[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Evolution of Cooperation in Prisoner’s Dilemma within Changeable External Environments

  • Published:
Dynamic Games and Applications Aims and scope Submit manuscript

Abstract

Rational individuals make a relatively dominant choice when faced with two opposing options of cooperation and betrayal in interactions. Without other interference, this often leads to a dilemma when social interests and personal interests are inconsistent, like the prisoner’s dilemma. However, in reality, an individual plays multiple games simultaneously. Members of a population may be influenced by other games to modify their strategic choices in the main game. In this paper, we consider intraspecific and interspecific games simultaneously from a multi-game perspective. How intraspecific cooperation evolves when different types of interspecific games as changeable environments is of great interest to us. We scrutinize the evolution of cooperation within the population in the prisoner’s dilemma by replicator dynamics with three different types of external environments of cooperation. The results show that the emergence and maintenance of cooperation in the intraspecific game are related to the relative size and the probability of cooperation in the external environment. Numerical experiences are provided to support the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Data availability

Not applicable.

References

  1. Chowdhury SN, Kundu S, Perc M, Ghosh D (2021) Complex evolutionary dynamics due to punishment and free space in ecological multigames. Proc R Soc A 477(2252):20210397

    Article  MathSciNet  Google Scholar 

  2. Colivicchi I, Iannucci G (2023) The environmental responsibility of firms and insurance coverage in an evolutionary game. Dyn Games Appl 13(3):801–818

    Article  MathSciNet  Google Scholar 

  3. Cressman R, Tao Y (2014) The replicator equation and other game dynamics. PNAS 111((supplement–3)):10810–10817

    Article  MathSciNet  Google Scholar 

  4. Cui G, Wang Z, Yang Y, Xu J, Yang H (2018) Evolution of cooperation under the influence of environments on individual-performed interactions. Int J Mod Phys C 29(08):1850070

    Article  Google Scholar 

  5. Gao L, Pan Q, He M (2023) Impact of peer pressure on the evolution of cooperation in prisoner’s dilemma game. Chaos Soliton Fract 174:113860

    Article  MathSciNet  Google Scholar 

  6. Gersbach H, Oberpriller Q, Scheffel M (2019) Double free-riding in innovation and abatement: a rules treaty solution. Environ Resour Econ 73:449–483

    Article  Google Scholar 

  7. Gokhale CS, Frean M, Rainey PB (2023) Eco-evolutionary logic of mutualisms. Dyn Games Appl 13(4):1066–1087

    Article  MathSciNet  Google Scholar 

  8. Guo H, Shen C, Hu S, Xing J, Tao P, Shi Y, Wang Z (2023) Facilitating cooperation in human-agent hybrid populations through autonomous agents. Iscience 26(11):108179

    Article  Google Scholar 

  9. Hashimoto K (2006) Unpredictability induced by unfocused games in evolutionary game dynamics. J Theor Biol 241(3):669–675

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  11. Hofbauer J, Sigmund K (2003) Evolutionary game dynamics. B Am Math Soc 40(4):479–519

    Article  MathSciNet  Google Scholar 

  12. Huang S, Hu Y, Liu X, Fu X (2024) The influence of environment-based reputation on cooperation in spatial prisoner’s dilemma game. Europhys Lett 144(6):62002

    Article  Google Scholar 

  13. Kleshnina M, Hilbe C, Šimsa Š, Chatterjee K, Nowak MA (2023) The effect of environmental information on evolution of cooperation in stochastic games. Nat Commun 14(1):4153

    Article  Google Scholar 

  14. Kurita K, Managi S (2022) Covid-19 and stigma: evolution of self-restraint behavior. Dyn Games Appl 12(1):168–182

    Article  MathSciNet  Google Scholar 

  15. Li WJ, Chen Z, Jin KZ et al (2022) Options for mobility and network reciprocity to jointly yield robust cooperation in social dilemmas. Appl Math Comput 435:127456

    MathSciNet  Google Scholar 

  16. Ma X, Quan J, Wang X (2023) Effect of reciprocity mechanisms on evolutionary dynamics in feedback-evolving games. Nonlinear Dyn 112(1):709–729

    Article  Google Scholar 

  17. May RM (1981) The evolution of cooperation. Nature 292(5821):291–292

    Article  Google Scholar 

  18. Milinski M, Semmann D, Krambeck HJ, Marotzke J (2006) Stabilizing the Earths climate is not a losing game: supporting evidence from public goods experiments. Proc Natl Acad Sci 103(11):3994–3998

