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Econ 145: Suggested Answer for Practice Questions II Ichiro Obara March 13, 2010 1. Consider the second price auction with three bidders . (a) If ( v 1 ,v 2 ,v 3 )=(0 . 5 , 0 . 7 , 0 , 2) , who wins and pays how much? Answer. Bidder 2 wins and pays 0 . 5 . (b) Compute the expected revenue for the seller in dominant strat- egy equilibrium when the distribution of each bidder’s value is uniform on [0 , 1] . Answer. We know the formula for this: 1 Z 1 0 G 2 , 3 ( v ) dv =1 Z 1 0 F ( v ) 3 +3(1 F ( v )) F ( v ) 2 dv =1 Z 1 0 v 3 +3(1 v ) v 2 dv =1 Z 1 0 3 v 2 2 v 3 dv =1 [ v 3 1 2 v 4 ] 1 0 = 1 2 In general (with n bidders), this is n 1 n +1 . (c) Suppose that a reserve price is set to 0 . 2, that is, each bidder needs to bid more than 0 . 2 to win. If no one’s bid is higher than 0 . 2 , then the object will not be sold. Otherwise the highest bidder wins and pays the higher value between the second highest bid and 0 . 2 . Compute the expected payment of a bidder when his or her value is 0 . 5 . 1

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Answer. Bidder v won’t participate in the auction if v< 0 . 2 . In this case, this bidder’s expected payment is 0 . Suppose that v 0 . 2 . Bidder v pays the larger value of 0 . 2o r v (1 , 2) (max { 0 . 2 ,v (1 , 2) } ) conditional on winning, i.e. v (1 , 2) v. So bidder v ’s expected payment is E [max { 0 . 2 ,v (1 , 2) }| v (1 , 2) v ]Pr ³ v (1 , 2) v ´ The probability to win is Pr ³ v (1 , 2) v ´ = G 1 , 2 ( v )= F ( v ) 2 = v 2 . Conditional on winning ( v (1 , 2) v ), v (1 , 2) is distributed ac- cording to the probability density function g 1 , 2 ( x ) G 1 , 2 ( v ) (= 2 x v 2 )on[0 ,v ]. 1 We can compute the conditional expectation of max { 0 . 2 ,v (1 , 2) } with respect to this pdf. Hence bidder v ’s expected payment is E [max { 0 . 2 ,v (1 , 2) }| v (1 , 2) v ]Pr ³ v (1 , 2) v ´ = Z v 0 max { 0 . 2 ,x } g 1 , 2 ( x ) G 1 , 2 ( v ) dx · G 1 , 2 ( v ) = Z v 0 max { 0 . 2 ,x } g 1 , 2 ( x ) dx = Z 0 . 2 0 0 . 2 g 1 , 2 ( x ) dx + Z v 0 . 2 xg 1 , 2 ( x ) dx = Z 0 . 2 0 0 . 4x dx + Z v 0 . 2 2 x 2 dx = 2 3 v 3 + 0 . 008 3 (d) Compute the seller’s expected revenue when the reserve price is 0 . 2 . Answer. The seller’s total expected revenue is nE 2 3 v 3 + 0 . 008 3 ¸ 1 This is a slight generalization of the conditional expectation formula. Let X 0bea random variable with CDF F. Let q be any function. We can compute the expected value of q ( X ) conditional on X k ( k is some f xed number) as follows: E [ q ( X ) | X k ]= Z k 0 q ( x ) f ( x ) F ( k ) dx. The formula we know: E [ X | X k ]= R k 0 x f ( x ) F ( k ) dx is a special case of this.
= n Z 1 0 2 3 v 3 dv + 0 . 008 3 ¸ = 1 . 016 6 n Note: In the class, we applied RET to derive the seller’s revenue. We derived each bidder’s payment and the seller’s expected rev-

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