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PS4 - Problemsets 4 5 1 Consider the first-price...

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Problemsets 4 & 5 1. Consider the first-price private-value auction where there are n bidders, each with a value drawn from a uniform distribution on [0 , ω ]. If everyone but player i follows a symmetric bidding strategy β ( ω i ), the expected utility of a bid b i is given by: Pr braceleftbig β 1 ( b i ) ω j bracerightbig n 1 [ ω i b i ] , the probability i ’s bid is the highest multiplied by the payoff if they win. Assume that β ( ω j ) = j . (a) Find the best-response bid b for bidder i when their value is ω i . (b) Find the value of a such that β ( ω i ) is a symmetric Bayesian Nash equilibrium. (c) Find the expected payment of bidder i . (d) What is the expected payment of bidder i in an all-pay common-value auction? 2. Consider a second-price common-value auction where the value to each player i is given by V i ( ω ) = αω i + (1 α ) 1 n 1 j negationslash = i ( ω j ). Each individual only knows their only component of the value, ω i , and makes a bid b i , and pays the second-highest bid. Again, assume that the individual signals ω i are distributed with a uniform distribution over [0 , ω ]. The expected utility is therefore Pr braceleftbig β 1 ( b i ) ω j
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