. 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.

. 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.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.7P

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