2. (Second-price auction) Consider the second price auction in the independent private values framework. The reserve price is 0, so the object is always sold. Two bidders i=1,2 have independent, uniformly drawn valuations ~ U[0, 1] and simultaneously place bids by. The winner is the highest bidder, but the price paid is the bid of the second bidder. The loser gets nothing. To summarize: if by > b2, payoffs are (v1-b2, 0). If by
2. (Second-price auction) Consider the second price auction in the independent private values framework. The reserve price is 0, so the object is always sold. Two bidders i=1,2 have independent, uniformly drawn valuations ~ U[0, 1] and simultaneously place bids by. The winner is the highest bidder, but the price paid is the bid of the second bidder. The loser gets nothing. To summarize: if by > b2, payoffs are (v1-b2, 0). If by
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![2. (Second-price anction) Consider the second price auction in the independent private values framework.
The reserve price is 0, so the object is always sold. Two bidders i = 1,2 have independent, uniformly
drawn valuations v;~ U(0, 1] and simultaneously place bids b,. The winner is the highest bidder, but
the price paid is the bid of the second bidder. The loser gets nothing.
To summarize: if by > b2, payoffs are (v – b2, 0). If bi < b2, the payoffs are (0, v2 - bi).
(a) An asymmetric BNE exists in which player 1 always gets the good. Determine the strategy profile
which makes this a BNE. Remember: in a BNE, no type of any player should be able to benefit
by deviating.
(b) A symmetric BNE also exists in which both players bid truthfully: they both play the strategy
b(v:) = v. To verify this, suppose that player 2 plays this strategy, and player 1 has a valuation
of vn E (0, 1).
(i) Why would player 1 do worse by deviating to bị > v1?
(ii) Why would player 1 do worse by deviating to by < v1?
(c) (Optional Challenge) What is the seller's expected revenue in the symmetric equilibrium above?
(i.e., what is the expectation of min{vi, v2}?)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ec7a4a2-6dfe-4257-a7a2-1065741fa033%2F476653b1-1c76-4d8f-abb6-fe7196ad4444%2Fe8bhb7w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. (Second-price anction) Consider the second price auction in the independent private values framework.
The reserve price is 0, so the object is always sold. Two bidders i = 1,2 have independent, uniformly
drawn valuations v;~ U(0, 1] and simultaneously place bids b,. The winner is the highest bidder, but
the price paid is the bid of the second bidder. The loser gets nothing.
To summarize: if by > b2, payoffs are (v – b2, 0). If bi < b2, the payoffs are (0, v2 - bi).
(a) An asymmetric BNE exists in which player 1 always gets the good. Determine the strategy profile
which makes this a BNE. Remember: in a BNE, no type of any player should be able to benefit
by deviating.
(b) A symmetric BNE also exists in which both players bid truthfully: they both play the strategy
b(v:) = v. To verify this, suppose that player 2 plays this strategy, and player 1 has a valuation
of vn E (0, 1).
(i) Why would player 1 do worse by deviating to bị > v1?
(ii) Why would player 1 do worse by deviating to by < v1?
(c) (Optional Challenge) What is the seller's expected revenue in the symmetric equilibrium above?
(i.e., what is the expectation of min{vi, v2}?)
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