A First Course in Probability (10th Edition)
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Author: Sheldon Ross
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![[Auctions and Reserve Prices] Consider a seller who must sell a single private value
good. There are two potential buyers, each with a valuation that can take on one of three
values, vi E {0,1,2}, each value occurring with an equal probability of 1/3. The players' values
are independently drawn.
The seller will offer the good using a second-price sealed-bid auction, but he can set a reserve
price of r≥ 0 that modifies the rules of the auction as follows. As seen in class, if both bids are
below r, then neither bidder obtains the good and it goes "poof"! If both bids are at or above r
then the regular auction rules prevail. If only one bid is at or above r then that bidder with the
highest bid obtains the good and pays r to the seller.
a. What is the expected revenue of the seller when r = 0 (i.e., no reserve price)?
b. Is it still a weakly dominant strategy for each player to bid his valuation when r >
0? What is the expected revenue of the seller when r = 1?
c. What is the optimal reserve price r for the seller, and what is the optimal
expected revenue of the seller? (You may assume r is an integer.)](https://content.bartleby.com/qna-images/question/b9e27b5d-0ab8-428e-954d-97d64fc14c61/49a679dc-96d5-4b4d-a15f-0f0539a6b05f/1e11h4a_processed.png)
Transcribed Image Text:[Auctions and Reserve Prices] Consider a seller who must sell a single private value
good. There are two potential buyers, each with a valuation that can take on one of three
values, vi E {0,1,2}, each value occurring with an equal probability of 1/3. The players' values
are independently drawn.
The seller will offer the good using a second-price sealed-bid auction, but he can set a reserve
price of r≥ 0 that modifies the rules of the auction as follows. As seen in class, if both bids are
below r, then neither bidder obtains the good and it goes "poof"! If both bids are at or above r
then the regular auction rules prevail. If only one bid is at or above r then that bidder with the
highest bid obtains the good and pays r to the seller.
a. What is the expected revenue of the seller when r = 0 (i.e., no reserve price)?
b. Is it still a weakly dominant strategy for each player to bid his valuation when r >
0? What is the expected revenue of the seller when r = 1?
c. What is the optimal reserve price r for the seller, and what is the optimal
expected revenue of the seller? (You may assume r is an integer.)
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