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Civilini A, Sadekar O, Battiston F, Gómez-Gardeñes J, Latora V. Explosive Cooperation in Social Dilemmas on Higher-Order Networks. PHYSICAL REVIEW LETTERS 2024; 132:167401. [PMID: 38701463 DOI: 10.1103/physrevlett.132.167401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2023] [Revised: 10/27/2023] [Accepted: 03/01/2024] [Indexed: 05/05/2024]
Abstract
Understanding how cooperative behaviors can emerge from competitive interactions is an open problem in biology and social sciences. While interactions are usually modeled as pairwise networks, the units of many real-world systems can also interact in groups of three or more. Here, we introduce a general framework to extend pairwise games to higher-order networks. By studying social dilemmas on hypergraphs with a tunable structure, we find an explosive transition to cooperation triggered by a critical number of higher-order games. The associated bistable regime implies that an initial critical mass of cooperators is also required for the emergence of prosocial behavior. Our results show that higher-order interactions provide a novel explanation for the survival of cooperation.
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Affiliation(s)
- Andrea Civilini
- School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
- Dipartimento di Fisica ed Astronomia, Università di Catania and INFN, Catania I-95123, Italy
| | - Onkar Sadekar
- Department of Network and Data Science, Central European University Vienna, Vienna 1100, Austria
| | - Federico Battiston
- Department of Network and Data Science, Central European University Vienna, Vienna 1100, Austria
| | - Jesús Gómez-Gardeñes
- Department of Condensed Matter Physics, University of Zaragoza, 50009 Zaragoza, Spain
- GOTHAM lab, Institute of Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, 50018 Zaragoza, Spain
| | - Vito Latora
- School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
- Dipartimento di Fisica ed Astronomia, Università di Catania and INFN, Catania I-95123, Italy
- Complexity Science Hub Vienna, A-1080 Vienna, Austria
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Pires DL, Broom M. More can be better: An analysis of single-mutant fixation probability functions under 2 × 2 games. Proc Math Phys Eng Sci 2022. [DOI: 10.1098/rspa.2022.0577] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Evolutionary game theory has proved to be a powerful tool to probe the self-organization of collective behaviour by considering frequency-dependent fitness in evolutionary processes. It has shown that the stability of a strategy depends not only on the payoffs received after each encounter but also on the population’s size. Here, we study
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games in well-mixed finite populations by analyzing the fixation probabilities of single mutants as functions of population size. We proved that nine of the 24 possible games always lead to monotonically decreasing functions, similarly to fixed fitness scenarios. However, fixation functions showed increasing regions under 12 distinct anti-coordination, coordination and dominance games. Perhaps counter-intuitively, this establishes that single-mutant strategies often benefit from being in larger populations. Fixation functions that increase from a global minimum to a positive asymptotic value are pervasive but may have been easily concealed by the weak selection limit. We obtained sufficient conditions to observe fixation increasing for small populations and three distinct ways this can occur. Finally, we describe fixation functions with the increasing regions bounded by two extremes under intermediate population sizes. We associate their occurrence with transitions from having one global extreme to other shapes.
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Affiliation(s)
- Diogo L. Pires
- Department of Mathematics, City, University of London, Northampton Square, London EC1V 0HB, UK
| | - Mark Broom
- Department of Mathematics, City, University of London, Northampton Square, London EC1V 0HB, UK
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Denisov S, Vershinina O, Thingna J, Hänggi P, Ivanchenko M. Quasi-stationary states of game-driven systems: A dynamical approach. CHAOS (WOODBURY, N.Y.) 2020; 30:123145. [PMID: 33380033 DOI: 10.1063/5.0019736] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2020] [Accepted: 12/02/2020] [Indexed: 06/12/2023]
Abstract
Evolutionary game theory is a framework to formalize the evolution of collectives ("populations") of competing agents that are playing a game and, after every round, update their strategies to maximize individual payoffs. There are two complementary approaches to modeling evolution of player populations. The first addresses essentially finite populations by implementing the apparatus of Markov chains. The second assumes that the populations are infinite and operates with a system of mean-field deterministic differential equations. By using a model of two antagonistic populations, which are playing a game with stationary or periodically varying payoffs, we demonstrate that it exhibits metastable dynamics that is reducible neither to an immediate transition to a fixation (extinction of all but one strategy in a finite-size population) nor to the mean-field picture. In the case of stationary payoffs, this dynamics can be captured with a system of stochastic differential equations and interpreted as a stochastic Hopf bifurcation. In the case of varying payoffs, the metastable dynamics is much more complex than the dynamics of the means.
