonstrate by increasing the degree to which payoffs from games affect birth rates. Since this kind of “stronger selection” has also been observed empirically, we discuss how it is incorporated into theoretical models more broadly.
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Affiliation(s)
- Alex McAvoy
- Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America
- Center for Mathematical Biology, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America
- * E-mail:
| | - Andrew Rao
- Department of Economics, Harvard University, Cambridge, Massachusetts, United States of America
| | - Christoph Hauert
- Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada
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Johnson T, Smirnov O. Temporal assortment of cooperators in the spatial prisoner's dilemma. Commun Biol 2021; 4:1283. [PMID: 34773077 PMCID: PMC8589994 DOI: 10.1038/s42003-021-02804-9] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2020] [Accepted: 10/25/2021] [Indexed: 11/09/2022] Open
Abstract
We study a spatial, one-shot prisoner's dilemma (PD) model in which selection operates on both an organism's behavioral strategy (cooperate or defect) and its decision of when to implement that strategy, which we depict as an organism's choice of one point in time, out of a set of discrete time slots, at which to carry out its PD strategy. Results indicate selection for cooperators across various time slots and parameter settings, including parameter settings in which cooperation would not evolve in an exclusively spatial model-as in work investigating exogenously imposed temporal networks. Moreover, in the presence of time slots, cooperators' portion of the population grows even under different combinations of spatial structure, transition rules, and update dynamics, though rates of cooperator fixation decline under pairwise comparison and synchronous updating. These findings indicate that, under certain evolutionary processes, merely existing in time and space promotes the evolution of cooperation.
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Affiliation(s)
- Tim Johnson
- Atkinson Graduate School of Management, Willamette University, Salem, OR, 97301, USA.
- Center for Governance and Public Policy Research, Willamette University, Salem, OR, 97301, USA.
| | - Oleg Smirnov
- Department of Political Science, Stony Brook University, Stony Brook, NY, 11794, USA
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Adaptation strategies and collective dynamics of extraction in networked commons of bistable resources. Sci Rep 2021; 11:21987. [PMID: 34753992 PMCID: PMC8578606 DOI: 10.1038/s41598-021-01314-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/06/2021] [Accepted: 10/18/2021] [Indexed: 11/24/2022] Open
Abstract
When populations share common-pool resources (CPRs), individuals decide how much effort to invest towards resource extraction and how to allocate this effort among available resources. We investigate these dual aspects of individual choice in networked games where resources undergo regime shifts between discrete quality states (viable or depleted) depending on collective extraction levels. We study the patterns of extraction that emerge on various network types when agents are free to vary extraction from each CPR separately to maximize their short-term payoffs. Using these results as a basis for comparison, we then investigate how results are altered if agents fix one aspect of adaptation (magnitude or allocation) while letting the other vary. We consider two constrained adaptation strategies: uniform adaptation, whereby agents adjust their extraction levels from all CPRs by the same amount, and reallocation, whereby agents selectively shift effort from lower- to higher-quality resources. A preference for uniform adaptation increases collective wealth on degree-heterogeneous agent-resource networks. Further, low-degree agents retain preferences for these constrained strategies under reinforcement learning. Empirical studies have indicated that some CPR appropriators ignore—while others emphasize—allocation aspects of adaptation; our results demonstrate that structural patterns of resource access can determine which behavior is more advantageous.
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Wechsler D, Bascompte J. Cheating in mutualisms promotes diversity and complexity. Am Nat 2021; 199:393-405. [DOI: 10.1086/717865] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
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Verma P, Reeves RG, Gokhale CS. A common gene drive language eases regulatory process and eco-evolutionary extensions. BMC Ecol Evol 2021; 21:156. [PMID: 34372763 PMCID: PMC8351217 DOI: 10.1186/s12862-021-01881-y] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2020] [Accepted: 07/12/2021] [Indexed: 02/08/2023] Open
Abstract
Background Synthetic gene drive technologies aim to spread transgenic constructs into wild populations even when they impose organismal fitness disadvantages. The extraordinary diversity of plausible drive mechanisms and the range of selective parameters they may encounter makes it very difficult to convey their relative predicted properties, particularly where multiple approaches are combined. The sheer number of published manuscripts in this field, experimental and theoretical, the numerous techniques resulting in an explosion in the gene drive vocabulary hinder the regulators’ point of view. We address this concern by defining a simplified parameter based language of synthetic drives. Results Employing the classical population dynamics approach, we show that different drive construct (replacement) mechanisms can be condensed and evaluated on an equal footing even where they incorporate multiple replacement drives approaches. Using a common language, it is then possible to compare various model properties, a task desired by regulators and policymakers. The generalization allows us to extend the study of the invasion dynamics of replacement drives analytically and, in a spatial setting, the resilience of the released drive constructs. The derived framework is available as a standalone tool. Conclusion Besides comparing available drive constructs, our tool is also useful for educational purpose. Users can also explore the evolutionary dynamics of future hypothetical combination drive scenarios. Thus, our results appraise the properties and robustness of drives and provide an intuitive and objective way for risk assessment, informing policies, and enhancing public engagement with proposed and future gene drive approaches.
