Game Theory Auctions Continued

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

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