The Dean is looking for a tenured professor to teach a course. Monetary incentives are needed to get someone interested, but how much? The Dean decides to use an auction to do the job. Two professors, equally qualified, applied for the position. The two professors are invited to covertly submit their bids to the Dean. The Dean will give the position to the professor who submits the lower bid (if there is a tie, the job is assigned randomly). The professor who gets the job will be paid his/her own bid. Each professorís reservation value for teaching the course is his/her private information. It is common knowledge that their reservation values are independently and uniformly distributed over [0,100]. So if a professor with a reservation value of 50 wins with a bid of 60, his payoff is 60 - 50 = 10. Please answer part a). (a) Find a Bayesian Nash equilibrium of the bidding game. (b) Suppose the two professorsíreservation values are 60 and 70, respectively. What are their bids in the Bayesian Nash equilibrium you computed in part (a)? Who is the winner? What is the payoff of the winner? (c) The faculty committee dislikes the auction rule. They propose to modify the auction slightly. Instead of the winner getting paid his/her own bid, the winner is paid his own bid plus a 10% premium. i. Find a Bayesian Nash equilibrium of the bidding game. ii. Suppose the two professors' reservation values are 60 and 70, respectively. What are their bids in the Bayesian Nash equilib- rium you computed in part (c)? Who is the winner? What is the payoff of the winner?
The Dean is looking for a tenured professor to teach
a course. Monetary incentives are needed to get someone
interested, but how much? The Dean decides to use an auction to do the
job. Two professors, equally qualified, applied for the position. The two
professors are invited to covertly submit their bids to the Dean. The Dean
will give the position to the professor who submits the lower bid (if there
is a tie, the job is assigned randomly). The professor who gets the job will
be paid his/her own bid. Each professorís reservation value for teaching
the course is his/her private information. It is common knowledge that
their reservation values are independently and uniformly distributed over
[0,100]. So if a professor with a reservation value of 50 wins with
a bid of 60, his payoff is 60 - 50 = 10.
Please answer part a).
(a) Find a Bayesian Nash equilibrium of the bidding game.
(b) Suppose the two professorsíreservation values are 60 and 70, respectively. What are their bids in the Bayesian Nash equilibrium you
computed in part (a)? Who is the winner? What is the payoff of the
winner?
(c) The faculty committee dislikes the auction rule. They propose to
modify the auction slightly. Instead of the winner getting paid his/her
own bid, the winner is paid his own bid plus a 10% premium.
i. Find a Bayesian Nash equilibrium of the bidding game.
ii. Suppose the two professors' reservation values are 60 and 70,
respectively. What are their bids in the Bayesian Nash equilib-
rium you computed in part (c)? Who is the winner? What is the
payoff of the winner?
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