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Affiliation(s)
- Prateek Verma
- Research Group for Theoretical Models of Eco-evolutionary Dynamics, Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, Plön, Germany.
| | - R Guy Reeves
- Department of Evolutionary Genetics, Max Planck Institute for Evolutionary Biology, Plön, Germany
| | - Chaitanya S Gokhale
- Research Group for Theoretical Models of Eco-evolutionary Dynamics, Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, Plön, Germany
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Consensus towards Partially Cooperative Strategies in Self-Regulated Evolutionary Games on Networks. GAMES 2021. [DOI: 10.3390/g12030060] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
Cooperation is widely recognized to be fundamental for the well-balanced development of human societies. Several different approaches have been proposed to explain the emergence of cooperation in populations of individuals playing the Prisoner’s Dilemma game, characterized by two concurrent natural mechanisms: the temptation to defect and the fear to be betrayed by others. Few results are available for analyzing situations where only the temptation to defect (Chicken game) or the fear to be betrayed (Stag-Hunt game) is present. In this paper, we analyze the emergence of full and partial cooperation for these classes of games. We find the conditions for which these Nash equilibria are asymptotically stable, and we show that the partial one is also globally stable. Furthermore, in the Chicken and Stag-Hunt games, partial cooperation has been found to be more rewarding than the full one of the Prisoner’s Dilemma game. This result highlights the importance of such games for understanding and sustaining different levels of cooperation in social networks.
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Wang G, Su Q, Wang L. Evolution of state-dependent strategies in stochastic games. J Theor Biol 2021; 527:110818. [PMID: 34181968 DOI: 10.1016/j.jtbi.2021.110818] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2021] [Revised: 06/06/2021] [Accepted: 06/22/2021] [Indexed: 10/21/2022]
Abstract
In a population of interacting individuals, the environment for interactions often changes due to individuals' behaviors, which in turn drive the evolution of individuals' behaviors. The interplay between the environment and individuals' behaviors has been demonstrated to remarkably influence the evolutionary outcomes. In reality, in highly cognitive species such as social primates and human beings, individuals are often capable of perceiving the environment change and then differentiate their strategies across different environment states. We propose a model of environmental feedback with state-dependent strategies: individuals have perceptions of distinct environment states and therefore take distinct sub-strategies under each of them; based on the sub-strategy, individuals then decide their behaviors; their behaviors subsequently modify the environment state. We use the theory of stochastic games and evolutionary dynamics to analyze this idea. We find that when environment changes slower than behaviors, state-dependent strategies (i.e. taking different sub-strategies under different environment states) can outperform state-independent strategies (i.e. taking an identical sub-strategy under all environment states), such as Win-Stay, Lose-Shift, the most leading strategy among state-independent strategies. The intuition is that delayed environmental feedback provides chances for individuals with state-dependent strategies to exploit those with state-independent strategies. Our results hold (1) in both well-mixed and structured populations; (2) when the environment switches between two or more states. Furthermore, the environment changing rate decides if state-dependent strategies benefit global cooperation. The evolution sees the rise of the cooperation level for fast environment switching and the decrease otherwise. Our work stresses that individuals' perceptions of different environment states are beneficial to their survival and social prosperity in a changing world.
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Affiliation(s)
- Guocheng Wang
- Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China
| | - Qi Su
- Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA 19104, USA; Department of Mathematics, University of Pennsylvania, Philadelphia, PA19104, USA; Department of Biology, University of Pennsylvania, Philadelphia, PA 19104, USA
| | - Long Wang
- Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China; Center for Multi-Agent Research, Institute for Artificial Intelligence, Peking University, Beijing 100871, China.
