(30 points) Suppose n bidders participate in a first-price auction. Each bidder's valuation is uniformly distributed...
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(30 points) Suppose n bidders participate in a first-price auction. Each bidder's valuation is uniformly distributed on [0, 1]. (a) (2 points) Suppose all other bidders adopt a bidding strategy whose inverse is o-¹(b). Express bidder i's expected profits in terms of vi, b, o-¹(b), and n. (b) (10 points) The first-order condition for the bidder's optimization problem is (n − 1) (v — b) (0−¹(b))"-² o−¹¹(b) – (0−¹(b))"−¹ = 0 -1 n-1 n-2 Suppose that o-¹(b) = kb. Plug o-¹ and o-¹ into the first-order condition and solve for v in terms of b and n. (c) (3 points) For which value of k is o-¹(b) = kb a symmetric Nash equilibrium? (d) (15 points) Find the seller's expected revenue in three different ways. i. (5 points) Calculate the expected revenue "directly" using what we know about order statistics for uniformly distributed random variables and the fact that o(v) = % 0. ii. (5 points) Calculate the expected revenue in a second-price auction with n bidders and Vi~ U[0, 1] and invoke the revenue equivalence theorem. iii. (5 points) Calculate the expected payment, T(v), for any symmetric auction in which bidders with a valuation of zero expect to pay zero-i.e. T(0) = 0 and the bidder with the highest valuation wins-i.e. P(v) = vn-1. Calculate the seller's expected revenue as n E[T(v)]. Hint: f cx dx = x²+¹ a+1 +100+1 = (30 points) Suppose n bidders participate in a first-price auction. Each bidder's valuation is uniformly distributed on [0, 1]. (a) (2 points) Suppose all other bidders adopt a bidding strategy whose inverse is o-¹(b). Express bidder i's expected profits in terms of vi, b, o-¹(b), and n. (b) (10 points) The first-order condition for the bidder's optimization problem is (n − 1) (v — b) (0−¹(b))"-² o−¹¹(b) – (0−¹(b))"−¹ = 0 -1 n-1 n-2 Suppose that o-¹(b) = kb. Plug o-¹ and o-¹ into the first-order condition and solve for v in terms of b and n. (c) (3 points) For which value of k is o-¹(b) = kb a symmetric Nash equilibrium? (d) (15 points) Find the seller's expected revenue in three different ways. i. (5 points) Calculate the expected revenue "directly" using what we know about order statistics for uniformly distributed random variables and the fact that o(v) = % 0. ii. (5 points) Calculate the expected revenue in a second-price auction with n bidders and Vi~ U[0, 1] and invoke the revenue equivalence theorem. iii. (5 points) Calculate the expected payment, T(v), for any symmetric auction in which bidders with a valuation of zero expect to pay zero-i.e. T(0) = 0 and the bidder with the highest valuation wins-i.e. P(v) = vn-1. Calculate the seller's expected revenue as n E[T(v)]. Hint: f cx dx = x²+¹ a+1 +100+1 =
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a Bidder is expected profits is equal to vi b1b This is because the expected value of a uniform dist... View the full answer
Related Book For
Managerial Economics A Problem Solving Approach
ISBN: 978-1133951483
3rd edition
Authors: Luke M. Froeb, Brian T. McCann, Mikhael Shor, Michael R. War
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