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Király B, Szabó G. Evolutionary games combining two or three pair coordinations on a square lattice. Phys Rev E 2017; 96:042101. [PMID: 29347482 DOI: 10.1103/physreve.96.042101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2017] [Indexed: 06/07/2023]
Abstract
We study multiagent logit-rule-driven evolutionary games on a square lattice whose pair interactions are composed of a maximal number of nonoverlapping elementary coordination games describing Ising-type interactions between just two of the available strategies. Using Monte Carlo simulations we investigate the macroscopic noise-level-dependent behavior of the two- and three-pair games and the critical properties of the continuous phase transtitions these systems exhibit. The four-strategy game is shown to be equivalent to a system that consists of two independent and identical Ising models.
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Affiliation(s)
- Balázs Király
- Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary
| | - György Szabó
- Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary
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Király B, Szabó G. Evolutionary games with coordination and self-dependent interactions. Phys Rev E 2017; 95:012303. [PMID: 28208366 DOI: 10.1103/physreve.95.012303] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2016] [Indexed: 06/06/2023]
Abstract
Multistrategy evolutionary games are studied on a square lattice when the pair interactions are composed of coordinations between strategy pairs and an additional term with self-dependent payoff. We describe a method for determining the strength of each elementary coordination component in n-strategy potential games. Using analytical and numerical methods, the presence and absence of Ising-type order-disorder phase transitions are studied when a single pair coordination is extended by some types of self-dependent elementary games. We also introduce noise-dependent three-strategy equivalents of the n-strategy elementary coordination games.
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Affiliation(s)
- Balázs Király
- Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary
| | - György Szabó
- Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
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Szabó G, Bodó KS, Samani KA. Separation of cyclic and starlike hierarchical dominance in evolutionary matrix games. Phys Rev E 2017; 95:012320. [PMID: 28208502 DOI: 10.1103/physreve.95.012320] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/10/2016] [Indexed: 06/06/2023]
Abstract
We study antisymmetric components of matrices characterizing pair interactions in multistrategy evolutionary games. Based on the dyadic decomposition of matrices we distinguish cyclic and starlike hierarchical dominance in the appropriate components. In the symmetric matrix games the strengths of these elementary components are determined. The general features and intrinsic symmetries of these interactions are represented by directed graphs. It is found that the variation of a single matrix component modifies simultaneously the strengths of two starlike hierarchical basis games and many other independent rock-paper-scissors type cyclic basis games. The application of the related concepts is illustrated by discussing the three-strategy voluntary prisoner's dilemma.
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Affiliation(s)
- György Szabó
- Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
| | - Kinga S Bodó
- Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary
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Szabó G, Király B. Extension of a spatial evolutionary coordination game with neutral options. Phys Rev E 2016; 93:052108. [PMID: 27300831 DOI: 10.1103/physreve.93.052108] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2016] [Indexed: 11/07/2022]
Abstract
The multiagent evolutionary games on a lattice are equivalent to a kinetic Ising model if the uniform pair interactions are defined by a two-strategy coordination game and the logit rule controls the strategy updates. Now we extend this model by allowing the players to use additional neutral strategies that provide zero payoffs for both players if one of them selects one of the neutral strategies. In the resulting n-strategy evolutionary games the analytical methods and numerical simulations indicate continuous order-disorder phase transitions when increasing the noise level if n does not exceed a threshold value. For larger n the system exhibits a first order phase transition at a critical noise level decreasing asymptotically as 2/ln(n).
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Affiliation(s)
- György Szabó
- Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
| | - Balázs Király
- Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary
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Szabó G, Bodó KS, Allen B, Nowak MA. Four classes of interactions for evolutionary games. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:022820. [PMID: 26382467 DOI: 10.1103/physreve.92.022820] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2015] [Indexed: 06/05/2023]
Abstract
The symmetric four-strategy games are decomposed into a linear combination of 16 basis games represented by orthogonal matrices. Among these basis games four classes can be distinguished as it is already found for the three-strategy games. The games with self-dependent (cross-dependent) payoffs are characterized by matrices consisting of uniform rows (columns). Six of 16 basis games describe coordination-type interactions among the strategy pairs and three basis games span the parameter space of the cyclic components that are analogous to the rock-paper-scissors games. In the absence of cyclic components the game is a potential game and the potential matrix is evaluated. The main features of the four classes of games are discussed separately and we illustrate some characteristic strategy distributions on a square lattice in the low noise limit if logit rule controls the strategy evolution. Analysis of the general properties indicates similar types of interactions at larger number of strategies for the symmetric matrix games.
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Affiliation(s)
- György Szabó
- Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
| | - Kinga S Bodó
- Budapest University of Technology and Economics, Budafoki t 8, H-1111 Budapest, Hungary
| | - Benjamin Allen
- Program for Evolutionary Dynamics, Harvard University, One Brattle Square, Cambridge, Massachusetts 02138, USA
| | - Martin A Nowak
- Program for Evolutionary Dynamics, Harvard University, One Brattle Square, Cambridge, Massachusetts 02138, USA
- Department of Mathematics, Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138, USA
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Vukov J, Varga L, Allen B, Nowak MA, Szabó G. Payoff components and their effects in a spatial three-strategy evolutionary social dilemma. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012813. [PMID: 26274231 DOI: 10.1103/physreve.92.012813] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2015] [Indexed: 06/04/2023]
Abstract
We study a three-strategy spatial evolutionary prisoner's dilemma game with imitation and logit update rules. Players can follow the always-cooperating, always-defecting or the win-stay-lose-shift (WSLS) strategies and gain their payoff from games with their direct neighbors on a square lattice. The friendliness parameter of the WSLS strategy-characterizing its cooperation probability in the first round-tunes the cyclic component of the game determining whether the game can be characterized by a potential. We measured and calculated the phase diagrams of the system for a wide range of parameters. When the game is a potential game and the logit rule is applied, the theoretically predicted phase diagram agrees very well with the simulation results. Surprisingly, this phase diagram can be accurate even in the nonpotential case if there are only two surviving strategies in the stationary state; this result harmonizes with the fact that all 2×2 games are potential games. For the imitation dynamics, we found that the effects of spatiality combined with the presence of two cooperative strategies are so strong that they suppress even substantial changes in the payoff matrix, thus the phase diagrams are independent of the cyclic component's intensity. At the same time, this type of strategy update mechanism supports the formation of cooperative clusters that results in a cooperative society in a wider parameter range compared to the logit dynamics.
