12.5 Not All That Glitters: A prospector owns a gold mine where he can dig to recover gold. His output depends on the amount of gold in the mine, denoted by x. The prospector knows the value of x, but the rest of the world knows only that the amount of gold is uniformly distributed on the interval [0, 1]. Before deciding to mine, the prospector can try to sell his mine to a large mining company, which is much more efficient in its extraction methods. The prospector can ask the company owner for any price p≥ 0, and the owner can reject (R) or accept (A) the offer. If the owner rejects the offer then the prospector is left to mine himself, and his payoff from self-mining is equal to 3x. If the owner accepts the offer then the prospector's payoff is the price p, while the owner's payoff is given by the net value 4x - p, and this is common knowledge. pter 12 Bayesian Games Show that for a given price p≥ 0 there is a threshold type x (p) = [0, 1] of prospector, such that types below x (p) will prefer to sell the mine, while types above x (p) will prefer to self-mine. b. Find the pure-strategy Bayesian Nash equilibrium of this game, and show that it is unique. What is the expected payoff of each type of prospector and of the company owner in the equilibrium you derived? a.

Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter11: Rational And Irrational Numbers
Section: Chapter Questions
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12.5
Not All That Glitters: A prospector owns a gold mine where he can dig to
recover gold. His output depends on the amount of gold in the mine, denoted
by x. The prospector knows the value of x, but the rest of the world knows
only that the amount of gold is uniformly distributed on the interval [0, 1].
Before deciding to mine, the prospector can try to sell his mine to a large
mining company, which is much more efficient in its extraction methods. The
prospector can ask the company owner for any price p≥ 0, and the owner
can reject (R) or accept (A) the offer. If the owner rejects the offer then the
prospector is left to mine himself, and his payoff from self-mining is equal to
3x. If the owner accepts the offer then the prospector's payoff is the price p,
while the owner's payoff is given by the net value 4x p, and this is common
knowledge.
apter 12 Bayesian Games
-
a.
Show that for a given price p≥ 0 there is a threshold type x (p) = [0, 1]
of prospector, such that types below x(p) will prefer to sell the mine,
while types above x (p) will prefer to self-mine.
b. Find the pure-strategy Bayesian Nash equilibrium of this game, and
show that it is unique. What is the expected payoff of each type of
prospector and of the company owner in the equilibrium you derived?
Transcribed Image Text:12.5 Not All That Glitters: A prospector owns a gold mine where he can dig to recover gold. His output depends on the amount of gold in the mine, denoted by x. The prospector knows the value of x, but the rest of the world knows only that the amount of gold is uniformly distributed on the interval [0, 1]. Before deciding to mine, the prospector can try to sell his mine to a large mining company, which is much more efficient in its extraction methods. The prospector can ask the company owner for any price p≥ 0, and the owner can reject (R) or accept (A) the offer. If the owner rejects the offer then the prospector is left to mine himself, and his payoff from self-mining is equal to 3x. If the owner accepts the offer then the prospector's payoff is the price p, while the owner's payoff is given by the net value 4x p, and this is common knowledge. apter 12 Bayesian Games - a. Show that for a given price p≥ 0 there is a threshold type x (p) = [0, 1] of prospector, such that types below x(p) will prefer to sell the mine, while types above x (p) will prefer to self-mine. b. Find the pure-strategy Bayesian Nash equilibrium of this game, and show that it is unique. What is the expected payoff of each type of prospector and of the company owner in the equilibrium you derived?
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