    Article  Google Scholar 

  19. Nowak MA (2006) Five rules for the evolution of cooperation. Science 314(5805):1560–1563

    Article  Google Scholar 

  20. Nowak MA (2006) Evolutionary dynamics: exploring the equations of life. Harvard University Press, Harvard

    Book  Google Scholar 

  21. Ohtsuki H, Nowak MA (2007) Direct reciprocity on graphs. J Theor Biol 247(3):462–470

    Article  MathSciNet  Google Scholar 

  22. Ohtsuki H, Nowak MA (2008) Evolutionary stability on graphs. J Theor Biol 251(4):698–707

    Article  MathSciNet  Google Scholar 

  23. Pennisi E (2005) How did cooperative behavior evolve? Science 309(5731):93–93

    Article  Google Scholar 

  24. Perc M, Jordan JJ, Rand DG, Wang Z, Boccaletti S, Szolnoki A (2017) Statistical physics of human cooperation. Phys Rep 687:1–51

    Article  MathSciNet  Google Scholar 

  25. Riehl J, Ramazi P, Cao M (2018) A survey on the analysis and control of evolutionary matrix games. Annu Rev Control 45:87–106

    Article  MathSciNet  Google Scholar 

  26. Salahshour M (2022) Interaction between games give rise to the evolution of moral norms of cooperation. PLoS Comput Biol 18(9):e1010429

    Article  Google Scholar 

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

    Article  Google Scholar 

  28. Sharma G, Guo H, Shen C, Tanimoto J (2023) Small bots, big impact: solving the conundrum of cooperation in optional prisoner’s dilemma game through simple strategies. Preprint at arXiv:2305.15818

  29. Shi F, Wang C, Yao C (2022) A new evolutionary game analysis for industrial pollution management considering the central government’s punishment. Dyn Games Appl 12:677–688

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  31. Stewart AJ, Plotkin JB (2014) Collapse of cooperation in evolving games. Proc Natl Acad Sci 111(49):17558–17563

    Article  Google Scholar 

  32. Szabó G, Fáth G (2007) Evolutionary games on graphs. Phys Rep 4(446):97–216

    Article  MathSciNet  Google Scholar 

  33. Szolnoki A, Perc M (2008) Coevolution of teaching activity promotes cooperation. New J Phys 10(4):043036

    Article  Google Scholar 

  34. Taylor PD, Jonker LB (1978) Evolutionary stable strategies and game dynamics. Math Biosci 40(1–2):145–156

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  36. Tarnita CE, Antal T, Ohtsuki H, Nowak MA (2009) Evolutionary dynamics in set structured populations. Proc Natl Acad Sci 106(21):8601–8604

    Article  Google Scholar 

  37. Venkateswaran VR, Gokhale CS (2019) Evolutionary dynamics of complex multiple games. Proc R Soc B 286(1905):20190900

    Article  Google Scholar 

  38. Weitz JS, Eksin C, Paarporn K, Brown SP, Ratcliff WC (2016) An oscillating tragedy of the commons in replicator dynamics with game-environment feedback. Proc Natl Acad Sci 113(47):E7518–E7525

    Article  Google Scholar 

  39. Wölfl B, Te Rietmole H, Salvioli M et al (2022) The contribution of evolutionary game theory to understanding and treating cancer. Dyn Games Appl 12(2):313–342

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  41. Yang X, Wang W, Zhang F, Qiao H (2017) Cooperation enhanced by habitat destruction in prisoner’s dilemma games. Phys A 486:668–673

    Article  Google Scholar 

Download references

Funding

This work is supported by a grant of the National Natural Science Foundation of China (NNSF) (No. 10972011) and the Science Foundation of Hebei Normal University, China (No. L2018B01).

Author information

Authors and Affiliations

  1. College of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China

    Aochong Xing, Gang Zhang & Haiyan Tian

  2. Hebei International Joint Research Center, Mathematics and Interdisciplinary Science, Shijiazhuang, 050024, China

    Gang Zhang

Authors
  1. Aochong Xing
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Haiyan Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

GZ and HT developed ideas for the model. AX analysed the results and wrote the manuscript. HT corrected the manuscript.

