Evolution of Cooperation in Public Goods Games with Stochastic Opting-Out
Abstract
:1. Introduction
2. Model and Methods
3. Results
3.1. Pairwise Invasion Dynamics in Finite Populations
- Natural selection favors cooperation over defection , if ;
- Neutral evolution , if ;
- Natural selection favors defection over cooperation , if .
3.2. Approximations of the Critical Threshold for Natural Selection to Favor Cooperation
3.3. Adaptive Dynamics in Finite Populations
4. Discussion and Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Derivation of πd
Appendix B. Derivation of πc
Appendix C. Transition Matrix
Appendix D. Inequalities
Appendix D.1. Proof of (15)
Appendix D.2. Proof of (16)
Appendix D.3. Proof of (17)
Appendix D.4. Lemma 1:
Appendix D.5. Lemma 2:
Appendix D.6. Proof that as N/n → 0, R(α) > N(1 − α) ≈ Rexp (α)
Appendix D.7. Lemma 3: Behavior of R(α) as α → 0
Appendix E. Proof That R(α) Is Strictly Decreasing on [0, 1)
Proof S(αinv)U(α) > S(α)U(αinv) for αinv < α
Appendix F. Justification of Approximations
Appendix F.1. Approximation for R(α) as N → ∞,
Appendix F.2. Approximation for R(α) for n ≫ N ≫ 0
Appendix F.3. Approximation for R(α) as N → ∞
Appendix F.4. Approximation for R(α) as N/n → 0 and α → 1
Appendix G. Derivation of πy
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Ginsberg, A.G.; Fu, F. Evolution of Cooperation in Public Goods Games with Stochastic Opting-Out. Games 2019, 10, 1. https://doi.org/10.3390/g10010001
Ginsberg AG, Fu F. Evolution of Cooperation in Public Goods Games with Stochastic Opting-Out. Games. 2019; 10(1):1. https://doi.org/10.3390/g10010001
Chicago/Turabian StyleGinsberg, Alexander G., and Feng Fu. 2019. "Evolution of Cooperation in Public Goods Games with Stochastic Opting-Out" Games 10, no. 1: 1. https://doi.org/10.3390/g10010001