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Aspiration dynamics generate robust predictions in heterogeneous populations. Nat Commun 2021; 12:3250. [PMID: 34059670 PMCID: PMC8166829 DOI: 10.1038/s41467-021-23548-4] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2020] [Accepted: 05/05/2021] [Indexed: 12/03/2022] Open
Abstract
Update rules, which describe how individuals adjust their behavior over time, affect the outcome of social interactions. Theoretical studies have shown that evolutionary outcomes are sensitive to model details when update rules are imitation-based but are robust when update rules are self-evaluation based. However, studies of self-evaluation based rules have focused on homogeneous population structures where each individual has the same number of neighbors. Here, we consider heterogeneous population structures represented by weighted networks. Under weak selection, we analytically derive the condition for strategy success, which coincides with the classical condition of risk-dominance. This condition holds for all weighted networks and distributions of aspiration levels, and for individualized ways of self-evaluation. Our findings recover previous results as special cases and demonstrate the universality of the robustness property under self-evaluation based rules. Our work thus sheds light on the intrinsic difference between evolutionary dynamics under self-evaluation based and imitation-based update rules. Social interaction outcomes can depend on the type of information individuals possess and how it is used in decision-making. Here, Zhou et al. find that self-evaluation based decision-making rules lead to evolutionary outcomes that are robust to different population structures and ways of self-evaluation.
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Zhang H. A game-theoretical dynamic imitation model on networks. J Math Biol 2021; 82:30. [PMID: 33683438 DOI: 10.1007/s00285-021-01573-7] [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: 08/13/2020] [Revised: 01/09/2021] [Accepted: 02/08/2021] [Indexed: 11/29/2022]
Abstract
A game-theoretical model is constructed to capture the effect of imitation on the evolution of cooperation. This imitation describes the case where successful individuals are more likely to be imitated by newcomers who will employ their strategies and social networks. Two classical repeated strategies 'always defect (ALLD)' and 'tit-for-tat (TFT)' are adopted. Mathematical analyses are mainly conducted by the method of coalescence theory. Under the assumption of a large population size and weak selection, the results show that the evolution of cooperation is promoted in this dynamic network. As we observed that the critical benefit-to-cost ratio is smaller compared to that in well-mixed populations. The critical benefit-to-cost ratio approaches a specific value which depends on three parameters, the repeated rounds of the game, the effective strategy mutation rate, and the effective link mutation rate. Specifically, for a very high value of the effective link mutation rate, the critical benefit-to-cost ratio approaches 1. Remarkably, for a low value of the effective link mutation rate, by letting the effective strategy mutation is nearly equal to zero, the critical benefit-to-cost ratio approaches [Formula: see text] for the resulting highly connected networks, which allows TFT to be evolutionary stable. It illustrates that dominance of TFTs is associated with more connected networks. This research can enrich the theory of the coevolution of game strategy and network structure with dynamic imitation.
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Affiliation(s)
- Hui Zhang
- School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an, 710072, Shaanxi, China.
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41
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West J, Ma Y, Kaznatcheev A, Anderson ARA. IsoMaTrix: a framework to visualize the isoclines of matrix games and quantify uncertainty in structured populations. Bioinformatics 2020; 36:5542-5544. [PMID: 33325501 PMCID: PMC8016459 DOI: 10.1093/bioinformatics/btaa1025] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2020] [Revised: 11/24/2020] [Accepted: 11/30/2020] [Indexed: 11/13/2022] Open
Abstract
Evolutionary game theory describes frequency-dependent selection for fixed, heritable strategies in a population of competing individuals using a payoff matrix. We present a software package to aid in the construction, analysis, and visualization of three-strategy matrix games. The IsoMaTrix package computes the isoclines (lines of zero growth) of matrix games, and facilitates direct comparison of well-mixed dynamics to structured populations on a lattice grid. IsoMaTrix computes fixed points, phase flow, trajectories, (sub)velocities, and uncertainty quantification for stochastic effects in spatial matrix games. We describe a result obtained via IsoMaTrix's spatial games functionality, which shows that the timing of competitive release in a cancer model (under continuous treatment) critically depends on the initial spatial configuration of the tumor. AVAILABILITY The code is available at: https://github.com/mathonco/isomatrix. SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.
