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Effect of reciprocity mechanisms on evolutionary dynamics in feedback-evolving games

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Abstract

The interplay between the strategies and environments has been shown to affect the evolution of cooperative behavior. Typically, it is assumed that the strategy-related changeable game environment alters the payoffs of strategic interactions, while the reciprocity mechanisms among interactions are usually ignored. Here we respectively study the feedback-evolving games with the direct and indirect mechanisms. This extension is facilitated by assuming the essential properties of reciprocity interactions are incorporated into the linear state-dependent payoff matrix. By the replicator dynamic process, it is found that except for the heteroclinic cycle or internal equilibrium present in the previous model, full cooperation and the highest level of the environment state can be dominant. Furthermore, by exploring the evolutionary dynamics in the local reciprocity-embedded feedback subsystems, we further stress that the most expected system states will be realized so long as cooperation is favored through the reciprocity manner in the circumstance that defection dominates. Even in terms of the internal equilibrium, the higher environmental state level can be enabled without the loss of cooperation. The results may explain the effectiveness of the reciprocity mechanism in avoiding the traps of social dilemmas within the time-invariant game interaction.

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Data availability

The datasets generated during the current study are not publicly available but are available from the corresponding author on reasonable request.

References

  1. Hofbauer, J., Sigmund, K.: Evolutionary games and population dynamics. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  2. Nowak, M.A., Sasaki, A., Taylor, C., Fudenberg, D.: Emergence of cooperation and evolutionary stability in finite populations. Nature 428(6983), 646–650 (2004)

    Google Scholar 

  3. Fang, L., Shi, G., Wang, L., Li, Y., et al.: Incentive mechanism for cooperative authentication: An evolutionary game approach. Inform. Sciences. 527, 369–381 (2020)

    MathSciNet  Google Scholar 

  4. Liang, H., Cui, Y., Ren, X., Wang, X.: Almost sure exponential stability of two-strategy evolutionary games with multiplicative noise. Inform. Sciences. 579, 888–903 (2021)

    MathSciNet  Google Scholar 

  5. Jusup, M., Holme, P., Kanazawa, K., Takayasu, M., et al.: Social physics. Phys. Rep. 948, 1–148 (2022)

    MathSciNet  Google Scholar 

  6. Hardin, G.: The Tragedy of the Commons. Science 162(3859), 1243–1248 (1968)

    Google Scholar 

  7. Sigmund, K., Hauert, C., Nowak, M.A.: Reward and punishment. Proc. Natl. Acad. Sci. U.S.A. 98(19), 10757–10762 (2001)

    Google Scholar 

  8. Lohse, J., Waichman, I.: The effects of contemporaneous peer punishment on cooperation with the future. Nat. Commun. 11(1), 1815 (2020)

    Google Scholar 

  9. Hauert, C., Monte, S.D., Hofbauer, J., Sigmund, K.: Volunteering as red queen mechanism for cooperation in public goods games. Science 296(5570), 1129–1132 (2002)

    Google Scholar 

  10. Santos, F.C., Santos, M.D., Pacheco, J.M.: Social diversity promotes the emergence of cooperation in public goods games. Nature 454(7201), 213–216 (2008)

    Google Scholar 

  11. Liu, L., Chen, X., Perc, M.: Evolutionary dynamics of cooperation in the public goods game with pool exclusion strategies. Nonlinear Dynam. 97(1), 749–766 (2019)

    Google Scholar 

  12. Quan, J., Chen, X., Yang, W., Wang, X.: Cooperation dynamics in spatial public goods games with graded punishment mechanism. Nonlinear Dynam. 111(9), 8837–8851 (2023)

    Google Scholar 

  13. Szolnoki, A., Perc, M., Mobilia, M.: Facilitators on networks reveal optimal interplay between information exchange and reciprocity. Phys. Rev. E 89(4), 042802 (2014)

    Google Scholar 

  14. Nowak, M.A.: Five rules for the evolution of cooperation. Science 314(5805), 1560–1563 (2006)

    Google Scholar 

  15. Taylor, C., Nowak, M.A.: Transforming the dilemma. Evolution 61(10), 2281–2292 (2007)

    Google Scholar 

  16. Nowak, M., Sigmund, K.: A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner’s Dilemma game. Nature 364(6432), 56–58 (1993)

    Google Scholar 

  17. Hilbe, C., Martinez-Vaquero, L.A., Chatterjee, K., Nowak, M.A.: Memory-n strategies of direct reciprocity. Proc. Natl. Acad. Sci. U.S.A. 114(18), 4715–4720 (2017)

    Google Scholar 

  18. Nowak, M.A., Sigmund, K.: Tit for tat in heterogeneous populations. Nature 355(6357), 250–253 (1992)

    Google Scholar 

  19. Chiong, R., Kirley, M.: Effects of iterated interactions in multiplayer spatial evolutionary games. IEEE Trans. Evol. Comput. 16(4), 537–555 (2012)

    Google Scholar 

  20. Nowak, M.A., Sigmund, K.: Evolution of indirect reciprocity by image scoring. Nature 393(6685), 573–577 (1998)

    Google Scholar 

  21. Quan, J., Nie, J., Chen, W., Wang, X.: Keeping or reversing social norms promote cooperation by enhancing indirect reciprocity. Chaos Soliton. Fract. 158, 111986 (2022)

    Google Scholar 

  22. Smith, J.M.: Group selection and kin selection. Nature 201(4924), 1145–1147 (1964)

    Google Scholar 

  23. Nowak, M.A., May, R.M.: Evolutionary games and spatial chaos. Nature 359(6398), 826–829 (1992)

    Google Scholar 

  24. Allen, B., Lippner, G., Chen, Y.T., Fotouhi, B., et al.: Evolutionary dynamics on any population structure. Nature 544(7649), 227–230 (2017)

