This document discusses the tragedy of the commons using a game theoretic model of farmers grazing goats on a common grassland. It describes how each farmer acts in their own self-interest to maximize the number of goats grazed, which leads to overutilization of the common resource through the Nash equilibrium. The Nash equilibrium results in more goats being grazed than the socially optimal equilibrium, demonstrating the suboptimal outcome of acting only on private incentives when involving a common resource.

1. Tragedy of the commons: A game theoretic approach

2. Introduction
Citizens- Respond to private incentives
Consequence-
1. Public Goods- Underprovided
2. Public/Common Resources- Over utilized
One current scenario is the overutilization of common property
resources- Tragedy of the Common

3. Common Property Game: Structure
• Suppose that there is a grassland outside a locality in which the
farmers can graze their goats
• Number of farmers- n
• 𝑔𝑖 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑜𝑎𝑡𝑠 𝑔𝑟𝑎𝑧𝑒𝑑 𝑏𝑦 𝑖𝑡ℎ 𝑓𝑎𝑟𝑚𝑒𝑟
• 𝐺 = 𝑔1 + −− − + 𝑔 𝑛 (total number of goats grazed in the
grassland)
• 𝑐 = 𝑐𝑜𝑠𝑡 𝑜𝑓 𝑏𝑢𝑦𝑖𝑛𝑔 𝑎𝑛𝑑 𝑐𝑎𝑟𝑖𝑛𝑔 𝑒𝑎𝑐ℎ 𝑔𝑜𝑎𝑡 𝑐 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
• 𝑣 = 𝑣𝑎𝑙𝑢𝑒 𝑒𝑛𝑗𝑜𝑦𝑒𝑑 𝑓𝑟𝑜𝑚 𝑔𝑜𝑎𝑡𝑠 𝑏𝑦 𝑡ℎ𝑒 𝑓𝑎𝑟𝑚𝑒𝑟 ∗ which definitely
depends upon the number of goat grazed**

4. Continued
• 𝑣 𝑖𝑠 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑢𝑝𝑜𝑛 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑜𝑎𝑡𝑠 𝑔𝑟𝑎𝑧𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑔𝑟𝑎𝑠𝑠𝑙𝑎𝑛𝑑
• 𝑣 = 𝑣 𝐺
• And 𝑣 𝐺 > 0 if 𝐺 < 𝐺 𝑚𝑎𝑥 *
• If 𝐺≥𝐺 𝑚𝑎𝑥 then 𝑣 𝐺 = 0
• Initially each goat gets plenty of grass but as the number of goats
increase there is a decline in average availability of grass
• So we can write
1. 𝑣/
𝐺 < 0
2. 𝑣//
𝐺 < 0
Provided 𝐺 < 𝐺 𝑚𝑎𝑥

5. Nash Equilibrium
• Farmer’s objective- to maximize his payoff with respect to the
no of goats (𝑔𝑖
∗
- optimum number of goat for the ith farmer)
• 𝑔𝑖 𝜖 0, ∞
• Now the total no of goats grazed can be expressed as
𝐺 = 𝑔1 + − − + 𝑔𝑖−1 + 𝑔𝑖 + 𝑔𝑖+1 + − − + 𝑔 𝑛
payoff of farmer-i will be expressed as
Π 𝑖 = 𝑔𝑖 𝑣𝑖 𝐺 − 𝑐𝑔𝑖
If each farmer seeks to maximize own payoff individually then
(𝑔1
∗
, ….., 𝑔 𝑛
∗ )→ 𝑁𝑎𝑠ℎ 𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚

6. Nash Equilibrium
• That means 𝑔𝑖
∗
would maximize the payoff of this farmer when
other farmers are choosing
(𝑔𝑖
∗
, … , 𝑔𝑖−1
∗
, 𝑔𝑖+1
∗
, … , 𝑔 𝑛
∗ )
So the objective function is
To maximize 𝑔𝑖 𝑣 𝑔𝑖 + 𝑔−𝑖
∗
− 𝑐𝑔𝑖
With respect to 𝑔𝑖
The first order condition requires
𝑣 𝑔𝑖 + 𝑔−𝑖
∗
+ 𝑔𝑖 𝑣/
𝑔𝑖 + 𝑔−𝑖
∗
− 𝑐 = 0
( for all, 𝑔−𝑖
∗
= 𝑔𝑖
∗
+ − − + 𝑔𝑖−1
∗
+𝑔𝑖+1
∗
+ − − + 𝑔 𝑛
∗ )

7. • Now if we sum up the equilibrium condition of all individual farmers
we get
𝑛𝑣 𝐺∗ + ∑𝑔𝑖 𝑣/ 𝐺∗ − 𝑛𝑐 = 0
Now if we divide both sides by n we get
𝑣 𝐺∗
+
𝐺∗
𝑛
𝑣/
𝐺∗
− 𝑛𝑐 = 0
If we consider the socially optimal equilibrium and it gives
𝐺 = 𝐺∗∗
The first order condition requires
𝑣 𝐺∗∗ +
𝐺∗∗
𝑛
𝑣/ 𝐺∗∗ − 𝑛𝑐 = 0

8. Sub Optimality
• The comparison generates that 𝐺∗ > 𝐺∗∗
• More goats are grazed in Nash Equilibrium compared to social
equilibrium
• There is always an incentive for a farmer to increase the number of
his goats as he does not consider its effect on the entire society.
• The common resource is over-utilized.