Suppose a website is auctioning an advertising slot to two potential advertisers, A and B. If advertiser i gets the slot, she gets a value equal to v_i that is uniformly distributed between 0 and 1. Both values are independently distributed. The total utility of bidder i is equal to v_i  T_i , where T_i is the payment to the auctioneer. Suppose first that the slot is auctioned off using a sealed-bid second-price auction.

Part a) What is advertiser i's utility as a function of her value and of both bids b_i and b_i? What is her best response as a function of the other advertiser's bid? Plot the best response as a function of b_i .

Part b) Now suppose that advertiser i's value is v_i and she does not know advertiser j's bid. What should she bid? Is this a dominant strategy?

Now suppose the slot is auctioned off using a sealed-bid first-price auction.

Part c) What is advertiser i's utility as a function of her value and of both bids b_i and b_i? What is her best response given the other advertiser's bid? Plot the best response as a function of b_i . (For this question, assume bids can only be in increments of one cent).

Part d) Now suppose that advertiser i's value is v_i . She does not know advertiser j's bid, but she knows that it has a uniform distribution between 0 and 0.5 . Show that if she bids bi [0, 0.5 ], her expected utility can be written as 2b_i(v_i b_i).

Part e) Find the first order condition for the expected utility from part d). Based on it, what is the optimal bid as a function of the bidder's value?

Part f) Show that the optimal bid you computed in part e) has a uniform distribution between 0 and 0.5 . Based on this, what are the equilibrium strategies?

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Lets break this down step by step Part a Advertiser i s utility is vibj Ti if i wi... View the full answer