Consider N farmers, each of whom can produce the same quantity of wheat at zero cost as
Question:
Consider N farmers, each of whom can produce the same quantity of wheat at zero cost as they wish.
If the kth farmer produces qk, the total quantity produced is q = q1 + q2 + ... qk + ... + qN. The price of wheat is determined by p = e-q.
a) What is the shape of the global recipe [q] = qe-q for q 0?
b) Using the previous point, show that the strategy of producing one unit of wheat is dominant for each farmer. Deduce that a farmer's corresponding profit is e-N.
c) Suppose the farmers agree to each produce 1/N units of wheat. Again based on the first point, show that total profit is then maximum. Verify that the profit for each farmer is 1/eN. can such an agreement be respected in the absence of an explicit contract?
d) Why is this N-player game a generalization of the prisoner's dilemma?
Note: This game is an illustration of the "tragedy of the commons".