Question

You owns a gold mine whose output depends on the amount of gold in the mine x. You know the value of x but what all the others know is only that x is uniformly distributed on the interval [0,1]. Before excavating the mine, you can sell the mine to a large mining company which has more advanced extraction methods. You can ask the company for any price p ≥ 0 and the company can reject or accept the offer. If the mining company rejects the offer then you are left to mine yourself, and the corresponding payoff  is 2x. If the company accepts the oer then your payoff is the price p while the company's payoff is given by the net value 3x−p, which is common knowledge.

(a)Show that for a given price p ≥ 0 there is a threshold type x(p) ∈ [0,1] such that any type below the threshold x < x(p) will prefer to sell the mine, while types above the threshold x > x(p) will prefer to self-mine.

(b)Find the pure strategy Bayesian Nash equilibrium of this game and show that it is unique.

(c)What is the expected payoff of each type x and that of mining company in the equilibrium you derived in (c)?

Answer 1

1. at x_p threshold value, the returns from either choices should be same.

Payoff(Selfmining) = Payoff(selling)

2x_p = p

x_p = 0.5p

Let's check

1. at x < x_p

Payoff while selfmining < p

Payoff while selling = p

So, selling will be preferred.

2. at x> x_p

Payoff while selfmining >p

payoff while selling = p

So, selfmining will be preferred.

b) BNE will occur if the seller sells it at x_p  and the mining firm accepts it as mining firm can only choose between zero profit or 3x-p and out of that, 3x-p will be maximum at x=x_p because above this point, seller won't sell it, He will selfmine it. Hence, max will occur at x=x_p. This is unique as below this point, seller will be desperate to sell it because he will get more money through selling it. while above this point, he will selfmine it, but company being the second player in this same will always choose 3x-p above zero or lesser profit. So, this is unique BNE.

c) Seller will get payoff = p, Company will get the payoff = 3(0.5p) - p = 0.5p