    Google Scholar 

  25. Wang, Z., Szolnoki, A., Perc, M.: Interdependent network reciprocity in evolutionary games. Sci. Rep. 3, 1183 (2013)

    Google Scholar 

  26. Traulsen, A., Nowak, M.A.: Evolution of cooperation by multilevel selection. Proc. Natl. Acad. Sci. U.S.A. 103(29), 10952–10955 (2006)

    Google Scholar 

  27. Szolnoki, A., Perc, M.: Correlation of positive and negative reciprocity fails to confer an evolutionary advantage: phase transitions to elementary strategies. Phys. Rev. X 3(4), 041021 (2013)

    Google Scholar 

  28. Hopkins, E.: A note on best response dynamics. Games Econ. Behav. 29(1–2), 138–150 (1999)

    MathSciNet  Google Scholar 

  29. Farahbakhsh, I., Bauch, C.T., Anand, M.: Best response dynamics improve sustainability and equity outcomes in common-pool resources problems, compared to imitation dynamics. J. Theor. Biol. 509, 110476 (2021)

    MathSciNet  Google Scholar 

  30. Taylor, D., P., Jonker, L.B.: Evolutionary stable strategies and game dynamics. Math. Biosci. 40, 145–156 (1978)

    MathSciNet  Google Scholar 

  31. Perc, M., Wang, Z.: Heterogeneous aspirations promote cooperation in the prisoner’s dilemma game. PLoS ONE 5(12), e15117 (2010)

    Google Scholar 

  32. Wu, B., Zhou, L.: Individualised aspiration dynamics: calculation by proofs. Plos Comput. Biol. 14(9), e1006035 (2018)

    Google Scholar 

  33. Zhou, L., Wu, B., Du, J., Wang, L.: Aspiration dynamics generate robust predictions in heterogeneous populations. Nat. Commun. 12(1), 3250 (2021)

    Google Scholar 

  34. Ashcroft, P., Altrock, P.M., Galla, T.: Fixation in finite populations evolving in fluctuating environments. J. R. Soc. Interface. 11(100), 20140663 (2014)

    Google Scholar 

  35. Mullon, C., Keller, L., Lehmann, L.: Social polymorphism is favoured by the co-evolution of dispersal with social behaviour. Nat. Ecol. Evol. 2(1), 132–140 (2018)

    Google Scholar 

  36. Bauch, C.T., Earn, D.J.D.: Vaccination and the theory of games. Proc. Natl. Acad. Sci. U.S.A. 101(36), 13391–13394 (2004)

    MathSciNet  Google Scholar 

  37. Taitelbaum, A., West, R., Assaf, M., Mobilia, M.: Population dynamics in a changing environment: random versus periodic switching. Phys. Rev. Lett. 125(4), 048105 (2020)

    MathSciNet  Google Scholar 

  38. Tilman, A.R., Watson, J.R., Levin, S.: Maintaining cooperation in social-ecological systems: effective bottom-up management often requires sub-optimal resource use. Theor. Ecol. 10(2), 155–165 (2017)

    Google Scholar 

  39. Estrela, S., Libby, E., Van Cleve, J., Debarre, F., et al.: Environmentally mediated social dilemmas. Trends Ecol. Evol. 34(1), 6–18 (2019)

    Google Scholar 

  40. Weitz, J.S., Eksin, C., Paarporn, K., Brown, S.P., et al.: An oscillating tragedy of the commons in replicator dynamics with game-environment feedback. Proc. Natl. Acad. Sci. U.S.A. 113(47), E7518–E7525 (2016)

    Google Scholar 

  41. Tilman, A.R., Plotkin, J.B., Akcay, E.: Evolutionary games with environmental feedbacks. Nat. Commun. 11(1), 915 (2020)

    Google Scholar 

  42. Wang, X., Fu, F.: Eco-evolutionary dynamics with environmental feedback: cooperation in a changing world. Europhys. Lett. 132(1), 10001 (2020)

    Google Scholar 

  43. Chen, X., Szolnoki, A.: Punishment and inspection for governing the commons in a feedback-evolving game. Plos Comput. Biol. 14(7), e1006347 (2018)

    Google Scholar 

  44. Yan, F., Chen, X., Qiu, Z., Szolnoki, A.: Cooperator driven oscillation in a time-delayed feedback-evolving game. New J. Phys. 23(5), 053017 (2021)

    MathSciNet  Google Scholar 

  45. Shao, Y., Wang, X., Fu, F.: Evolutionary dynamics of group cooperation with asymmetrical environmental feedback. Europhys. Lett. 126(4), 40005 (2019)

    Google Scholar 

  46. Das Bairagya, J., Mondal, S.S., Chowdhury, D., Chakraborty, S.: Game-environment feedback dynamics in growing population: Effect of finite carrying capacity. Phys. Rev. E 104(4), 044407 (2021)

    MathSciNet  Google Scholar 

  47. Cao, L., Wu, B.: Eco-evolutionary dynamics with payoff-dependent environmental feedback. Chaos Soliton. Fract. 150, 111088 (2021)

    MathSciNet  Google Scholar 

  48. Li, B., Wu, Z., Guan, J.: Alternating rotation of coordinated and anti-coordinated action due to environmental feedback and noise. Chaos Soliton. Fract. 164, 112689 (2022)

    MathSciNet  Google Scholar 

  49. Gong, L., Yao, W., Gao, J., Cao, M.: Limit cycles analysis and control of evolutionary game dynamics with environmental feedback. Automatica 145, 110536 (2022)

