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Dynamical reciprocity in interacting games: Numerical results and mechanism analysis

Rizhou Liang, Qinqin Wang, Jiqiang Zhang, Guozhong Zheng, Lin Ma, and Li Chen
Phys. Rev. E 105, 054302 – Published 3 May 2022
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  • INTRODUCTION
  • GENERAL FORMULATION
  • PRELIMINARY RESULTS FOR TWO INTERACTING…
  • A MEAN-FIELD ANALYSIS
  • DYNAMICAL MECHANISM
  • ROBUSTNESS STUDIES
  • INTERACTING GAMES ON COMPLEX NETWORKS
  • MORE GAMES
  • SUMMARY AND DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We study the evolution of two mutually interacting pairwise games on different topologies. On two-dimensional square lattices, we reveal that the game-game interaction can promote the cooperation prevalence in all cases, and the cooperation-defection phase transitions even become absent and fairly high cooperation is expected when the interaction becomes very strong. A mean-field theory is developed that points out dynamical routes arising therein. Detailed analysis shows indeed that there are rich categories of interactions in either the individual or bulk scenario: invasion, neutral, and catalyzed types; their combination puts cooperators at a persistent advantage position, which boosts the cooperation. The robustness of the revealed reciprocity is strengthened by the studies of model variants, including the public goods game, asymmetrical or time-varying interactions, games of different types, games with timescale separation, different updating rules, etc. The structural complexities of the underlying population, such as Newman-Watts small world networks, Erdős-Rényi random networks, and Barabási-Albert networks, also do not alter the working of the dynamical reciprocity. In particular, as the number of games engaged increases, the cooperation level continuously improves in general. However, our analysis shows that the dynamical reciprocity works only in structured populations, otherwise the game-game interaction has no any impact on the cooperation at all. In brief, our work uncovers a cooperation mechanism in the structured populations, which indicates the great potential for human cooperation since concurrent issues are so often seen in the real world.

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    • Received 3 February 2021
    • Revised 28 February 2022
    • Accepted 12 April 2022

    DOI:https://doi.org/10.1103/PhysRevE.105.054302

    ©2022 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Patterns in complex systemsPopulation dynamicsSocial systems
    NetworksInterdisciplinary PhysicsStatistical Physics & Thermodynamics

    Authors & Affiliations

    Rizhou Liang1, Qinqin Wang1, Jiqiang Zhang2,3, Guozhong Zheng1, Lin Ma1, and Li Chen1,4,*

    • 1School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, People's Republic of China
    • 2School of Physics and Electronic-Electrical Engineering, Ningxia University, Yinchuan 750021, People's Republic of China
    • 3Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100191, People's Republic of China
    • 4Robert Koch-Institute, Nordufer 20, 13353 Berlin, Germany

    • *chenl@snnu.edu.cn

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    Issue

    Vol. 105, Iss. 5 — May 2022

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    Images

    • Figure 1
      Figure 1

      Modeling two interacting games. (a) Consider a group of networked players: they play two games G1,2 simultaneously, and two payoffs are obtained accordingly, which can be interpreted as the fitness in their evolution. (b) When two neighboring individuals, say, players x and y, are to update their strategies with respect to a given game (e.g., game G1 here), the update depends on not only the payoffs obtained in G1 (with weight 1−θ) but also the one in the other game G2 (with θ), and this combination is termed the effective payoff; see Eq. (3).

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    • Figure 2
      Figure 2

      The evolution of two symmetrically interacting PD games on the 2D square lattice. (a) The phase transitions of cooperation prevalence regarding game G1 (fCG1=fCC+fCD) for different interaction strengths θ, and the temptation b is the control parameter. (b) Time series for b=1.1. Note that fCG2≈fCG1 due to the symmetry and is not shown. Parameters are L=1024 and K=0.1, the random initial condition for both games, and each point is averaged over 50 ensembles after transient in (a).

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    • Figure 3
      Figure 3

      Color-coded fraction of cooperators regarding the first game (fCG1=fCC+fCD) for the general pairwise game within the S−T parameter space with θ=0, 0.5, and 1 on the 2D square lattice, respectively shown in (a)–(c). Due to the symmetry, fCG2≈fCG1. Four quadrants correspond to four different games (defined in Sec. 2). Parameters: R=1, P=0, L=128.

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    • Figure 4
      Figure 4

      Color-coded fraction of cooperation prevalence fCG1 in θ−r̂ parameter space for two symmetrically interacting public goods games. r̂=r/(k+1) is the normalized gain factor, and k+1 is the number of games the individual is evolved. Also fCG2≈fCG1 due to the symmetry. Here L=128 and K=0.5.

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    • Figure 5
      Figure 5

      The cooperator fraction fCG1 for the two interacting pairwise games in the well-mixed population within S−T parameter space for θ=0.5. The resulting cooperation prevalence exactly corresponds to the solution of single game; the less significant bistability in SH game is simply due to the random initial conditions, where fC≈fD≈1/2. Due to the symmetry, fCG1≈fCG2 in all cases, except very few case along the line S=T−1 in SH, depending on their initial conditions. Other parameters: R=1, P=0, N=214.

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    • Figure 6
      Figure 6

      Time series of all four fractions for two symmetrically cross-played (θ=1) PD games starting from random initial conditions. All six binary compositions are included: (a) CC-DD, (b) CC-CD, (c) CC-DC, (d) CD-DD, (e) DC-DD, (f) DC-DC. (a), (b)–(e), and (f) correspond to category (i)–(iii), respectively. Parameters: S=0, T=1.1, R=1, P=0, L=1024 for the 2D square lattice.

