Consider a second-price sealed-bid auction of a single object with two bidders. Both bidders have the same value v for the object. Bidder 1 knows the exact value of v while bidder 2 knows only that v is uniformly distributed on [0, 1]. (a) Identify all weakly dominated strategies for each player. Note that a strategy for bidder 1 consists of a bid b(v) for each v € [0, 1], while a strategy for bidder 2 consists only of a single bid b. We say that a strategy is weakly dominated for bidder 1 if there is some v € [0, 1] for which the bid b(v) is weakly dominated.
Consider a second-price sealed-bid auction of a single object with two bidders. Both bidders have the same value v for the object. Bidder 1 knows the exact value of v while bidder 2 knows only that v is uniformly distributed on [0, 1]. (a) Identify all weakly dominated strategies for each player. Note that a strategy for bidder 1 consists of a bid b(v) for each v € [0, 1], while a strategy for bidder 2 consists only of a single bid b. We say that a strategy is weakly dominated for bidder 1 if there is some v € [0, 1] for which the bid b(v) is weakly dominated.
Chapter1: Making Economics Decisions
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![. Consider a second-price sealed-bid auction of a single object with two bidders. Both bidders
have the same value v for the object. Bidder 1 knows the exact value of v while bidder 2
knows only that v is uniformly distributed on [0, 1].
(a) Identify all weakly dominated strategies for each player. Note that a strategy for bidder
1 consists of a bid b(v) for each v € [0, 1], while a strategy for bidder 2 consists only of
a single bid b. We say that a strategy is weakly dominated for bidder 1 if there is some
v € [0, 1] for which the bid b(v) is weakly dominated.
Solution: Since bidder 1 knows the value, bidding b(v) = v weakly dominates every other
strategy. For bidder 2, any bid b> 1 is weakly dominated by b = 1; no bid b = [0, 1] is
weakly dominated.
(b) Find all Nash equilibria in which neither bidder uses a weakly dominated strategy.
Solution: By part (a), bidder 1 must use the strategy b(v) = v. Given this strategy,
bidder 2 is indifferent among all bids b = [0, 1]: whenever he wins the auction, he pays
exactly his value and receives a payoff of 0. Therefore, there is a Nash equilibrium for
each b = [0, 1] given by bidder 1 bidding b(v) = v and bidder 2 bidding b.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0542aa5b-cdbd-4921-99ff-0dadb1c30f45%2Fbb9dfd92-4f33-4439-8507-bb41258827b6%2Fdyualhz9_processed.png&w=3840&q=75)
Transcribed Image Text:. Consider a second-price sealed-bid auction of a single object with two bidders. Both bidders
have the same value v for the object. Bidder 1 knows the exact value of v while bidder 2
knows only that v is uniformly distributed on [0, 1].
(a) Identify all weakly dominated strategies for each player. Note that a strategy for bidder
1 consists of a bid b(v) for each v € [0, 1], while a strategy for bidder 2 consists only of
a single bid b. We say that a strategy is weakly dominated for bidder 1 if there is some
v € [0, 1] for which the bid b(v) is weakly dominated.
Solution: Since bidder 1 knows the value, bidding b(v) = v weakly dominates every other
strategy. For bidder 2, any bid b> 1 is weakly dominated by b = 1; no bid b = [0, 1] is
weakly dominated.
(b) Find all Nash equilibria in which neither bidder uses a weakly dominated strategy.
Solution: By part (a), bidder 1 must use the strategy b(v) = v. Given this strategy,
bidder 2 is indifferent among all bids b = [0, 1]: whenever he wins the auction, he pays
exactly his value and receives a payoff of 0. Therefore, there is a Nash equilibrium for
each b = [0, 1] given by bidder 1 bidding b(v) = v and bidder 2 bidding b.
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