    MathSciNet  Google Scholar 

  50. Shu, L., Fu, F.: Eco-evolutionary dynamics of bimatrix games. P. Roy. Soc. A. 478(2267), 20220567 (2022)

    MathSciNet  Google Scholar 

  51. Hauert, C., Saade, C., McAvoy, A.: Asymmetric evolutionary games with environmental feedback. J. Theor. Biol. 462, 347–360 (2019)

    MathSciNet  Google Scholar 

  52. Arefin, M.R., Tanimoto, J.: Imitation and aspiration dynamics bring different evolutionary outcomes in feedback-evolving games. P. Roy. Soc. A. 477(2251), 20210240 (2021)

    MathSciNet  Google Scholar 

  53. Szolnoki, A., Chen, X.: Environmental feedback drives cooperation in spatial social dilemmas. Europhys. Lett. 120(5), 58001 (2017)

    Google Scholar 

  54. Liu, H., Wang, X., Liu, L., Li, Z.: Co-evolutionary game dynamics of competitive cognitions and public opinion environment. Front. Phys. 9, 658130 (2021)

    Google Scholar 

  55. Liu, F., Wu, B.: Environmental quality and population welfare in markovian eco-evolutionary dynamics. Appl. Math. Comput. 431, 127309 (2022)

    MathSciNet  Google Scholar 

  56. Liu, Y., Cao, L., Wu, B.: General non-linear imitation leads to limit cycles in eco-evolutionary dynamics. Chaos Soliton. Fract. 165, 112817 (2022)

    MathSciNet  Google Scholar 

  57. Hilbe, C., Simsa, S., Chatterjee, K., Nowak, M.A.: Evolution of cooperation in stochastic games. Nature 559(7713), 246–249 (2018)

    Google Scholar 

  58. Su, Q., McAvoy, A., Wang, L., Nowak, M.A.: Evolutionary dynamics with game transitions. Proc. Natl. Acad. Sci. U.S.A. 116(51), 25398–25404 (2019)

    MathSciNet  Google Scholar 

  59. Wang, G., Su, Q., Wang, L.: Evolution of state-dependent strategies in stochastic games. J. Theor. Biol. 527, 110818 (2021)

    MathSciNet  Google Scholar 

  60. Wang, X., Zheng, Z., Fu, F.: Steering eco-evolutionary game dynamics with manifold control. P. Roy. Soc. A. 476(2233), 20190643 (2020)

    MathSciNet  Google Scholar 

  61. Paarporn, K., Eksin, C., Weitz, J.S., Wardi, Y.: Optimal control policies for evolutionary dynamics with environmental feedback. In: IEEE 57th Conference on Decision and Control (CDC), pp. 1905–1910. IEEE (2018)

  62. Xia, C., Wang, J., Perc, M., Wang, Z.: Reputation and reciprocity. Phys. Life. Rev. 46, 8–45 (2023)

    Google Scholar 

  63. Hilbe, C., Chatterjee, K., Nowak, M.A.: Partners and rivals in direct reciprocity. Nat. Hum. Behav. 2(7), 469–477 (2018)

    Google Scholar 

  64. Ohtsuki, H., Iwasa, Y.: How should we define goodness?—reputation dynamics in indirect reciprocity. J. Theor. Biol. 231(1), 107–120 (2004)

    MathSciNet  Google Scholar 

  65. Santos, F.P., Santos, F.C., Pacheco, J.M.: Social norm complexity and past reputations in the evolution of cooperation. Nature 555(7695), 242–245 (2018)

    Google Scholar 

  66. Jiang, Y., Wang, X., Liu, L., Wei, M., et al.: Nonlinear eco-evolutionary games with global environmental fluctuations and local environmental feedbacks. Plos Comput. Biol. 19(6), e1011269 (2023)

    Google Scholar 

  67. Capraro, V., Perc, M.: Mathematical foundations of moral preferences. J. R. Soc. Interface. 18(175), 20200880 (2021)

    Google Scholar 

Download references

Funding

This research was supported by the National Natural Science Foundation of China (No. 72371193, 72031009, 71871173), and the Chinese National Funding of Social Sciences (No.20&ZD058).

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Contributions

All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by X.M., J.Q. The first draft of the manuscript was written by X.M. and J.Q., and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Ji Quan.

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Appendices

Appendix 1

Stability analysis in the direct reciprocity embedded-feedback model

The conditions for stability of the fixed point in the feedback-evolving games with the direct reciprocity mechanism are analyzed. The coupled payoff matrix is defined by

$$ A_{1(n)} = (1 - n)\left[ {\begin{array}{*{20}c} {\frac{{R_{0} }}{1 - w}} & {S_{0} + \frac{{wP_{0} }}{1 - w}} \\ {T_{0} + \frac{{wP_{0} }}{1 - w}} & {\frac{{P_{0} }}{1 - w}} \\ \end{array} } \right] + n\left[ {\begin{array}{*{20}c} {\frac{{R_{1} }}{1 - w}} & {S_{1} + \frac{{wP_{1} }}{1 - w}} \\ {T_{1} + \frac{{wP_{1} }}{1 - w}} & {\frac{{P_{1} }}{1 - w}} \\ \end{array} } \right], $$
(14)

and the replicator dynamics for this system can be described as

$$ \begin{gathered} \dot{x} = x(1 - x)\left\{ {\left. \begin{gathered} (1 - n)\left[ {(\frac{{R_{0} + (1 - 2w)P_{0} }}{1 - w} - S_{0} - T_{0} )x + S_{0} - P_{0} } \right] \hfill \\ + n\left[ {(\frac{{R_{1} + (1 - 2w)P_{1} }}{1 - w} - S_{1} - T_{1} )x + S_{1} - P_{1} } \right] \hfill \\ \end{gathered} \right\}} \right. \hfill \\ \dot{n} = \varepsilon n(1 - n)(\theta x - (1 - x)) \hfill \\ \end{gathered} $$
(15)

