Suppose you want to buy a house that is sold by way of a first-price sealed bid auction. In contrast to the model in the lecture, there are more than 2 players. Players simultaneously and independently submit bids bi, i = 1,..., n. The house is awarded to the highest bidder who must pay his bid, denoted x. Your conjecture is that each player bids a fraction a of her valuation, i.e. b; = avi, where vi is a player's valuation. Moreover, suppose that you have the following information: You know the number of persons n that bid for the house. You believe the other bidders' values are independently and uniformly distributed over the interval [0, 1], where 1 stands for 1.000.000. a) What is the probability that your bid x exceeds the bid of player 2? b) What is the probability that your bid x exceeds the bids of player 2 and 3? c) Now, what is the probability that your bid exceeds that of all other bidders? d) What bid should you choose? 1 e) Explain intuitively how your bid changes with the number of bidders n. f) Now suppose that your own valuation for the house is 800.000 and that there are 5 other bidders. How much should you bid?