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Affiliation(s)
- Jeromos Vukov
- Research Center for Educational and Network Studies, Centre for Social Sciences, Hungarian Academy of Sciences, P. O. Box 20, H-1250 Budapest, Hungary
| | - Levente Varga
- Babeş-Bolyai University, Faculty of Physics, RO-400084 Cluj-Napoca, Romania
| | - Benjamin Allen
- Program for Evolutionary Dynamics, Harvard University, One Brattle Square, Cambridge, Massachusetts 02138, USA
| | - Martin A Nowak
- Program for Evolutionary Dynamics, Harvard University, One Brattle Square, Cambridge, Massachusetts 02138, USA
- Department of Mathematics, Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138, USA
| | - György Szabó
- Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary
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Szabó G, Szolnoki A. Congestion phenomena caused by matching pennies in evolutionary games. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032110. [PMID: 25871057 DOI: 10.1103/physreve.91.032110] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/18/2014] [Indexed: 06/04/2023]
Abstract
Evolutionary social dilemma games are extended by an additional matching-pennies game that modifies the collected payoffs. In a spatial version players are distributed on a square lattice and interact with their neighbors. First, we show that the matching-pennies game can be considered as the microscopic force of the Red Queen effect that breaks the detailed balance and induces eddies in the microscopic probability currents if the strategy update is analogous to the Glauber dynamics for the kinetic Ising models. The resulting loops in probability current breaks symmetry between the chessboardlike arrangements of strategies via a bottleneck effect occurring along the four-edge loops in the microscopic states. The impact of this congestion is analogous to the application of a staggered magnetic field in the Ising model; that is, the order-disorder critical transition is wiped out by noise. It is illustrated that the congestion induced symmetry breaking can be beneficial for the whole community within a certain region of parameters.
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Affiliation(s)
- György Szabó
- Institute of Technical Physics and Materials Science, Research Centre for Natural Sciences, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
- Regional Knowledge Centre, Eötvös University, Irányi Dániel utca 4, H-8000 Székesfehérvár, Hungary
| | - Attila Szolnoki
- Institute of Technical Physics and Materials Science, Research Centre for Natural Sciences, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
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Yang HX, Wu ZX, Rong Z, Lai YC. Peer pressure: enhancement of cooperation through mutual punishment. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:022121. [PMID: 25768472 DOI: 10.1103/physreve.91.022121] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/20/2014] [Indexed: 06/04/2023]
Abstract
An open problem in evolutionary game dynamics is to understand the effect of peer pressure on cooperation in a quantitative manner. Peer pressure can be modeled by punishment, which has been proved to be an effective mechanism to sustain cooperation among selfish individuals. We investigate a symmetric punishment strategy, in which an individual will punish each neighbor if their strategies are different, and vice versa. Because of the symmetry in imposing the punishment, one might intuitively expect the strategy to have little effect on cooperation. Utilizing the prisoner's dilemma game as a prototypical model of interactions at the individual level, we find, through simulation and theoretical analysis, that proper punishment, when even symmetrically imposed on individuals, can enhance cooperation. Also, we find that the initial density of cooperators plays an important role in the evolution of cooperation driven by mutual punishment.
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Affiliation(s)
- Han-Xin Yang
- Department of Physics, Fuzhou University, Fuzhou 350108, China
| | - Zhi-Xi Wu
- Institute of Computational Physics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Zhihai Rong
- CompleX Lab, Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 610054, China
- Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tucson, Arizona 85287, USA
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Varga L, Vukov J, Szabó G. Self-organizing patterns in an evolutionary rock-paper-scissors game for stochastic synchronized strategy updates. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042920. [PMID: 25375580 DOI: 10.1103/physreve.90.042920] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/15/2014] [Indexed: 06/04/2023]
Abstract
We study a spatial evolutionary rock-paper-scissors game with synchronized strategy updating. Players gain their payoff from games with their four neighbors on a square lattice and can update their strategies simultaneously according to the logit rule, which is the noisy version of the best-response dynamics. For the synchronized strategy update two types of global oscillations (with an ordered strategy arrangement and periods of three and six generations) can occur in this system in the zero noise limit. At low noise values, all nine oscillating phases are present in the system by forming a self-organizing spatial pattern due to the comprising invasion and speciation processes along the interfaces separating the different domains.
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Affiliation(s)
- Levente Varga
- Institute of Technical Physics and Materials Science, Research Centre for Natural Sciences, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary and Babeş-Bolyai University, RO-400084 Cluj-Napoca, Romania
| | - Jeromos Vukov
- Institute of Technical Physics and Materials Science, Research Centre for Natural Sciences, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
| | - György Szabó
- Institute of Technical Physics and Materials Science, Research Centre for Natural Sciences, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
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