The corner points and one interior fixed point can be obtained. The calculation of the eigenvalue of the Jacobian matrix related to each fixed point yields:

$$ {\mathbf{J}} = \left( {\begin{array}{*{20}c} {\partial \dot{x}/\partial x} & {\partial \dot{x}/\partial n} \\ {\partial \dot{n}/\partial x} & {\partial \dot{n}/\partial n} \\ \end{array} } \right), $$
(16)

then the conditions for the stability of the corner points can be derived:

  1. (1)

    \(x^{ * } = 0,n^{ * } = 0\)

eigenvalues are \(\lambda_{1} = S_{0} - P_{0} > 0,\lambda_{2} = - \varepsilon < 0\), thus, it is a saddle point.

  1. (2)

    \(x^{ * } = 0,n^{ * } = 1\)

eigenvalues are \(\lambda_{1} = S_{1} - P_{1} < 0,\lambda_{2} = \varepsilon > 0\), thus, it is a saddle point.

  1. (3)

    \(x^{ * } = 1,n^{ * } = 0\)

eigenvalues are \(\lambda_{1} = T_{0} - \frac{{R_{0} }}{1 - w} + \frac{{wP_{0} }}{1 - w},\lambda_{2} = \varepsilon \theta > 0\), the fixed point is an unstable or a saddle point when \(\lambda_{1} > 0\) or \(\lambda_{1} < 0\).

  1. (4)

    \(x^{ * } = 1,n^{ * } = 1\)

eigenvalues are \(\lambda_{1} = T_{1} - \frac{{R_{1} }}{1 - w} + \frac{{wP_{1} }}{1 - w},\lambda_{2} = - \varepsilon \theta < 0\). If \(T_{1} - \frac{{R_{1} }}{1 - w} + \frac{{wP_{1} }}{1 - w} < 0\), then the fixed point is stable.

Assuming \((x_{i}^{ * } ,n_{i}^{ * } )\) is an arbitrary internal fixed point. The associated Jacobian matrix is:

$$ J = \left[ {\begin{array}{*{20}c} {x(1 - x)\frac{\partial g(x,n)}{{\partial x}}} & {x(1 - x)\frac{\partial g(x,n)}{{\partial n}}} \\ {\varepsilon n(1 - n)(1 + \theta )} & 0 \\ \end{array} } \right]_{{(x_{i}^{ * } ,n_{i}^{ * } )}} , $$
(17)

where \(g(x,n) = \pi_{1} (x,n) - \pi_{2} (x,n)\) denotes the differences in payoffs. Here \(\partial g/\partial n < 0\) for all cases. The stability depends on the sign of the trace of \(J\). It is equivalent to the sign of \(\partial g/\partial x\), where

$$ \begin{gathered} \frac{\partial g}{{\partial x}} = \frac{{\partial \pi_{1} }}{\partial x} - \frac{{\partial \pi_{2} }}{\partial x} \hfill \\ \, = Z(n)[1,1] - Z(n)[2,1] - Z(n)[1,2] + Z(n)[2,2]. \hfill \\ \end{gathered} $$
(18)

Given the condition that \(\pi_{1} = \pi_{2}\) at the fixed point, then \(x_{i}^{ * } = 1/(1 + \theta )\), and the following condition can be satisfied:

$$ Z(n)[1,1]x_{i}^{ * } + Z(n)[1,2](1 - x_{i}^{ * } ) = Z(n)[2,1]x_{i}^{ * } + Z(n)[2,2](1 - x_{i}^{ * } ), $$
(19)

where,

$$ Z(n)[1,1] = \frac{{(1 - n)R_{0} + nR_{1} }}{1 - w}, $$
(20)
$$ Z(n)[1,2] = (1 - n)\left( {S_{0} + \frac{{wP_{0} }}{1 - w}} \right) + n\left( {S_{1} + \frac{{wP_{1} }}{1 - w}} \right), $$
(21)
$$ Z(n)[2,1] = (1 - n)\left( {T_{0} + \frac{{wP_{0} }}{1 - w}} \right) + n\left( {T_{1} + \frac{{wP_{1} }}{1 - w}} \right), $$
(22)
$$ Z(n)[2,2] = \frac{{(1 - n)P_{0} + nP_{1} }}{1 - w}. $$
(23)

Then the interior fixed point is

$$ \left\{ {\begin{array}{*{20}l} {x_{i}^{*} = 1/(1 + \theta )} \hfill \\ {n_{i}^{*} = \frac{{\left( {T_{0} - \frac{{R_{0} }}{{1 - w}}} \right) + \theta (P_{0} - S_{0} ) + \frac{{wP_{0} }}{{1 - w}}}}{{\left( {T_{0} - \frac{{R_{0} }}{{1 - w}}} \right) + \theta (P_{0} - S_{0} ) + \frac{{wP_{0} }}{{1 - w}} + \left( {\frac{{R_{1} }}{{1 - w}} - T_{1} } \right) + \theta (S_{1} - P_{1} ) - \frac{{wP_{1} }}{{1 - w}}}}} \hfill \\ \end{array} } \right.. $$
(24)