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Affiliation(s)
- Sergey Denisov
- Department of Computer Science, Oslo Metropolitan University, N-0130 Oslo, Norway
| | - Olga Vershinina
- Department of Applied Mathematics, Lobachevsky University, 603950 Nizhny Novgorod, Russia
| | - Juzar Thingna
- Center for Theoretical Physics of Complex Systems (IBS), Daejeon 34126, South Korea
| | - Peter Hänggi
- Institut für Physik, Universität Augsburg, D-86135 Augsburg, Germany
| | - Mikhail Ivanchenko
- Department of Applied Mathematics, Lobachevsky University, 603950 Nizhny Novgorod, Russia
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Zhou L, Li A, Wang L. Coevolution of nonlinear group interactions and strategies in well-mixed and structured populations. J Theor Biol 2018; 440:32-41. [PMID: 29221892 DOI: 10.1016/j.jtbi.2017.12.004] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2017] [Revised: 11/29/2017] [Accepted: 12/03/2017] [Indexed: 11/15/2022]
Abstract
In microbial populations and human societies, the rule of nonlinear group interactions strongly affects the intraspecific evolutionary dynamics, which leads to the variation of the strategy composition eventually. The consequence of such variation may retroact to the rule of the interactions. This correlation indicates that the rule of nonlinear group interactions may coevolve with individuals' strategies. Here, we develop a model to investigate such coevolution in both well-mixed and structured populations. In our model, positive and negative correlations between the rule and the frequency of cooperators are considered, with local and global information. When the correlation refers to the global information, we show that in well-mixed populations, the coevolutionary outcomes cover the scenarios of defector dominance, coexistence, and bi-stability. Whenever the population structure is considered, its impact on the coevolutionary dynamics depends on the type of the correlation: with a negative (positive) correlation, population structure promotes (inhibits) the evolution of cooperation. Furthermore, when the correlation is based on the more accessible local information, we reveal that a negative correlation pushes cooperators into a harsh situation whereas a positive one lowers the barriers for cooperators to occupy the population. All our analytical results are validated by numerical simulations. Our results shed light on the power of the coevolution of nonlinear group interactions and evolutionary dynamics on generating various evolutionary outcomes, implying that the coevolutionary framework may be more appropriate than the traditional cases for understanding the evolution of cooperation in both structureless and structured populations.
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Affiliation(s)
- Lei Zhou
- Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China; Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544, USA.
| | - Aming Li
- Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China; Center for Complex Network Research and Department of Physics, Northeastern University, Boston, MA 02115, USA
| | - Long Wang
- Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China.
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Liu X, Pan Q, He M. Promotion of cooperation in evolutionary game dynamics with local information. J Theor Biol 2018; 437:1-8. [PMID: 29031517 DOI: 10.1016/j.jtbi.2017.10.013] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2017] [Revised: 09/07/2017] [Accepted: 10/12/2017] [Indexed: 11/15/2022]
Abstract
In this paper, we propose a strategy-updating rule driven by local information, which is called Local process. Unlike the standard Moran process, the Local process does not require global information about the strategic environment. By analyzing the dynamical behavior of the system, we explore how the local information influences the fixation of cooperation in two-player evolutionary games. Under weak selection, the decreasing local information leads to an increase of the fixation probability when natural selection does not favor cooperation replacing defection. In the limit of sufficiently large selection, the analytical results indicate that the fixation probability increases with the decrease of the local information, irrespective of the evolutionary games. Furthermore, for the dominance of defection games under weak selection and for coexistence games, the decreasing of local information will lead to a speedup of a single cooperator taking over the population. Overall, to some extent, the local information is conducive to promoting the cooperation.
