(1) At Wylie's Auction House in Geneva, NY, a rare copy of Varian's textbook, first edition, is being sold by auction. There are 5 bidders in attendance: Austin, Dylan, John, Michael, and Tom. The book is worth $100 to Austin, $20 to Dylan, and $5 to each of the others. The bidders do not collude and they don't know each others' valuations. a. If the auctioneer sells it in an English auction, who would get the book, and approximately how much would the buyer pay? b. If the auctioneer sells it in a sealed-bid, second-price auction, and if no bidder knows the others' values for the book, how much should Austin bid in order to maximize his expected gain? How much should Dylan bid? How much would each of the others bid? Who would get the book, and for how much money? (2) Late in the day at an antique rug auction, there are only two bidders left, Anna and Griffin. The last rug is brought out and the seller says she will accept sealed bids from each bidder and will sell the rug to the highest bidder at the highest bidder's bid amount. Each bidder believes that the other is equally likely to value the rug at any amount between 0 and $1000. Therefore for any number X between 0 and 1000, the probability that the other bidder values the rug at less than X is X111 000. The rug is actually worth $800 to Anna. If she gets the rug, her gain will be the difference between $800 and what she pays for it, and if she doesn't get the rug, her gain will be zero. She wants to make her bid in such a way as to maximize her expected gain. a. Suppose Anna thinks that Griffin will bid exactly what the rug is worth to him. If she bids $700 for the rug, what is the probability that she will get the rug? If she gets the rug for $700, what is her gain? What is her expected gain if she bids $700? b. Again suppose that Griffin will bid exactly what the rug is worth to him. If Anna bids $600 for the rug, what is the probability that she will get the rug? If she gets the rug for $600, what is her gain? What is her expected gain if she bids $600? c. Again suppose that Griffin will bid exactly what the rug is worth to him. If Anna bids $x for the rug (where $x is between 0 and $1000), what is the probability that she will get the rug? If she gets the rug, what is her gain? Write a formula for her expected gain if she bids $x. Find the $x that maximizes her expected gain. [hint: take a derivative] d. Let us go a little further toward finding a general answer. Suppose that the value of the rug to Anna is $1! and she believes that Griffin will bid exactly what the rug is worth to him. Write a formula that expresses her expected gain in terms of the variables V and x if she bids $x. Now find the $x (as a function of V) that maximizes her expected gain. [same hint]