# The Tragedy of the Commons from a Game-Theoretic Perspective

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## Abstract

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## 1. A High-Impact Publication

#### 1.1. Illustrative Examples of a Coordination and a Cooperation Game

## 2. Game Theory and Common Property

As a rationale being, each herdsman seeks to maximize his gain. Explicitly or implicitly, more or less consciously, he asks, “what is the utility to me of adding one more animal to my herd?” This utility has a negative and a positive component.(1) The positive component is a function of the increment of one animal. Since the herdsman receives all the proceeds from the sale of the additional animal, the positive utility is nearly +1.(2) The negative component is a function of the additional overgrazing created by one more animal. Since, however, the effects of overgrazing are shared by all [...], the negative utility for any particular decision-making herdsman is only a fraction of −1.Adding together the component particular utilities, the rational herdsman concludes that the only sensible course for him to pursue is to add another animal to his herd. And another; and another... But this is the conclusion reached by each and every rational herdsman sharing a commons. Therein is the tragedy. Each man is locked into a system that compels him to increase his herd without limit—in a world that is limited. Ruin is the destination towards which all men rush, each pursuing his own best interest in a society that believes in the freedom of the commons. Freedom in a commons brings ruin to all. [1, p. 1244]

_{max}. Effort causes cost according to some function c(E) and produces, for a given resource stock S, an amount H of harvest that can be sold at price p. Current profits (the term in the squared brackets) are exponentially discounted at rate r. The stock of the resource is governed by the dynamic Equation (2), where g(S) is the “natural” growth function of the resource. So, when growth is larger than harvest, the stock will increase, and when growth is smaller than harvest, the stock will decline. When re-growth and harvest are equal, the stock will not change over time. The Net-Present-Value is the discounted sum of profits: revenue (the harvest multiplied by its price per unit) minus the cost of effort that is required for harvesting. To keep things very simple it is common to assume that cost and harvest are proportional to effort and that the latter depends on the existing state of the resource according to some function q(S), so that H(S, E) = q(S)E. The Hamiltonian of this problem can then be written as:

**Figure 4.**Stock development under optimal and non-cooperative management. (

**a**) Social planner; (

**b**) Two symmetric players.

^{i }<c

^{j}). Suppose the game starts and the players harvest the stock down to a level where the less efficient player makes no more profits. He or she would then leave the fishery. Would the other player then want to deplete the stock further? No, of course not; but player i cannot let the stock increase either because then the less efficient player would want to enter again. So in effect, when the players are asymmetric in this simple game, some rents remain in equilibrium (but less than in the social optimum). The situation is illustrated in Figure 5a, where λ refers to the marginal profit of the most efficient player.

**Figure 5.**Stock development under non-cooperative management. (

**a**) Two asymmetric players; (

**b**) Three asymmetric players.

## 3. Cooperation in the Commons

**Figure 6.**Illustration of two-phase punishment scheme in [10]. (

**1**) cooperation; (

**2**) defection; (

**3**) punishment; (

**4**) cooperation.

## 4. Game Theory, Cooperation, and Climate Change

## 5. Conclusions

## Acknowledgments

## References

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Diekert, F.K. The Tragedy of the Commons from a Game-Theoretic Perspective. *Sustainability* **2012**, *4*, 1776-1786.
https://doi.org/10.3390/su4081776

**AMA Style**

Diekert FK. The Tragedy of the Commons from a Game-Theoretic Perspective. *Sustainability*. 2012; 4(8):1776-1786.
https://doi.org/10.3390/su4081776

**Chicago/Turabian Style**

Diekert, Florian K. 2012. "The Tragedy of the Commons from a Game-Theoretic Perspective" *Sustainability* 4, no. 8: 1776-1786.
https://doi.org/10.3390/su4081776