Summary of "Lecture-08: Tragedy of Commons- Introduction and Analysis"

Main Ideas and Concepts

Introduction to the Tragedy of the Commons:
The lecture discusses the "Tragedy of the Commons," a scenario in which multiple agents compete for a shared resource, leading to its overexploitation. Examples of common resources include forests, fisheries, and mineral mines.
Modeling Competition:
The focus is on modeling the strategic interaction between two timber agencies that log trees in a forest. Each agency can exert a certain effort (denoted as e1 and e2) to log trees, and their payoffs depend on their own efforts as well as the combined efforts of both agencies.
Payoff Functions:
The payoff for each agency is modeled as:
- Agency 1: u1(e1, e2) = e1(1 - e1 - e2)
- Agency 2: u2(e2, e1) = e2(1 - e1 - e2)
The payoffs are proportional to their own efforts but decrease with the total effort exerted by both agencies, reflecting the depletion of the resource.
Best Response and Nash Equilibrium:
The best response for each agency is derived by maximizing their respective Payoff Functions. For Agency 1, the best response is given by:
- e1* = (1 - e2) / 2
For Agency 2, the best response is:
- e2* = (1 - e1) / 2
The Nash Equilibrium occurs where both agencies are playing their best response strategies simultaneously.
Calculating the Nash Equilibrium:
By solving the simultaneous equations derived from the best responses, the Nash Equilibrium efforts are found to be:
- e1* = 1/3
- e2* = 1/3
The Nash payoff for both agencies at equilibrium is:
- u1(e1*, e2*) = u2(e2*, e1*) = 1/9
Implications and Extensions:
The lecture emphasizes that while the example considers only two agencies, the model can be extended to include more agents, leading to further insights into the tragedy of commons scenario.

Methodology/Instructions

Modeling the Game:
- Identify the common resource and the competing agents.
- Define the effort levels for each agent.
- Establish the Payoff Functions based on individual efforts and joint efforts.
Finding Best Responses:
- Differentiating the Payoff Functions with respect to each agent's effort.
- Setting the derivatives equal to zero to find the best response functions.
Determining Nash Equilibrium:
- Solve the system of equations formed by the best response functions to find the equilibrium efforts.
- Calculate the corresponding payoffs at the Nash Equilibrium.

Speakers or Sources Featured

The lecture appears to be conducted by an unnamed speaker, likely a professor or instructor in a course on Game Theory. The content is educational and based on theoretical concepts in economics and strategic interaction.