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Affiliation(s)
- Xuesong Liu
- School of Mathematical Science, Dalian University of Technology, Dalian 116024, China
| | - Qiuhui Pan
- School of Mathematical Science, Dalian University of Technology, Dalian 116024, China; School of Innovation and Entrepreneurship, Dalian University of Technology, Dalian 116024, China.
| | - Mingfeng He
- School of Mathematical Science, Dalian University of Technology, Dalian 116024, China
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Zheng XD, Li C, Yu JR, Wang SC, Fan SJ, Zhang BY, Tao Y. A simple rule of direct reciprocity leads to the stable coexistence of cooperation and defection in the Prisoner's Dilemma game. J Theor Biol 2017; 420:12-17. [PMID: 28259660 DOI: 10.1016/j.jtbi.2017.02.036] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2016] [Revised: 02/23/2017] [Accepted: 02/27/2017] [Indexed: 11/25/2022]
Abstract
The long-term coexistence of cooperation and defection is a common phenomenon in nature and human society. However, none of the theoretical models based on the Prisoner's Dilemma (PD) game can provide a concise theoretical model to explain what leads to the stable coexistence of cooperation and defection in the long-term even though some rules for promoting cooperation have been summarized (Nowak, 2006, Science 314, 1560-1563). Here, based on the concept of direct reciprocity, we develop an elementary model to show why stable coexistence of cooperation and defection in the PD game is possible. The basic idea behind our theoretical model is that all players in a PD game prefer a cooperator as an opponent, and our results show that considering strategies allowing opting out against defection provide a general and concise way of understanding the fundamental importance of direct reciprocity in driving the evolution of cooperation.
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Affiliation(s)
- Xiu-Deng Zheng
- Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing, PR China
| | - Cong Li
- Department of Mathematics and Statistics, University of Montreal, Montreal, Canada
| | - Jie-Ru Yu
- College of Resources and Environmental Sciences, Gansu Agricultural University, Lanzhou, PR China
| | - Shi-Chang Wang
- Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing, PR China
| | - Song-Jia Fan
- Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing, PR China
| | - Bo-Yu Zhang
- School of Mathematical Science, Beijing Normal University, Beijing, PR China.
| | - Yi Tao
- Key Laboratory of Animal Ecology and Conservation Biology, Centre for Computational and Evolutionary Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing, PR China.
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Wu B, Mao S, Wang J, Zhou D. Control of epidemics via social partnership adjustment. Phys Rev E 2017; 94:062314. [PMID: 28085324 PMCID: PMC7217516 DOI: 10.1103/physreve.94.062314] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2016] [Indexed: 11/07/2022]
Abstract
Epidemic control is of great importance for human society. Adjusting interacting partners is an effective individualized control strategy. Intuitively, it is done either by shortening the interaction time between susceptible and infected individuals or by increasing the opportunities for contact between susceptible individuals. Here, we provide a comparative study on these two control strategies by establishing an epidemic model with nonuniform stochastic interactions. It seems that the two strategies should be similar, since shortening the interaction time between susceptible and infected individuals somehow increases the chances for contact between susceptible individuals. However, analytical results indicate that the effectiveness of the former strategy sensitively depends on the infectious intensity and the combinations of different interaction rates, whereas the latter one is quite robust and efficient. Simulations are shown to verify our analytical predictions. Our work may shed light on the strategic choice of disease control.
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Affiliation(s)
- Bin Wu
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People's Republic of China
| | - Shanjun Mao
- School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China
| | - Jiazeng Wang
- Department of Mathematics, Beijing Technology and Business University, Beijing 100048, People's Republic of China
| | - Da Zhou
- School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, Xiamen University, Xiamen 361005, People's Republic of China
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Getz WM, Salter R, Seidel DP, van Hooft P. Sympatric speciation in structureless environments. BMC Evol Biol 2016; 16:50. [PMID: 26922946 PMCID: PMC4770699 DOI: 10.1186/s12862-016-0617-0] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2015] [Accepted: 02/12/2016] [Indexed: 11/10/2022] Open
Abstract
BACKGROUND Darwin and the architects of the Modern Synthesis found sympatric speciation difficult to explain and suggested it is unlikely to occur. Increasingly, evidence over the past few decades suggest that sympatric speciation can occur under ecological conditions that require at most intraspecific competition for a structured resource. Here we used an individual-based population model with variable foraging strategies to study the evolution of mating behavior among foraging strategy types. Initially, individuals were placed at random on a structureless resource landscape, with subsequent spatial variation induced through foraging activity itself. The fitness of individuals was determined by their biomass at the end of each generational cycle. The model incorporates three diallelic, codominant foraging strategy genes, and one mate-choice or m-trait (i.e. incipient magic trait) gene, where the latter is inactive when random mating is assumed. RESULTS Under non-random mating, the m-trait gene promotes increasing levels of either disassortative or assortative mating when the frequency of m respectively increases or decreases from 0.5. Our evolutionary simulations demonstrate that, under initial random mating conditions, an activated m-trait gene evolves to promote assortative mating because the system, in trying to fit a multipeak adaptive landscape, causes heterozygous individuals to be less fit than homozygous individuals. CONCLUSION Our results extend our theoretical understanding that sympatric speciation can evolve under nicheless or gradientless resource conditions: i.e. the underlying resource is monomorphic and initially spatially homogeneous. Further the simplicity and generality of our model suggests that sympatric speciation may be more likely than previously thought to occur in mobile, sexually-reproducing organisms.