Further, it is stable when \(\partial g(x,n)/\partial x < 0\) hold:

$$ \frac{{\left( {T_{0} - \frac{{R_{0} }}{1 - w}} \right) + \theta (P_{0} - S_{0} ) + \frac{{wP_{0} }}{1 - w}}}{{\left( {T_{0} - \frac{{R_{0} }}{1 - w}} \right) + \theta (P_{0} - S_{0} ) + \frac{{wP_{0} }}{1 - w} + \left( {\frac{{R_{1} }}{1 - w} - T_{1} } \right) + \theta (S_{1} - P_{1} ) - \frac{{wP_{1} }}{1 - w}}} < \frac{{S_{0} - P_{0} }}{{(S_{0} - P_{0} ) + (P_{1} - S_{1} )}}, $$
(25)

further, it is equivalent to

$$ \frac{{(T_{0} - R_{0} ) + w(P_{0} - T_{0} )}}{{(T_{0} - R_{0} ) + (R_{1} - T_{1} ) + w(P_{0} - T_{0} ) + w(T_{1} - P_{1} )}} < \frac{{S_{0} - P_{0} }}{{(S_{0} - P_{0} ) + (P_{1} - S_{1} )}}. $$
(26)

Appendix 2

Stability analysis in the indirect reciprocity embedded-feedback model

The stability of the fixed point in the feedback model with indirect reciprocity is further explored. The coupled payoff matrix is defined by:

$$ A_{2(n)} = (1 - n)\left[ {\begin{array}{*{20}c} {R_{0} } & {(1 - q)S_{0} + qP_{0} } \\ {(1 - q)T_{0} + qP_{0} } & {P_{0} } \\ \end{array} } \right] + n\left[ {\begin{array}{*{20}c} {R_{1} } & {(1 - q)S_{1} + qP_{1} } \\ {(1 - q)T_{1} + qP_{1} } & {P_{1} } \\ \end{array} } \right], $$
(27)

and the replicator dynamics for this system can be described as

$$ \begin{gathered} \dot{x} = x(1 - x)\left\{ {\left. \begin{gathered} (1 - n)\left[ {(R_{0} - S_{0} + qS_{0} - qP_{0} - T_{0} + qT_{0} - qP_{0} + P_{0} )x + S_{0} - qS_{0} + qP_{0} - P_{0} } \right] \hfill \\ + n\left[ {(R_{1} - S_{1} + qS_{1} - qP_{1} - T_{1} + qT_{1} - qP_{1} + P_{1} )x + S_{1} - qS_{1} + qP_{1} - P_{1} } \right] \hfill \\ \end{gathered} \right\}} \right.. \hfill \\ \dot{n} = \varepsilon n(1 - n)(\theta x - (1 - x)) \hfill \\ \end{gathered} $$
(28)

Five fixed points in the coevolution dynamics can be obtained, and the conditions for the stability of each point can be derived with the aid of Jacobian matrices:

(1) \(x^{ * } = 0,n^{ * } = 0\),

eigenvalues are \(\lambda_{1} = (1 - q)(S_{0} - P_{0} ) > 0,\lambda_{2} = - \varepsilon < 0\), thus, it is a saddle point.

(2) \(x^{ * } = 0,n^{ * } = 1\),

eigenvalues are \(\lambda_{1} = (1 - q)(S_{1} - P_{1} ) < 0,\lambda_{2} = \varepsilon > 0\), thus, it is a saddle point.

(3) \(x^{ * } = 1,n^{ * } = 0\),

eigenvalues are \(\lambda_{1} = T_{0} - R_{0} + q(P_{0} - T_{0} ),\lambda_{2} = \varepsilon \theta > 0\), thus, the fixed point is an unstable or a saddle point when \(\lambda_{1} > 0\) or \(\lambda_{1} < 0\).

(4) \(x^{ * } = 1,n^{ * } = 1\),

eigenvalues are \(\lambda_{1} = qP_{1} - qT_{1} + T_{1} - R_{1} ,\lambda_{2} = - \varepsilon \theta < 0\). If \(qP_{1} - qT_{1} + T_{1} - R_{1} < 0\), then the fixed point is stable.

The interior fixed point can be obtained by:

$$ \left\{ {\begin{array}{*{20}l} {x^{ * } = 1/(1 + \theta )} \hfill \\ {n^{ * } = \frac{{(T_{0} - R_{0} ) + \theta (1 - q)(P_{0} - S_{0} ) - qT_{0} + qP_{0} }}{{(T_{0} - R_{0} ) + (R_{1} - T_{1} ) + \theta (1 - q)(S_{1} - P_{1} ) + \theta (1 - q)(P_{0} - S_{0} ) + qP_{0} - qT_{0} + qT_{1} - qP_{1} }}} \hfill \\ \end{array} } \right.. $$
(29)

The stability of the internal equilibria can be satisfied when \(\partial g(x,n)/\partial x < 0\) holds:

$$ \begin{gathered} \frac{{(T_{0} - R_{0} ) + \theta (1 - q)(P_{0} - S_{0} ) - qT_{0} + qP_{0} }}{{(T_{0} - R_{0} ) + (R_{1} - T_{1} ) + \theta (1 - q)(S_{1} - P_{1} ) + \theta (1 - q)(P_{0} - S_{0} ) + qP_{0} - qT_{0} + qT_{1} - qP_{1} }}. \hfill \\ < \frac{{(S_{0} - P_{0} )}}{{(S_{0} - P_{0} ) + (P_{1} - S_{1} )}} \hfill \\ \end{gathered} $$
(30)

The condition can be further rewritten as

$$ \frac{{(T_{0} - R_{0} ) + q(P_{0} - T_{0} )}}{{(T_{0} - R_{0} ) + (R_{1} - T_{1} ) + q(P_{0} - T_{0} ) + q(T_{1} - P_{1} )}} < \frac{{(S_{0} - P_{0} )}}{{(S_{0} - P_{0} ) + (P_{1} - S_{1} )}}. $$
(31)