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Affiliation(s)
- Jeffrey West
- Integrated Mathematical Oncology Department, H. Lee Moffitt Cancer Center & Research Institute, Tampa, 33612, Florida
| | - Yongqian Ma
- Department of Physics & Astronomy, University of Southern California, Los Angeles, CA, 90089-1191
| | - Artem Kaznatcheev
- Department of Translational Hematology & Oncology Research, Cleveland Clinic, Cleveland, OH, USA
| | - Alexander R A Anderson
- Integrated Mathematical Oncology Department, H. Lee Moffitt Cancer Center & Research Institute, Tampa, 33612, Florida
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Karlsson CJ, Rowlett J. Decisions and disease: a mechanism for the evolution of cooperation. Sci Rep 2020; 10:13113. [PMID: 32753581 PMCID: PMC7403384 DOI: 10.1038/s41598-020-69546-2] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2020] [Accepted: 07/13/2020] [Indexed: 01/01/2023] Open
Abstract
In numerous contexts, individuals may decide whether they take actions to mitigate the spread of disease, or not. Mitigating the spread of disease requires an individual to change their routine behaviours to benefit others, resulting in a 'disease dilemma' similar to the seminal prisoner's dilemma. In the classical prisoner's dilemma, evolutionary game dynamics predict that all individuals evolve to 'defect.' We have discovered that when the rate of cooperation within a population is directly linked to the rate of spread of the disease, cooperation evolves under certain conditions. For diseases which do not confer immunity to recovered individuals, if the time scale at which individuals receive accurate information regarding the disease is sufficiently rapid compared to the time scale at which the disease spreads, then cooperation emerges. Moreover, in the limit as mitigation measures become increasingly effective, the disease can be controlled; the number of infections tends to zero. It has been suggested that disease spreading models may also describe social and group dynamics, indicating that this mechanism for the evolution of cooperation may also apply in those contexts.
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Affiliation(s)
- Carl-Joar Karlsson
- Department of Mathematical Sciences, Chalmers University of Technology and The University of Gothenburg, 41296, Gothenburg, Sweden
| | - Julie Rowlett
- Department of Mathematical Sciences, Chalmers University of Technology and The University of Gothenburg, 41296, Gothenburg, Sweden.
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Abstract
Cooperation is prevalent in nature, not only in the context of social interactions within the animal kingdom but also on the cellular level. In cancer, for example, tumour cells can cooperate by producing growth factors. The evolution of cooperation has traditionally been studied for well-mixed populations under the framework of evolutionary game theory, and more recently for structured populations using evolutionary graph theory (EGT). The population structures arising due to cellular arrangement in tissues, however, are dynamic and thus cannot be accurately represented by either of these frameworks. In this work, we compare the conditions for cooperative success in an epithelium modelled using EGT, to those in a mechanical model of an epithelium—the Voronoi tessellation (VT) model. Crucially, in this latter model, cells are able to move, and birth and death are not spatially coupled. We calculate fixation probabilities in the VT model through simulation and an approximate analytic technique and show that this leads to stronger promotion of cooperation in comparison with the EGT model.
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Affiliation(s)
- Jessie Renton
- Department of Mathematics, University College London , Gower Street, London WC1E 6BT , UK
| | - Karen M Page
- Department of Mathematics, University College London , Gower Street, London WC1E 6BT , UK
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Madeo D, Mocenni C. Self-regulation versus social influence for promoting cooperation on networks. Sci Rep 2020; 10:4830. [PMID: 32179794 PMCID: PMC7075901 DOI: 10.1038/s41598-020-61634-7] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/11/2019] [Accepted: 02/27/2020] [Indexed: 11/09/2022] Open
Abstract
Cooperation is a relevant and controversial phenomenon in human societies. Indeed, although it is widely recognized essential for tackling social dilemmas, finding suitable policies for promoting cooperation can be arduous and expensive. More often, it is driven by pre-established schemas based on norms and punishments. To overcome this paradigm, we highlight the interplay between the influence of social interactions on networks and spontaneous self-regulating mechanisms on individuals behavior. We show that the presence of these mechanisms in a prisoner's dilemma game, may oppose the willingness of individuals to defect, thus allowing them to behave cooperatively, while interacting with others and taking conflicting decisions over time. These results are obtained by extending the Evolutionary Game Equations over Networks to account for self-regulating mechanisms. Specifically, we prove that players may partially or fully cooperate whether self-regulating mechanisms are sufficiently stronger than social pressure. The proposed model can explain unconditional cooperation (strong self-regulation) and unconditional defection (weak self-regulation). For intermediate self-regulation values, more complex behaviors are observed, such as mutual defection, recruiting (cooperate if others cooperate), exploitation of cooperators (defect if others cooperate) and altruism (cooperate if others defect). These phenomena result from dynamical transitions among different game structures, according to changes of system parameters and cooperation of neighboring players. Interestingly, we show that the topology of the network of connections among players is crucial when self-regulation, and the associated costs, are reasonably low. In particular, a population organized on a random network with a Scale-Free distribution of connections is more cooperative than on a network with an Erdös-Rényi distribution, and, in turn, with a regular one. These results highlight that social diversity, encoded within heterogeneous networks, is more effective for promoting cooperation.
