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145hw4W10

# 145hw4W10 - n bidders when each bidder’s value is...

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Econ 145: Problem Set IV Ichiro Obara February 24, 2010 March 9, 2010 Student Name: Student ID: Score Problem 1 Problem 2 Problem 3 Problem 4 Total 1

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1. (A.15-1, SSS) You are one of n bidders who draw i.i.d. valuations from F ( x ) . The probability that your draw x is the highest is F ( x ) n 1 . Then what is the probability that your draw is k th highest? What are these probabilities when each draws follow the uniform distribution on [0 , 1]? 2
2. Consider a second price auction with n bidders. (a) Compute the seller’s expected revenue (in symmetric Bayesian Nash equilibrium) when each bidder’s value is independently and uniformly distributed on [0 , 2] . 3

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(b) Compute the seller’s expected revenue when each bidder’s value is independently distributed on [0 , 1] according to CDF F ( x )= x 2 . 4
3. Consider a f rst price auction with

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Unformatted text preview: n bidders when each bidder’s value is independently distributed on [0 , 1] according to CDF F ( x ) = x 2 . (a) What is the symmetric BNE? 5 (b) What is the seller’s expected revenue in equilibrium? 6 4. All-pay auction is an auction in which the highest bidder wins and every bidder must pay his or her own bid whether winning or losing the auction. Consider an all-pay auction with n bidders, whose values are independently and uniformly distributed over [0 , 1] . (a) Derive the symmetric BNE. 7 (b) Find the optimal reserve price to maximize the seller’s revenue as a function of n (hint: you may use the revenue equivalence theo-rem). Also derive the symmetric BNE with this optimal reserve price. 8...
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145hw4W10 - n bidders when each bidder’s value is...

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