Corresponding author

Correspondence to Haiyan Tian.

Ethics declarations

Conflict of interest

The authors declare no Conflict of interest.

Ethical Approval

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A The External Environment is Cooperation Type I

Proof of Theorem 1

To ensure \(\bar{\rho _{C} } \in (0,1)\), we get

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} \theta (S_{2} +(1-S_{2}-T_{2})\varphi )+S_{1}>0,\\ \theta (S_{2} +(1-S_{2}-T_{2})\varphi )+S_{1}<S_{1}+T_{1}-1. \end{array}\right. } \Rightarrow {\left\{ \begin{array}{ll} \varphi < \frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})},\\ \varphi > \frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})}. \end{array}\right. } \end{aligned} \end{aligned}$$
(A1)

Considering \(\varphi \in (0,1)\), the following conditions need to be satisfied:

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} \frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})}\ge 0,\\ \frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})}\le 1. \end{array}\right. } \end{aligned} \Rightarrow \theta \ge \frac{T_{1}-1}{S_{2}}. \end{aligned}$$
(A2)

So, when \(\theta \ge \frac{T_{1}-1}{S_{2}}\) and \(\frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})}<\varphi < \frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})}\), internal equilibrium point \(\bar{\rho _{C}}\) exist. Considering

$$\begin{aligned} \begin{aligned}&{g}'(\bar{\rho _{C}})=\frac{1}{(1+\theta )^{2}} \left( S_{1} +\theta (S_{2}+(1-S_{2}-T_{2})\varphi )\right) \\&\qquad \qquad \quad \left( \frac{S_{1} +\theta (S_{2}+(1-S_{2}-T_{2})\varphi )}{S_{1}+T_{1}-1 }-1\right) <0,\\&{g}'(0)=\frac{1}{(1+\theta )^{2}} \left( S_{1} +\theta (S_{2}+(1-S_{2}-T_{2})\varphi )\right)>0,\\&{g}'(1)=-\frac{1}{(1+\theta )^{2}} \left( 1-T_{1} +\theta (S_{2}+(1-S_{2}-T_{2})\varphi )\right) >0. \end{aligned} \end{aligned}$$
(A3)

Therefore, internal equilibrium point \(\bar{\rho _{C}}\) is stable.

When \(\bar{\rho _{C}}\le 0\) or \(\bar{\rho _{C}}\ge 1\), internal equilibrium point \(\bar{\rho _{C}}\) doesn’t exist. We get

$$\begin{aligned} \begin{aligned}&S_{1} +\theta (S_{2}+(1-S_{2}-T_{2})\varphi )\le 0\\&\Rightarrow \theta > \frac{-S_{1}}{S_{2}}~~and~~\frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})}\le \varphi < 1. \end{aligned} \end{aligned}$$
(A4)

Now,

$$\begin{aligned} & {g}'(0)<0, {g}'(1)>0. \end{aligned}$$
(A5)
$$\begin{aligned} & \begin{aligned}&S_{1} +\theta (S_{2}+(1-S_{2}-T_{2})\varphi )\ge S_{1}+T_{1}-1\\&\Rightarrow \theta > \frac{T_{1}-1}{S_{2}}~~ and~~ 0< \varphi \le \frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})}. \end{aligned} \end{aligned}$$
(A6)

Now,

$$\begin{aligned} {g}'(1)<0, {g}'(0)>0. \end{aligned}$$
(A7)

When \(0< \theta \le \frac{-S_{1}}{S_{2}}\) and \(0<\varphi < 1\), then

$$\begin{aligned} (1-T_{2})\frac{-S_{1} }{S_{2}}+S_{1}<S_{1} +\theta (S_{2}+(1-S_{2}-T_{2})\varphi )<0. \end{aligned}$$
(A8)

So, \(\bar{\rho _{C}}<0\), internal equilibrium point \(\bar{\rho _{C}}\) doesn’t exist. Now,

$$\begin{aligned} {g}'(0)<0, {g}'(1)>0. \end{aligned}$$
(A9)

When \(\frac{-S_{1}}{S_{2}}< \theta < \frac{T_{1}-1}{S_{2}}\) and \(0< \varphi <\frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})} \), \(\bar{\rho _{C}}\) exist. Now,

$$\begin{aligned} {g}'(\bar{\rho _{C}})<0, {g}'(0)>0,{g}'(1)>0. \end{aligned}$$
(A10)