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Affiliation(s)
- Dario Madeo
- Department of Information Engineering and Mathematics, University of Siena, Via Roma 56, 53100, Siena, Italy.
| | - Chiara Mocenni
- Department of Information Engineering and Mathematics, University of Siena, Via Roma 56, 53100, Siena, Italy
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45
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Chen X, Brännström Å, Dieckmann U. Parent-preferred dispersal promotes cooperation in structured populations. Proc Biol Sci 2020; 286:20181949. [PMID: 30963948 DOI: 10.1098/rspb.2018.1949] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Dispersal is a key process for the emergence of social and biological behaviours. Yet, little attention has been paid to dispersal's effects on the evolution of cooperative behaviour in structured populations. To address this issue, we propose two new dispersal modes, parent-preferred and offspring-preferred dispersal, incorporate them into the birth-death update rule, and consider the resultant strategy evolution in the prisoner's dilemma on random-regular, small-world, and scale-free networks, respectively. We find that parent-preferred dispersal favours the evolution of cooperation in these different types of population structures, while offspring-preferred dispersal inhibits the evolution of cooperation in homogeneous populations. On scale-free networks when the strength of parent-preferred dispersal is weak, cooperation can be enhanced at intermediate strengths of offspring-preferred dispersal, and cooperators can coexist with defectors at high strengths of offspring-preferred dispersal. Moreover, our theoretical analysis based on the pair-approximation method corroborates the evolutionary outcomes on random-regular networks. We also incorporate the two new dispersal modes into three other update rules (death-birth, imitation, and pairwise comparison updating), and find that similar results about the effects of parent-preferred and offspring-preferred dispersal can again be observed in the aforementioned different types of population structures. Our work, thus, unveils robust effects of preferential dispersal modes on the evolution of cooperation in different interactive environments.
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Affiliation(s)
- Xiaojie Chen
- 1 School of Mathematical Sciences, University of Electronic Science and Technology of China , Chengdu 611731 , People's Republic of China.,2 Evolution and Ecology Program, International Institute for Applied Systems Analysis (IIASA) , Schlossplatz 1, 2361 Laxenburg , Austria
| | - Åke Brännström
- 2 Evolution and Ecology Program, International Institute for Applied Systems Analysis (IIASA) , Schlossplatz 1, 2361 Laxenburg , Austria.,3 Department of Mathematics and Mathematical Statistics, Umeå University , 90187 Umeå , Sweden
| | - Ulf Dieckmann
- 2 Evolution and Ecology Program, International Institute for Applied Systems Analysis (IIASA) , Schlossplatz 1, 2361 Laxenburg , Austria
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Wang X, Gu C, Zhao J, Quan J. Evolutionary game dynamics of combining the imitation and aspiration-driven update rules. Phys Rev E 2019; 100:022411. [PMID: 31574646 DOI: 10.1103/physreve.100.022411] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/14/2019] [Indexed: 06/10/2023]
Abstract
So far, most studies on evolutionary game dynamics in finite populations have concentrated on a single update rule. However, given the impacts of the environment and individual cognition, individuals may use different update rules to change their current strategies. In light of this, the current paper reports on a study that constructed a mixed stochastic evolutionary game dynamic by combining the imitation and aspiration-driven update processes. The target was to clarify the influences of the aspiration-driven process on the evolution of the level of cooperation by considering the behavior of a population in which individuals have two strategies available: cooperation and defection. Through a numerical analysis of unstructured populations and simulation analyses of structured populations and of the random-matching model, the following results were found. First, the mean fraction of cooperators varied alongside the probability with which the individual adopted the aspiration-driven update rule. In the Prisoner's Dilemma and coexistence games, the aspiration-driven update process promoted cooperation in the well-mixed population but inhibited it in structured ones and the random-matching model; however, in the coordination game, the aspiration-driven update process was seen to exert the opposite effect on cooperation by inhibiting the latter in a homogeneously mixed population but promoting it in structured ones and in the random-matching model. Second, the mean fraction of cooperators changed with the aspiration level in the differently structured populations and random-matching model, and there appeared a phase transition point. Third, the evolutionary characteristics of the mean fraction of cooperators maintained robustness in the differently structured populations and random-matching model. These results extend evolutionary game theory.