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Affiliation(s)
- Wayne M Getz
- Department ESPM, University of California, Berkeley, CA, 94720-3114, USA. .,School of Mathematical Sciences, University of KwaZulu-Natal, PB X54001, Durban, 4000, South Africa.
| | - Richard Salter
- Computer Science Department, Oberlin College, Oberlin, OH, 44074, USA.
| | - Dana Paige Seidel
- Department ESPM, University of California, Berkeley, CA, 94720-3114, USA.
| | - Pim van Hooft
- Resource Ecology Group, Wageningen University, Droevendaalsesteeg 3a, 6708 PB, Wageningen, The Netherlands.
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The influence of demographic stochasticity on evolutionary dynamics and stability. Theor Popul Biol 2013; 88:47-56. [DOI: 10.1016/j.tpb.2013.06.005] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2012] [Revised: 04/30/2013] [Accepted: 06/06/2013] [Indexed: 11/19/2022]
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Abstract
We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth-death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.
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Wang JZ, Qian M, Qian H. Circular stochastic fluctuations in SIS epidemics with heterogeneous contacts among sub-populations. Theor Popul Biol 2012; 81:223-31. [PMID: 22273833 DOI: 10.1016/j.tpb.2012.01.002] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/07/2011] [Revised: 01/06/2012] [Accepted: 01/06/2012] [Indexed: 11/29/2022]
Abstract
The conceptual difference between equilibrium and non-equilibrium steady state (NESS) is well established in physics and chemistry. This distinction, however, is not widely appreciated in dynamical descriptions of biological populations in terms of differential equations in which fixed point, steady state, and equilibrium are all synonymous. We study NESS in a stochastic SIS (susceptible-infectious-susceptible) system with heterogeneous individuals in their contact behavior represented in terms of subgroups. In the infinite population limit, the stochastic dynamics yields a system of deterministic evolution equations for population densities; and for very large but finite systems a diffusion process is obtained. We report the emergence of a circular dynamics in the diffusion process, with an intrinsic frequency, near the endemic steady state. The endemic steady state is represented by a stable node in the deterministic dynamics. As a NESS phenomenon, the circular motion is caused by the intrinsic heterogeneity within the subgroups, leading to a broken symmetry and time irreversibility.
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Affiliation(s)
- Jia-Zeng Wang
- Department of Mathematics, Beijing Technology and Business University, Beijing 100048, PR China.
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Zhou D, Qian H. Fixation, transient landscape, and diffusion dilemma in stochastic evolutionary game dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:031907. [PMID: 22060403 DOI: 10.1103/physreve.84.031907] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2011] [Revised: 04/12/2011] [Indexed: 05/31/2023]
Abstract
Agent-based stochastic models for finite populations have recently received much attention in the game theory of evolutionary dynamics. Both the ultimate fixation and the pre-fixation transient behavior are important to a full understanding of the dynamics. In this paper, we study the transient dynamics of the well-mixed Moran process through constructing a landscape function. It is shown that the landscape playing a central theoretical "device" that integrates several lines of inquiries: the stable behavior of the replicator dynamics, the long-time fixation, and continuous diffusion approximation associated with asymptotically large population. Several issues relating to the transient dynamics are discussed: (i) multiple time scales phenomenon associated with intra- and inter-attractoral dynamics; (ii) discontinuous transition in stochastically stationary process akin to Maxwell construction in equilibrium statistical physics; and (iii) the dilemma diffusion approximation facing as a continuous approximation of the discrete evolutionary dynamics. It is found that rare events with exponentially small probabilities, corresponding to the uphill movements and barrier crossing in the landscape with multiple wells that are made possible by strong nonlinear dynamics, plays an important role in understanding the origin of the complexity in evolutionary, nonlinear biological systems.
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Affiliation(s)
- Da Zhou
- School of Mathematical Sciences, Peking University, Beijing 100871, China.
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