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Question 4. [25 points] Consider a variant of the sealed-bid auctions with imperfect information considered in class in which the players are risk-averse. Specifically, assume that the payoff of a player with valuation v who wins the object and pays the price p is (v ? p)^1/m, where m > 1. (In class we considered the case m = 1, in which the bidders are "risk neutral". For m > 1, the bidders are "risk averse".)

1. [7 points] Find an equilibrium of the second-price auction.
2. [12points]Suppose that there are two players and each player's valuation is drawn independently from a uniform distribution on [0, 1] (as we assumed in class). Find an equilibrium of the first- price auction. (Hint: Assume that the player strategy is a linear function of their evaluation, i.e. when player 2's valuation is v2 she bids ?v2, where ? is a constant. Find the best response of player 1 to this strategy of player 2 when player 1's valuation is v1.)
3. [6points]Following the environment of part (2) (two players , uniform distribution of valuations), compare the expected value of the price paid by a winner with valuation v in the equilibrium of a second-price and a first-price auction. How does the auctioneer's revenue differ between the two auctions?.

Question 4. [25 points] Consider a variant of the sealed-bid auctions with imperfect information considered in class in which the players are risk-averse. Specifically, assume that the payoff of a player with valuation v who wins the object and pays the price p is (v -p) /", where m > 1. (In class we considered the case m = 1, in which the bidders are "risk neutral". For m > 1, the bidders are "risk averse".) 1. [7 points] Find an equilibrium of the second-price auction. 2. [12 points] Suppose that there are two players and each player's valuation is drawn independently from a uniform distribution on [0, 1] (as we assumed in class). Find an equilibrium of the first- price auction. (Hint: Assume that the player strategy is a linear function of their evaluation, i.e. when player 2's valuation is vz she bids Buz, where / is a constant. Find the best response of player 1 to this strategy of player 2 when player 1's valuation is v1.) 3. [6 points]Following the environment of part (2) (two players, uniform distribution of valuations), compare the expected value of the price paid by a winner with valuation v in the equilibrium of a second-price and a first-price auction. How does the auctioneer's revenue differ between the two auctions