Game Theory Auctions Continued at HelpWithAssignment.com

As we have discussed about Auction games in our previous articles, we shall now speak about the background of these auction games.

We have discussed that there are 4 types of auctions

First Price Auction;

Second Price Auction:

All-Pay Auction:

Lottery:

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.

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.

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.

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. 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.

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.

We shall retain the assumptions of symmetric, independent and private values. Each player’s type, v_i represents his or her value for the seller’s good. Given her value for the good v_i, the quantity of the good that she receives, q_i, and the payment that she must make p_i, each buyer’s payoff was assumed to be:

U_i (q_i p_i/ v₁ v₂) = v_iq_i - p_i

The goal of mechanical design is to represent any auction in a simple way. The rules of any auction can be potentially very complicated, but ultimately all of the rules boil down to just a quantity of the goods sold to player i, q_i (b₁, b₂) given the bids of both players and a payment required of player i, p_i (b₁, b₂) given the bids of both players.

q₁ (b₁; b₂) = 1 if b₁ > b₂

= 0 if b1 < b₂

p₁ (b₁; b₂) = b₁ if b₁ > b₂

= 0 if b₁ < b₂

First Price Auction

so that the high bidder wins the auction and pays his or her bid. 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

Lottery

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

Tullock Lottery

This is in continuation with our previous articles on Economics, Auctions, Prisoners' Dilemma and Cutting a Cake

For more details you can visit our website at http://www.helpwithassignment.com/economics-assignment-help and http://www.helpwiththesis.com

Game Theory - Auctions at HelpWithAssignment.com

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. 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. There are different types of auctions available.

• The First Price Auction: In a first price auction, each bidder simultaneously writes down his name and a number on a slip of paper and submits the paper to the auctioneer. The auctioneer turns over all the of the papers, the bidder who wrote down the highest number is given the item and pays the amount that is written on his slip. Under certain assumptions, this procedure is identical to the so-called Dutch auction, or descending price auction. The price begins at some incredibly high amount and drops at a predetermined rate, for example 10 cents per second. The auction ends when one of the bidders presses a button which keeps the price from falling further and pays an amount of money, equal to the posted price when he stopped the clock. The other bidders receive nothing and pay nothing.
• The second price auction: In second price auction, each bidder simultaneously writes down his name and a number on a slip of paper and submits the papers to an auctioneer. The auctioneer turns over all the papers and the bidder who wrote down the highest number is given the item and pays the second highest amount that was written by other bidders. Under certain conditions and assumptions the auction is similar to that of Button auction and auction of fish in Japan. The bidders will have their names on a board with lights next to their names. The auction begins with zero and ascends at a predetermined rate. As long as the lights next to the name of a bidder is still on, the participant is still in the auction. Each bidder can press a button which will turn off the lights. The last bidder to be in the auction with lights on will win the item and pays the price at which his previous bidder dropped out. This is similar to the selling procedure on eBay.
• The All-Pay Auction: In all-pay price auction, each bidder simultaneously writes downs his name and a number on a slip of paper and submits the papers to an auctioneer. The auctioneer turns over all of the papers, the bidder who wrote down the highest number is given the item but all bidders who wrote down the highest number is given the item but all bidders have to pay their bids.
• The Lottery: In a lottery, each bidder can write down only one of the two bids, “In” or “Out”. All those players that are “in” are required to pay a price p, and have an equal chance of winning the item. Those that are “out” pay nothing and do not have the potential to win anything. We can think of these situations as Bayesian Games. In these games, the set of players will consist of a number of bidders, usually two. Given their type, each player chooses a bid, that is what to write down on the slip of paper and submit to the auctioneer. Given this outcome, each player’s payoff is

Ui (Qi, Pi/ V1, V2) = ViQi – Pi

Thus, each player’s payoff is his or her value for the good multiplied by the quantity of the good that he or she receives, minus any payment she has to make. Crucially, each buyer’s payoff only on his or her own value, any other bidder’s value for the item is irrelevant to each bidder. If two buyers are interested in acquiring a new painting from a young artist. If one art dealer has a very high valuation, it may mean, for example, that he or she has a client lined up for this piece, and although the other dealer may not know the identity of the client for certain, the fact that the other dealer places a high value on the painting may be important to a given art dealer’s assessment of its value.

This article is in continuation with our previous articles on Economics and Game Theory which include Prisoners' Dilemma, Battle of the sexes, Cutting a Cake, Solow's Growth Model

For more details you can visit our website at http://www.helpwithassignment.com/economics-assignment-help and http://www.helpwiththesis.com