\(\square \)

Proof of Theorem 2

When \(S_{1}+\theta (S_{2} +(1-S_{2}-T_{2})\varphi ) < 0\), \({g}'(0)<0.\) Considering \(\varphi \in (0,1)\), we get \(\theta \ge \frac{-S_{1}}{S_{2}}\) and \(\frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})}<\varphi <1\). The same reason can be used to obtain when \(S_{1}+\theta (S_{2} +(1-S_{2}-T_{2})\varphi ) > 0\). When \(0<\theta <\frac{-S_{1}}{S_{2}} \) and \(0<\varphi <1\), the proof is the same as in Case1. \(\square \)

Proof of Theorem 3

To ensure \(\bar{\rho _{C} } \in (0,1)\), we get

$$\begin{aligned} \frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})}< \varphi <\frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})} \end{aligned}$$
(A11)

Considering \(\varphi \in (0,1)\), the following conditions need to be satisfied:

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} \frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})} \ge 0\\ \frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})} \le 1. \end{array}\right. } \end{aligned} \Rightarrow \theta \ge \frac{-S_{1}}{S_{2}}. \end{aligned}$$
(A12)

Thus, when \(\theta \ge \frac{-S_{1}}{S_{2}}\) and \( \frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})}< \varphi <\frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})}\), internal equilibrium point \(\bar{\rho _{C}}\) exist.Considering

$$\begin{aligned} \begin{aligned}&{g}'(\bar{\rho _{C}})> 0,\\&{g}'(0)< 0,\\&{g}'(1)< 0. \end{aligned} \end{aligned}$$
(A13)

Therefore, internal equilibrium point \(\bar{\rho _{C}}\) is not stable. Equilibrium point \(\rho _{C} =0\), \(\rho _{C} =1\) are stable.

When \(\bar{\rho _{C}}\le 0\) or \(\bar{\rho _{C}}\ge 1\), internal equilibrium point \(\bar{\rho _{C}}\) doesn’t exist. We get

$$\begin{aligned} & \theta > \frac{-S_{1}}{S_{2}}~~ and~~ 0< \varphi \le \frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})} \Rightarrow {g}'(1)<0. \end{aligned}$$
(A14)
$$\begin{aligned} & \theta > \frac{T_{1}-1}{S_{2}}~~ and~~ \frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})}\le \varphi< 1 \Rightarrow {g}'(0)<0. \end{aligned}$$
(A15)

When \(0< \theta \le \frac{T_{1}-1}{S_{2}}\) and \(0<\varphi < 1\), then \(\bar{\rho _{C}}<0\), internal equilibrium point \(\bar{\rho _{C}}\) doesn’t exist. Now,

$$\begin{aligned} {g}'(0)<0, {g}'(1)>0. \end{aligned}$$
(A16)

When \(\frac{T_{1}-1}{S_{2}}< \theta < \frac{-S_{1}}{S_{2}}\) and \(0< \varphi <\frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})}\), \(\bar{\rho _{C}}\) exist. We have

$$\begin{aligned} {g}'(\bar{\rho _{C}})>0, {g}'(0)<0,{g}'(1)<0. \end{aligned}$$
(A17)

\(\square \)

Appendix B The External Environment is Cooperation Type II

Proof of Theorem 4

To ensure \(\bar{\rho _{C} } \in (0,1)\), we have

$$\begin{aligned} \frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})}< \varphi < \frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})} \end{aligned}$$
(B18)

Considering \(\varphi \in (0,1)\), the following conditions need to be satisfied:

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} \frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})}\ge 0,\\ \frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})}\le 1. \end{array}\right. } \end{aligned} \Rightarrow \theta \ge \frac{T_{1}-1}{1-T_{2}}. \end{aligned}$$
(B19)

Then, \({g}'(0)>0\), \({g}'(1)>0\), \({g}'(\bar{\rho _{C}})<0\).

