1. Game Theory Auctions at HelpWithAssignment.com www.HelpWithAssignment.com

2. Auctions are among the oldest methods of selling items, with recorded occurrences as early as 500 BC. While historically not as common as sale by fixed price or bargaining, the development and proliferation of the internet and information technology has made auctions into a common means of selling and buying items, both large and small. Game Theory Auctions www.HelpWithAssignment.com

3. At its most basic, an auction is a set of rules by which an item is sold to one of several potential buyers. In this set of rules by which an item is sold to one of the several potential buyers. Game Theory Auctions www.HelpWithAssignment.com

4. We have discussed that there are 4 types of auctions First Price Auction Second Price Auction All-Pay Auction Lottery Game Theory Auctions www.HelpWithAssignment.com

5. In these type of auctions we have seen that in the First Price Auction, the item is sold to the bidder who quotes the highest price. In the Second Price Auction, the item is sold to the bidder who quotes the highest amount but he will pay the second highest price that was quoted. Game Theory Auctions www.HelpWithAssignment.com

6. In All-Pay Auction, all the bidders will write down their numbers but, the item is sold to the highest quotation. But all the bidders will have to pay even though they have not received the item. Game Theory Auctions www.HelpWithAssignment.com

7. In Lottery, all the bidders will have an equal chance of winning the item. All of the bidders will pay their share of the price, but only one person will receive the item. So, there is a chance that anyone in the bid can win the lottery. Game Theory Auctions www.HelpWithAssignment.com

8. Here, we have discovered a fact: the first three types of auctions, all resulted in the same revenue for the seller, but the fourth auction type did not result in the same revenue. Game Theory Auctions www.HelpWithAssignment.com

9. Furthermore, the first three auction types always sold the good to the buyer who had the highest value of an item, but in lottery, if both buyers enter, there is a chance that the buyer whose value is lower can also win the lottery. Game Theory Auctions www.HelpWithAssignment.com

10. We will use the technique of mechanism design and apply this technique for the design of auctions. We will use techniques to determine which auction format will raise the most revenues for the seller among all the conceivable auction formats. Auctions Mechanical Design www.HelpWithAssignment.com

11. We shall retain the assumptions of symmetric, independent and private values. Each player’s type, vi represents his or her value for the seller’s good. Auctions Mechanical Design www.HelpWithAssignment.com

12. Given her value for the good vi, the quantity of the good that she receives, qi, and the payment that she must make pi, each buyer’s payoff was assumed to be: Ui(qi pi/ v1 v2) = viqi- pi Auctions Mechanical Design www.HelpWithAssignment.com

13. The goal of mechanical design is to represent any auction in a simple way. The rules of any auction can be potentially very complicated. Auctions Mechanical Design www.HelpWithAssignment.com

14. But ultimately all of the rules boil down to just a quantity of the goods sold to player i, qi(b1, b2) given the bids of both players and a payment required of player i, pi (b1, b2) given the bids of both players. Auctions Mechanical Design www.HelpWithAssignment.com

15. First Price Auction q1(b1; b2) = 1 if b1 > b2 = 0 if b1 < b2 p1(b1; b2) = b1 if b1 > b2 = 0 if b1 < b2 Auctions Mechanical Design www.HelpWithAssignment.com

16. The lottery, for example, could be written q1(b1; b2) = 1 if b1 = 1 and b2 = 0 = 1/2 if b1 = 1 and b2 = 1 = 0 if b1 = 0 p1(b1; b2) = p if b1 = 1 = 0 if b1 = 0 Auctions Mechanical Design www.HelpWithAssignment.com

17. Another type of auction, which we have not considered is the "Tullock Lottery," in which your chance of winning the good is just your share of the total of the bids q1(b1; b2) = b1/b1+b2 p1(b1; b2) = pb1 Auctions Mechanical Design www.HelpWithAssignment.com

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