Appendix 3

Stability analysis of the feedback-evolving games with local reciprocity

In this section, the stability of the fixed points is analyzed given the reciprocity mechanisms are only considered in the circumstances where cooperation is favored or not. Firstly, we consider the direct reciprocity mechanism in the payoff matrix \(A_{1}\), where the mutual defection is favored, and the \(2 \times 2\) game matrix is defined by:

$$ A_{11(n)} = (1 - n)\left[ {\begin{array}{*{20}c} {R_{0} } & {S_{0} } \\ {T_{0} } & {P_{0} } \\ \end{array} } \right] + n\left[ {\begin{array}{*{20}c} {\frac{{R_{1} }}{1 - w}} & {S_{1} + \frac{{wP_{1} }}{1 - w}} \\ {T_{1} + \frac{{wP_{1} }}{1 - w}} & {\frac{{P_{1} }}{1 - w}} \\ \end{array} } \right], $$
(32)

and the corresponding replicator dynamics are:

$$ \begin{aligned} \dot{x} & = x(1 - x)\left\{ {\left. \begin{aligned} & (1 - n)\left[ {(R_{0} + P_{0} - S_{0} - T_{0} )x + S_{0} - P_{0} } \right] \hfill \\ & + n\left[ {(\frac{{R_{1} + (1 - 2w)P_{1} }}{1 - w} - S_{1} - T_{1} )x + S_{1} - P_{1} } \right] \hfill \\ \end{aligned} \right\}} \right.. \hfill \\ \dot{n} & = \varepsilon n(1 - n)(\theta x - (1 - x)) \hfill \\ \end{aligned} $$
(33)

The interior fixed point is given by \((x_{i}^{ * } ,n_{i}^{ * } )\):

$$ \left\{ {\begin{array}{*{20}l} {x_{i}^{ * } = 1/(1 + \theta )} \hfill \\ {n_{i}^{ * } = \frac{{(T_{0} - R_{0} ) + \theta (P_{0} - S_{0} )}}{{(T_{0} - R_{0} ) + \theta (P_{0} - S_{0} ) + (\frac{{R_{1} }}{1 - w} - T_{1} ) + \theta (S_{1} - P_{1} ) - \frac{{wP_{1} }}{1 - w}}}} \hfill \\ \end{array} } \right.. $$
(34)

The Jacobin matrices at the corner points are:

$$ J(0,0) = \left[ {\begin{array}{*{20}c} {S_{0} - P_{0} } & 0 \\ 0 & { - \varepsilon } \\ \end{array} } \right],\;J(0,1) = \left[ {\begin{array}{*{20}c} {S_{1} - P_{1} } & 0 \\ 0 & \varepsilon \\ \end{array} } \right], $$
$$ J(1,0) = \left[ {\begin{array}{*{20}c} {T_{0} - R_{0} } & 0 \\ 0 & {\varepsilon \theta } \\ \end{array} } \right],\;J(1,1) = \left[ {\begin{array}{*{20}c} {T_{1} - \frac{{R_{1} }}{1 - w} + \frac{{wP_{1} }}{1 - w}} & 0 \\ 0 & { - \varepsilon \theta } \\ \end{array} } \right], $$

The three corner equilibria, that is \((0,0)\), \((0,1)\) and \((1,0)\) are both unstable points. The corner point \((1,1)\) is stable if \(T_{1} - \frac{{R_{1} }}{1 - w} + \frac{{wP_{1} }}{1 - w} < 0\), which is consistent with the global reciprocity feedbacking model. The stability of the internal point can be satisfied when the following condition holds:

$$ \frac{{(T_{0} - R_{0} ) + \theta (P_{0} - S_{0} )}}{{(T_{0} - R_{0} ) + \theta (P_{0} - S_{0} ) + (\frac{{R_{1} }}{1 - w} - T_{1} ) + \theta (S_{1} - P_{1} ) - \frac{{wP_{1} }}{1 - w}}} < \frac{{S_{0} - P_{0} }}{{(S_{0} - P_{0} ) + (P_{1} - S_{1} )}}, $$
(35)

further with

$$ \frac{{(T_{0} - R_{0} )}}{{(T_{0} - R_{0} ) + (\frac{{R_{1} }}{1 - w} - T_{1} ) - \frac{{wP_{1} }}{1 - w}}} < \frac{{S_{0} - P_{0} }}{{(S_{0} - P_{0} ) + (P_{1} - S_{1} )}}, $$
(36)

that is,

$$ \frac{{T_{0} - R_{0} + w(R_{0} - T_{0} )}}{{T_{0} - R_{0} + w(R_{0} - T_{0} ) + R_{1} - T_{1} + w(T_{1} - P_{1} )}} < \frac{{S_{0} - P_{0} }}{{(S_{0} - P_{0} ) + (P_{1} - S_{1} )}}. $$
(37)

Then, consider the reciprocity mechanism embedded in the payoff matrix \(A_{0}\), where mutual cooperation is favored. The payoff matrix of this sub-system is:

$$ A_{10(n)} = (1 - n)\left[ {\begin{array}{*{20}c} {\frac{{R_{0} }}{1 - w}} & {S_{0} + \frac{{wP_{0} }}{1 - w}} \\ {T_{0} + \frac{{wP_{0} }}{1 - w}} & {\frac{{P_{0} }}{1 - w}} \\ \end{array} } \right] + n\left[ {\begin{array}{*{20}c} {R_{1} } & {S_{1} } \\ {T_{1} } & {P_{1} } \\ \end{array} } \right], $$
(38)

and the corresponding replicator dynamics are:

$$ \begin{gathered} \dot{x} = x(1 - x)\left\{ {\left. \begin{gathered} (1 - n)\left[ {(\frac{{R_{0} + (1 - 2w)P_{0} }}{1 - w} - S_{0} - T_{0} )x + S_{0} - P_{0} } \right] \hfill \\ + n\left[ {(R_{1} + P_{1} - S_{1} - T_{1} )x + S_{1} - P_{1} } \right] \hfill \\ \end{gathered} \right\}} \right.. \hfill \\ \dot{n} = \varepsilon n(1 - n)(\theta x - (1 - x)) \hfill \\ \end{gathered} $$
(39)

