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 (vp)^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. Find an equilibrium of the second-price auction.

2. Suppose that there are two players and each players 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 2s 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 1s valuation is v₁.)

3. 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 auctioneers revenue differ between the two auctions?.