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    • Figure 7
      Figure 7

      Time series of all four fractions for two symmetrically cross-played (θ=1) PD games starting from half-half patched initial conditions. (a) CC-DD, (b) CC-CD, (c) CC-DC, (d) CD-DD, (e) DC-DD, (f) CD-DC. (a), (b)–(e), and (f) correspond to category (i)–(iii), respectively. Parameters: S=0, T=1.1, R=1, P=0, L=1024 for the 2D square lattice.

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    • Figure 8
      Figure 8

      Time evolution of interacting pair proportions for two symmetrically interacting PD games. Both random (a, c) and half-half patched (b, d) initial conditions are considered. Panels (a) and (b) and panels (c) and (d) correspond to θ=0.5 and 1, respectively. Pr=PCD−DC/PCC−DD to compares the relative proportion for CD–DC and CD–DD pairs. Parameters: S=0, T=1.1, R=1, P=0, L=1024 for the 2D square lattice.

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    • Figure 9
      Figure 9

      Cluster size distribution for two symmetrically interacting PD games starting from random initial conditions. (a) and (b) are time evolution of the average cluster size of the four species for θ=0.5 and 1, respectively. (c) and (d) are probability function distributions of cluster size at t=1000 for θ=0.5 and 1, respectively. In (d), there is a giant CC cluster with the size comparable to the population size (∼N), which is not shown. Parameters: S=0, T=1.1, R=1, P=0, L=1024 for the 2D square lattice.

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    • Figure 10
      Figure 10

      Cluster compactness analysis. (a, b) Time evolution of compactness of the four species for θ=0.5 and 1, respectively. (c, d) Probability function distributions of cluster compactness at t=1000 for θ=0.5 and 1, respectively. Same settings as in Fig. 9.

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    • Figure 11
      Figure 11

      Color-coded cooperation prevalences for two asymmetrically interacting PD games on a 2D square lattice within θ1−θ2 parameter space; (a) and (b) are for game G1,2, respectively. Other parameters: S=0, T=1.1, R=1, P=0, L=128.

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    • Figure 12
      Figure 12

      Color-coded cooperation prevalence in S−T parameter space for two interacting games; one is PD and the other is SD, on the 2D square lattice, for θ=0,0.5,1 (a–c). They have opposite sign in S, e.g., the SD at top left corner (T=1,S=1) interacts with PD with (T=1,S=−1) at the bottom left. Other parameters: R=1, P=0, and L=128 for the lattice.

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    • Figure 13
      Figure 13

      The impact of timescale separation of the two interacting PD games. The cooperation prevalence fc vs the timescale ratio Tr for θ=0.5 (a) and θ=1 (b). No cooperation is seen for θ=0 for the given parameters. Other parameters: S=0, T=1.1, R=1, P=0, L=1024 for the 2D square lattice.

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    • Figure 14
      Figure 14

      Cooperation phase diagram for fCG1 with other four update rules with AU, for θ=0,0.5,1. The update rules are the following: replicator rule (first row, a–c), multiple replicator rule (second row, d–f), Moran rule (third row, g–i), and unconditional imitation (bottom row, j–l). Random initial conditions are adopted. The four numbers are the average cooperation prevalence for the corresponding quadrants. fCG2≈fCG1 due to the symmetrical settings. Parameters: R=1, P=0, and L=128 for the 2D square lattice.

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    • Figure 15
      Figure 15

      Cooperation phase diagram with other four update rules but with SU, for three interaction strengths θ=0,0.5,1. Other settings are exactly the same as in Fig. 14.

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    • Figure 16
      Figure 16

      Two symmetrically interacting PD games on Newman-Watts small-world networks. (a) The cooperation prevalence fc vs the temptation T=b for a couple of interaction strengths θ. (b) fc vs ϕ for three temptation values b with θ=0.5. fCG2≈fCG1 due to the symmetrical settings. Parameters: P=S=0, R=1 for both games, the network size N=220 with ϕ=0.01.

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    • Figure 17
      Figure 17

      Two symmetrically interacting PD games on Erdős-Rényi random networks. (a) The cooperation prevalence fc vs the temptation T=b for a couple of interaction strengths θ. (b) fc vs the average degree for three temptation values b with θ=0.5. fCG2≈fCG1 due to the symmetrical settings. Parameters: P=S=0, R=1 for both games, the network size N=220 with 〈k〉=4.

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    • Figure 18
      Figure 18

      Two symmetrically interacting PD games on BA scale-free networks. The cooperation prevalence fc vs the temptation T=b for a couple of interaction strengths θ. fCG2≈fCG1 due to the symmetrical settings. Parameters: P=S=0, R=1 for both games, the network size N=220 with 〈k〉=4. K=0.025 due to the average payoff scheme; the flipping noise pf=0.01.

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    • Figure 19
      Figure 19

      Three interacting games. (a) Phase transitions of the three cooperation prevalences vs the interaction strength θ′ by assuming θ1′=θ2′=θ′. (b) Color-coded fraction of cooperators regarding game G1 (fCG1=∑X,Y∈S1fCXY) within the interaction space θ1′−θ2′. Due to the restriction of θ1′+θ2′≤1, the upper right half is unphysical. Note that the cooperation prevalence for the three games fCG1,2,3 is approximately identical due to the symmetrical setting, as shown in (a). Other parameters: S=0, R=1, P=0, T=1.1, L=1024 for the 2D square lattice.

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