# The Tragedy of the Commons as a Prisoner’s Dilemma. Its Relevance for Sustainability Games

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## Abstract

**:**

## 1. Introduction

## 2. Modeling the Tragedy

#### 2.1. Diekerts’s Representation of the Tragedy of the Commons

#### 2.2. Shortcomings of Diekerts’s Representation and an Alternative Formulation Removing Them

## 3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. A Running Example with N = 3 Agents

**Table A1.**Explicit 3-player game-theoretic representation of the Tragedy of the Commons. Different colors and styles are used to denote the payoffs of the various agents: “Player 1” (black), “Player 2” (red), “Player 3” (bolded blue).

All Other 2 Players | |||||
---|---|---|---|---|---|

Player 2 Cooperates Player 3 Cooperates | Player 2 Defects, Player 3 Cooperates | Player 2 Cooperates Player 3 Defects | Player 2 Defects Player 3 Defects | ||

Player 1 | Cooperate | $1,1,1$ | $\frac{1}{3},\frac{4}{3},\frac{1}{3}$ | $\frac{1}{3},\frac{1}{3},\frac{4}{3}$ | $-\frac{1}{3},\frac{2}{3},\frac{2}{3}$ |

Defect | $\frac{4}{3},\frac{1}{3},\frac{1}{3}$ | $\frac{2}{3},\frac{2}{3},-\frac{1}{3}$ | $\frac{2}{3},-\frac{1}{3},\frac{2}{3}$ | $0,0,0$ |

#### Appendix A.2. A Rigorous Proof of the Main Result

- (i)
- ${\pi}_{n}\u2329{\pi}_{0}\le 1\text{}\forall n\u232a0$;
- (ii)
- $\frac{\partial {\pi}_{n}}{\partial n}<0\text{}\forall n{n}^{crit\text{}}$;
- (iii)
- $\frac{\partial {\pi}_{n}}{\partial n}=0\text{}\forall n{n}^{crit\text{}}$;
- (iv)
- ${\pi}_{{n}^{crit}}=0$.

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**Table 1.**Payoff matrix of Hardin’s game-theoretic representation of the Tragedy of the Commons according to Diekert’s Figure 3 (Diekert [24], p. 1779).

All Other Players | ||
---|---|---|

Player 1 | Cooperate | Defect |

Cooperate | $1,1$ | $1-\epsilon ,\text{}2-\epsilon $ |

Defect | $2-\epsilon ,1-\epsilon $ | $0,0$ |

**Table 2.**Payoff matrix of the “corrected” game-theoretic representation of the Tragedy of the Commons when each player imposes a cost $\mathsf{\epsilon}$ on the others.

All Other Players | ||
---|---|---|

Player 1 | Cooperate | Defect |

Cooperate | $1,1$ | $-\frac{N-2}{N},\text{}\frac{2}{N}$ |

Defect | $2-\frac{2}{N},1-\frac{2}{N}$ | $0,0$ |

**Table 3.**The explicit N-player game-theoretic representation of the Tragedy of the Commons based on splitting the ($N-1$ ) “other players” into subgroups of cooperators and defectors. Different colors and styles are used to denote the payoffs of the various agents: “Player 1” (black), “other players” that do cooperate (red), “other players” that defect (bolded blue).

All Other Players $(\mathit{N}-1)$ | |||||
---|---|---|---|---|---|

$\mathit{N}-1$ Cooperate (No One Defects) | $\mathit{N}-2$ Cooperate, 1 Defects | … | All Defect | ||

Player 1 | Cooperate | $1,1,\dots 1$ | $1-\frac{2}{N},1-\frac{2}{N}\dots ,1-\frac{2}{N},$ $2-\frac{2}{N}$ | … | $-\frac{N-2}{N}$, $\frac{2}{N},\dots ,\frac{2}{N}$ |

Defect | $2-\frac{2}{N},$ $1-\frac{2}{N},\dots ,1-\frac{2}{N}$ | $2-\frac{4}{N},$ $1-\frac{4}{N},\dots ,1-\frac{4}{N},$ $2-\frac{4}{N}$ | … | $0,$ $0,\dots ,0$ |

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Carrozzo Magli, A.; Della Posta, P.; Manfredi, P.
The Tragedy of the Commons as a Prisoner’s Dilemma. Its Relevance for Sustainability Games. *Sustainability* **2021**, *13*, 8125.
https://doi.org/10.3390/su13158125

**AMA Style**

Carrozzo Magli A, Della Posta P, Manfredi P.
The Tragedy of the Commons as a Prisoner’s Dilemma. Its Relevance for Sustainability Games. *Sustainability*. 2021; 13(15):8125.
https://doi.org/10.3390/su13158125

**Chicago/Turabian Style**

Carrozzo Magli, Alessio, Pompeo Della Posta, and Piero Manfredi.
2021. "The Tragedy of the Commons as a Prisoner’s Dilemma. Its Relevance for Sustainability Games" *Sustainability* 13, no. 15: 8125.
https://doi.org/10.3390/su13158125