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Affiliation(s)
- Xianjia Wang
- Economics and Management School, Wuhan University, Wuhan 430072, China
- Institute of Systems Engineering, Wuhan University, Wuhan 430072, China
| | - Cuiling Gu
- Institute of Systems Engineering, Wuhan University, Wuhan 430072, China
| | - Jinhua Zhao
- Economics and Management School, Wuhan University, Wuhan 430072, China
| | - Ji Quan
- School of Management, Wuhan University of Technology, Wuhan 430070, China
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Su Q, Zhou L, Wang L. Evolutionary multiplayer games on graphs with edge diversity. PLoS Comput Biol 2019; 15:e1006947. [PMID: 30933968 PMCID: PMC6459562 DOI: 10.1371/journal.pcbi.1006947] [Citation(s) in RCA: 20] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2018] [Revised: 04/11/2019] [Accepted: 03/12/2019] [Indexed: 11/20/2022] Open
Abstract
Evolutionary game dynamics in structured populations has been extensively explored in past decades. However, most previous studies assume that payoffs of individuals are fully determined by the strategic behaviors of interacting parties, and social ties between them only serve as the indicator of the existence of interactions. This assumption neglects important information carried by inter-personal social ties such as genetic similarity, geographic proximity, and social closeness, which may crucially affect the outcome of interactions. To model these situations, we present a framework of evolutionary multiplayer games on graphs with edge diversity, where different types of edges describe diverse social ties. Strategic behaviors together with social ties determine the resulting payoffs of interactants. Under weak selection, we provide a general formula to predict the success of one behavior over the other. We apply this formula to various examples which cannot be dealt with using previous models, including the division of labor and relationship- or edge-dependent games. We find that labor division can promote collective cooperation markedly. The evolutionary process based on relationship-dependent games can be approximated by interactions under a transformed and unified game. Our work stresses the importance of social ties and provides effective methods to reduce the calculating complexity in analyzing the evolution of realistic systems.
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Affiliation(s)
- Qi Su
- Center for Systems and Control, College of Engineering, Peking University, Beijing, China
- Center for Polymer Studies, Department of Physics, Boston University, Boston, Massachusetts, United States of America
| | - Lei Zhou
- Center for Systems and Control, College of Engineering, Peking University, Beijing, China
| | - Long Wang
- Center for Systems and Control, College of Engineering, Peking University, Beijing, China
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Hauert C, Saade C, McAvoy A. Asymmetric evolutionary games with environmental feedback. J Theor Biol 2019; 462:347-360. [DOI: 10.1016/j.jtbi.2018.11.019] [Citation(s) in RCA: 25] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2018] [Revised: 11/18/2018] [Accepted: 11/20/2018] [Indexed: 10/27/2022]
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49
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Spatial heterogeneity and evolution of fecundity-affecting traits. J Theor Biol 2018; 454:190-204. [DOI: 10.1016/j.jtbi.2018.06.005] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2018] [Revised: 06/01/2018] [Accepted: 06/04/2018] [Indexed: 11/23/2022]
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50
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Cassese D. Replicator equation on networks with degree regular communities. APPLIED NETWORK SCIENCE 2018; 3:29. [PMID: 30839808 PMCID: PMC6214305 DOI: 10.1007/s41109-018-0083-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/08/2018] [Accepted: 07/13/2018] [Indexed: 06/09/2023]
Abstract
The replicator equation is one of the fundamental tools to study evolutionary dynamics in well-mixed populations. This paper contributes to the literature on evolutionary graph theory, providing a version of the replicator equation for a family of connected networks with communities, where nodes in the same community have the same degree. This replicator equation is applied to the study of different classes of games, exploring the impact of the graph structure on the equilibria of the evolutionary dynamics.
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Affiliation(s)
- Daniele Cassese
- Department of Mathematics, University of Namur, NaXys, Rempart de la Vierge 8, Namur, Belgium
- ICTEAM, University of Louvain, Av Georges Lemaître, Louvain-la-Neuve, Belgium
- Oxford Mathematical Institute, Woodstock Road, Oxford, OX2 6GG UK
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