# Evolution of Cooperation in Public Goods Games with Stochastic Opting-Out

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## Abstract

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## 1. Introduction

## 2. Model and Methods

## 3. Results

#### 3.1. Pairwise Invasion Dynamics in Finite Populations

- Natural selection favors cooperation over defection ${x}_{1}>1/n>{y}_{1}$, if ${\pi}_{c}-{\pi}_{d}>0$;
- Neutral evolution ${x}_{1}=1/n={y}_{1}$, if ${\pi}_{c}-{\pi}_{d}=0$;
- Natural selection favors defection over cooperation ${x}_{1}<1/n<{y}_{1}$, if ${\pi}_{c}-{\pi}_{d}<0$.

#### 3.2. Approximations of the Critical Threshold $R(\alpha )$ 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 π_{d}

## Appendix B. Derivation of π_{c}

## 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: $1-{\alpha}^{N-1}N+{\alpha}^{N}(N-1)>1$

#### Appendix D.5. Lemma 2: $1-[1-{\alpha}^{N}]/\left[N(1-\alpha )\right]>0$

#### Appendix D.6. Proof that as N/n → 0, R(α) > N(1 − α) ≈ R_{exp} (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(α_{inv})U(α) > S(α)U(α_{inv}) for α_{inv} < α

## Appendix F. Justification of Approximations

#### Appendix F.1. Approximation for R(α) as N → ∞, $\frac{N}{n-1}=c$

#### 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 π_{y}

## References

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**Figure 1.**Model schematic of stochastic opting-out. Cooperators (blue) and defectors (red) are represented by dots. A fixed number of agents are randomly drawn from the population to participate in a PGG, represented by the small tan rectangular area. While most selected agents are able to make it to the game, some are not. Selected agents then return to the general populace, where no game is occurring. The fitness of agents is determined by the average payoffs they obtain from PGG interactions (cooperators vs. defectors) as well as from non-participation. Natural selection drives the co-evolutionary dynamics of opting-out behavior (the probability of non-participation, $\alpha $), and cooperation (the probability to cooperate in the PGG, $\beta $).

**Figure 2.**Pairwise invasion dynamics in finite populations. Shown are the graphs of fixation probabilities for ${\pi}_{c}-{\pi}_{d}>0$, as in panels (

**a**,

**b**), and for ${\pi}_{c}-{\pi}_{d}<0$, as in panels (

**c**,

**d**). If ${\pi}_{c}-{\pi}_{d}>0$, the fixation probability starting with one cooperator ${x}_{1}$ is always larger $1/n$ (neutral drift) which is in turn always larger than that of one defector, ${y}_{1}$. On the other hand, if ${\pi}_{c}-{\pi}_{d}<0$, then the situation is reversed, that is, ${y}_{1}>1/n>{x}_{1}$. We confirm that the specific values chosen for ${\pi}_{c}-{\pi}_{d}$ are admissible for the given values of population size n.

**Figure 3.**Critical threshold $R(\alpha )$ of the PGG investment return (i.e., enhancement factor) r required for cooperation to be favored. The shaded areas represent combinations of the PGG enhancement factor r and the probability of non-participation (i.e., opting-out) $\alpha $ which promote cooperation for game size $N=5$ and $n=10$ in (

**a**), $N=500$ and $n=1000$ in (

**b**), $N=5$ and $n=7$ in (

**c**), and $N=500$ and n = 1,000,000 in (

**d**). The yellow line in each panel is the approximation ${R}_{exp}(\alpha )$ as given in Equation (27).

**Figure 4.**Coevolution of cooperation and stochastic opting-out. Shown in Panels (

**a**–

**c**) are the adaptive dynamics using the StreamPlot function of Mathematica in a finite population of size $n=5$ for the selection pressure $\gamma =1$ and various values of return on investment, r, and payoff for non-participants, $\sigma $. Following the arrows leads to the most likely path the population will take in the strategy space. Please note that if $\sigma <r-1$, there exists a critical threshold of cooperativity ${\beta}^{*}=\sigma /(r-1)$ such that increasing likelihood of participation is more beneficial for agents with cooperativity $\beta >\sigma /(r-1)$ whereas participating cooperators are always prone to exploitation by others as shown in Panels (

**b.2**) and (

**c.2**). Hence, for $r<(2n-2)/(n-2)$, the only evolutionarily stable strategy (ESS) is $(1,0)$. However, if $r>(2n-2)/(n-2)>2$ where the game is no longer a social dilemma, $(0,1)$ is an ESS.

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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

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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 Style**

Ginsberg, 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