By the computation of the Jacobian matrix, we find that all four corner points are unstable, and the point \((1,1)\) cannot be stable as well. At the four corner points, we have:

$$ J(0,0) = \left[ {\begin{array}{*{20}c} {S_{0} - P_{0} } & 0 \\ 0 & { - \varepsilon } \\ \end{array} } \right],\;J(0,1) = \left[ {\begin{array}{*{20}c} {S_{1} - P_{1} } & 0 \\ 0 & \varepsilon \\ \end{array} } \right], $$
$$ J(1,0) = \left[ {\begin{array}{*{20}c} {T_{0} - \frac{{R_{0} - wP_{0} }}{1 - w}} & 0 \\ 0 & {\varepsilon \theta } \\ \end{array} } \right],\;J(1,1) = \left[ {\begin{array}{*{20}c} {T_{1} - R_{1} } & 0 \\ 0 & { - \varepsilon \theta } \\ \end{array} } \right]. $$

The stability of the internal equilibria can be satisfied when the following condition holds:

$$ \frac{{\left( {T_{0} - \frac{{R_{0} }}{1 - w}} \right) + \theta (P_{0} - S_{0} ) + \frac{{wP_{0} }}{1 - w}}}{{\left( {T_{0} - \frac{{R_{0} }}{1 - w}} \right) + \theta (P_{0} - S_{0} ) + (R_{1} - T_{1} ) + \theta (S_{1} - P_{1} ) + \frac{{wP_{0} }}{1 - w}}} < \frac{{S_{0} - P_{0} }}{{(S_{0} - P_{0} ) + (P_{1} - S_{1} )}}, $$
(40)

further with

$$ \frac{{T_{0} - R_{0} + w(P_{0} - T_{0} )}}{{T_{0} - R_{0} + w(P_{0} - T_{0} ) + R_{1} - T_{1} + w(T_{1} - R_{1} )}} < \frac{{S_{0} - P_{0} }}{{(S_{0} - P_{0} ) + (P_{1} - S_{1} )}}. $$
(41)

The outcomes in this case show the unavailable of full cooperation. The system converges to the stable point in the interior at most.

We further consider the indirect reciprocity mechanism is coupled in the circumstances where cooperation is favored or not. To do so, \(A_{0}\) is fixed, and the mechanism is integrated into \(A_{1}\), and the payoff matrix of this subsystem is:

$$ A_{21(n)} = (1 - n)\left[ {\begin{array}{*{20}c} {R_{0} } & {S_{0} } \\ {T_{0} } & {P_{0} } \\ \end{array} } \right] + n\left[ {\begin{array}{*{20}c} {R_{1} } & {(1 - q)S_{1} + qP_{1} } \\ {(1 - q)T_{1} + qP_{1} } & {P_{1} } \\ \end{array} } \right]. $$
(42)

The replicator dynamics of this feedback model are:

$$ \begin{gathered} \dot{x} = x(1 - x)\left\{ {\left. \begin{gathered} (1 - n)\left[ {(R_{0} - S_{0} - T_{0} + P_{0} )x + S_{0} - P_{0} } \right] \hfill \\ + n\left[ {(R_{1} - S_{1} + qS_{1} - qP_{1} - T_{1} + qT_{1} - qP_{1} + P_{1} )x + S_{1} - qS_{1} + qP_{1} - P_{1} } \right] \hfill \\ \end{gathered} \right\}} \right.. \hfill \\ \dot{n} = \varepsilon n(1 - n)(\theta x - (1 - x)) \hfill \\ \end{gathered} $$
(43)

The Jacobian matrixes at the four corner points are, respectively, as:

$$ J(0,0) = \left[ {\begin{array}{*{20}c} {S_{0} - P_{0} } & 0 \\ 0 & { - \varepsilon } \\ \end{array} } \right],\;J(0,1) = \left[ {\begin{array}{*{20}c} {(1 - q)(S_{1} - P_{1} )} & 0 \\ 0 & \varepsilon \\ \end{array} } \right], $$
$$ J(1,0) = \left[ {\begin{array}{*{20}c} {T_{0} - R_{0} } & 0 \\ 0 & {\varepsilon \theta } \\ \end{array} } \right],\;J(1,1) = \left[ {\begin{array}{*{20}c} {T_{1} - R_{1} + qP_{1} - qT_{1} } & 0 \\ 0 & { - \varepsilon \theta } \\ \end{array} } \right]. $$

By the computation of the Jacobian matrix, we also find the three corner equilibria, that is \((0,0)\),\((0,1)\) and \((1,0)\) are both unstable points. The corner point \((1,1)\) is stable if \(T_{1} - R_{1} + qP_{1} - qT_{1} < 0\), which is consistent with the global reciprocity feedbacking model. The interior fixed point can be obtained and given by \((x_{i}^{ * } ,n_{i}^{ * } )\):

$$ \left\{ {\begin{array}{*{20}l} {x_{i}^{ * } = 1/(1 + \theta )} \hfill \\ {n_{i}^{ * } = \frac{{(T_{0} - R_{0} ) + \theta (P_{0} - S_{0} )}}{{(T_{0} - R_{0} ) + (R_{1} - T_{1} ) + \theta (S_{1} - P_{1} ) + \theta (P_{0} - S_{0} ) + \theta q(P_{1} - S_{1} ) + qT_{1} - qP_{1} }}} \hfill \\ \end{array} } \right.. $$
(44)

