Tragedy of the commons

Tragedy of the commons

• 1.
  Tragedy of thecommons: A game theoretic approach
• 2.
  Introduction Citizens- Respond toprivate 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 𝑖𝑠𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 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’sobjective- 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 • Thatmeans 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 ifwe 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 • Thecomparison 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.