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Unformatted text preview: 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 ) = aω 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]....
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