When \(\bar{\rho _{C}}\le 0\) or \(\bar{\rho _{C}}\ge 1\), internal equilibrium point \(\bar{\rho _{C}}\) doesn’t exist. We have

$$\begin{aligned} & \begin{aligned} \theta > \frac{-S_{1}}{1-T_{2}}~~ and~~ 0< \varphi \le \frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})} \end{aligned} \Rightarrow {g}'(0)<0. \end{aligned}$$
(B20)
$$\begin{aligned} & \begin{aligned} \theta > \frac{T_{1}-1}{1-T_{2}}~~ and~~ \frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})} \le \varphi< 1 \end{aligned} \Rightarrow {g}'(1)<0. \end{aligned}$$
(B21)

When \(0< \theta \le \frac{-S_{1}}{1-T_{2}}\) and \(0<\varphi < 1\), then \(\bar{\rho _{C}}<0\), internal equilibrium point \(\bar{\rho _{C}}\) doesn’t exist. Now,

$$\begin{aligned} {g}'(0)<0, {g}'(1)>0. \end{aligned}$$
(B22)

When \(\frac{-S_{1}}{1-T_{2}}< \theta < \frac{T_{1}-1}{1-T_{2}}\) and \(\frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})}< \varphi < 1\), \(\bar{\rho _{C}}\) exist. Now,

$$\begin{aligned} {g}'(\bar{\rho _{C}})<0, {g}'(0)>0, {g}'(1)>0. \end{aligned}$$
(B23)

\(\square \)

Proof of Theorem 5

When \(S_{1}+\theta (S_{2} +(1-S_{2}-T_{2})\varphi ) < 0\), \({g}'(0)<0.\) Considering \(\varphi \in (0,1)\), we get \(\theta \ge \frac{-S_{1}}{1-T_{2}}\) and \(0<\varphi <\frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})}\). The same reason can be used to obtain when \(S_{1}+\theta (S_{2} +(1-S_{2}-T_{2})\varphi ) > 0\). When \(0< \theta < \frac{-S_{1}}{1-T_{2}} \) and \(0<\varphi < 1\), the proof is the same as in Case1. \(\square \)

Proof of Theorem 6

To ensure \(\bar{\rho _{C} } \in (0,1)\), we have

$$\begin{aligned} \frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})}< \varphi < \frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})}. \end{aligned}$$
(B24)

Considering \(\varphi \in (0,1)\), the following conditions need to be satisfied:

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} \frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})}\ge 0\\ \frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})}\le 1. \end{array}\right. } \end{aligned} \Rightarrow \theta \ge \frac{-S_{1}}{1-T_{2}}. \end{aligned}$$
(B25)

Then, \({g}'(0)<0\), \({g}'(1)<0\), \({g}'(\bar{\rho _{C}})>0\).

When \(\bar{\rho _{C}}\le 0\) or \(\bar{\rho _{C}}\ge 1\), internal equilibrium point \(\bar{\rho _{C}}\) doesn’t exist. We have

$$\begin{aligned} & \theta > \frac{-S_{1}}{1-T_{2}}~~ and~~ \frac{-S_{1}-\theta S_{2} }{\theta (1-S_{2}-T_{2})} \le \varphi< 1 \Rightarrow {g}'(1)<0. \end{aligned}$$
(B26)
$$\begin{aligned} & \theta > \frac{T_{1}-1}{1-T_{2}}~~ and~~ 0< \varphi \le \frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})} \Rightarrow {g}'(0)<0. \end{aligned}$$
(B27)

When \(0< \theta \le \frac{T_{1}-1}{1-T_{2}}\) and \(0<\varphi < 1\), then \(\bar{\rho _{C}}<0\), internal equilibrium point \(\bar{\rho _{C}}\) doesn’t exist. Now,

$$\begin{aligned} {g}'(0)<0, {g}'(1)>0. \end{aligned}$$
(B28)

When \(\frac{T_{1}-1}{1-T_{2}}< \theta < \frac{-S_{1}}{1-T_{2}}\) and \(\frac{T_{1}-1-\theta S_{2} }{\theta (1-S_{2}-T_{2})}< \varphi <1\), \(\bar{\rho _{C}}\) exist. We have

$$\begin{aligned} {g}'(\bar{\rho _{C}})>0, {g}'(0)<0,{g}'(1)<0. \end{aligned}$$
(B29)

\(\square \)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xing, A., Zhang, G. & Tian, H. Evolution of Cooperation in Prisoner’s Dilemma within Changeable External Environments. Dyn Games Appl (2024). https://doi.org/10.1007/s13235-024-00607-1

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13235-024-00607-1

Keywords

Search

Navigation