The stability of the internal equilibria can be satisfied as:

$$ \frac{{(T_{0} - R_{0} ) + \theta (P_{0} - S_{0} )}}{{(T_{0} - R_{0} ) + (R_{1} - T_{1} ) + \theta (S_{1} - P_{1} ) + \theta (P_{0} - S_{0} ) + \theta q(P_{1} - S_{1} ) + qT_{1} - qP_{1} }} < \frac{{(S_{0} - P_{0} )}}{{(S_{0} - P_{0} ) + (P_{1} - S_{1} )}}, $$
(45)

further with:

$$ \frac{{T_{0} - R_{0} }}{{(T_{0} - R_{0} ) + (R_{1} - T_{1} ) + qT_{1} - qP_{1} }} < \frac{{(S_{0} - P_{0} )}}{{(S_{0} - P_{0} ) + (P_{1} - S_{1} )}}. $$
(46)

Considering the reciprocity mechanism is embedded in the payoff matrix \(A_{0}\). The payoff matrix of the subsystem is:

$$ A_{22(n)} = (1 - n)\left[ {\begin{array}{*{20}c} {R_{0} } & {(1 - q)S_{0} + qP_{0} } \\ {(1 - q)T_{0} + qP_{0} } & {P_{0} } \\ \end{array} } \right] + n\left[ {\begin{array}{*{20}c} {R_{1} } & {S_{1} } \\ {T_{1} } & {P_{1} } \\ \end{array} } \right], $$
(47)

and the replicator dynamics can be described as:

$$ \begin{gathered} \dot{x} = x(1 - x)\left\{ {\left. \begin{gathered} (1 - n)\left[ {(R_{0} - S_{0} + qS_{0} - qP_{0} - T_{0} + qT_{0} - qP_{0} + P_{0} )x + S_{0} - qS_{0} + qP_{0} - P_{0} } \right] \hfill \\ + n\left[ {(R_{1} - S_{1} - T_{1} + P_{1} )x + S_{1} - P_{1} } \right] \hfill \\ \end{gathered} \right\}} \right.. \hfill \\ \dot{n} = \varepsilon n(1 - n)(\theta x - (1 - x)) \hfill \\ \end{gathered} $$
(48)

At the four corner fixed points, we have:

$$ J(0,0) = \left[ {\begin{array}{*{20}c} {(1 - q)(S_{0} - P_{0} )} & 0 \\ 0 & { - \varepsilon } \\ \end{array} } \right],\;J(0,1) = \left[ {\begin{array}{*{20}c} {(S_{1} - P_{1} )} & 0 \\ 0 & \varepsilon \\ \end{array} } \right], $$
$$ J(1,0) = \left[ {\begin{array}{*{20}c} {T_{0} - R_{0} + qP_{0} - qT_{0} } & 0 \\ 0 & {\varepsilon \theta } \\ \end{array} } \right],\;J(1,1) = \left[ {\begin{array}{*{20}c} {T_{1} - R_{1} } & 0 \\ 0 & { - \varepsilon \theta } \\ \end{array} } \right]. $$

All four corner points are unstable through the calculation of the Jacobian matrix. The interior point \((x_{i}^{ * } ,n_{i}^{ * } )\) is:

$$ \left\{ {\begin{array}{*{20}l} {x_{i}^{ * } = 1/(1 + \theta )} \hfill \\ {n_{i}^{ * } = \frac{{(T_{0} - R_{0} ) + \theta (P_{0} - S_{0} ) + \theta q(S_{0} - P_{0} ) - qT_{0} + qP_{0} }}{{(T_{0} - R_{0} ) + (R_{1} - T_{1} ) + \theta (S_{1} - P_{1} ) + \theta (P_{0} - S_{0} ) + \theta q(S_{0} - P_{0} ) + qP_{0} - qT_{0} }}} \hfill \\ \end{array} } \right.. $$
(49)

The stability of the internal equilibria can be satisfied when the following condition holds:

$$ \frac{{(T_{0} - R_{0} ) + \theta (P_{0} - S_{0} ) + \theta q(S_{0} - P_{0} ) - qT_{0} + qP_{0} }}{{(T_{0} - R_{0} ) + (R_{1} - T_{1} ) + \theta (S_{1} - P_{1} ) + \theta (P_{0} - S_{0} ) + \theta q(S_{0} - P_{0} ) + qP_{0} - qT_{0} }} < \frac{{(S_{0} - P_{0} )}}{{(S_{0} - P_{0} ) + (P_{1} - S_{1} )}}, $$
(50)

and it can be further rewritten as

$$ \frac{{(T_{0} - R_{0} ) - qT_{0} + qP_{0} }}{{(T_{0} - R_{0} ) + (R_{1} - T_{1} ) + qP_{0} - qT_{0} }} < \frac{{(S_{0} - P_{0} )}}{{(S_{0} - P_{0} ) + (P_{1} - S_{1} )}}. $$
(51)

To sum up, a single, stable fixed point in the interior can be obtained.

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Ma, X., Quan, J. & Wang, X. Effect of reciprocity mechanisms on evolutionary dynamics in feedback-evolving games. Nonlinear Dyn 112, 709–729 (2024). https://doi.org/10.1007/s11071-023-09052-y

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