Suppose that two players participate in the following auction: Both players have valuations vi for an object, where vi is distributed uniformly on [0, 1]. After observing their own valuation, both players simultaneously submit a bid bi . Whoever has the highest bid wins the object and pays the auctioneer bi 2 ; i.e. the winner pays half of their bid.
Find the Bayesian Nash Equilibrium of this auction. Assume that player 2 uses the bid function of the form b2 = cv2.