1st Price Auctions

Assumptions

1. There is a single unit of the good, a rare painting, to be sold by A. Vant Garde.

2. There are two buyers: Kristy and Sutheby.
a. Kristy values the good at . She thinks that there is probability 1/2 that Sutheby also values the good at .
b.  If Kristy wins the painting then her payoff is  '- price'.
c.  Sutheby has the same attributes as Kristy, but he values the painting at . He thinks that there is probability 1/2 that Kristy also values the painting at .   If he wins the auction then his payoff is -price.
d. In analyzing the auction we know Kristy and Sutheby know this as well, but do not know each other's type.

3. The seller of the painting wants to maximize the expected selling price.

Rules

Neither player knows the other's valuation, nor may they communicate this information to one another.

Each player submits a sealed bid.  The sealed bids are opened and announced by the auctioneer.

The higher bidder gets the object and pays his or her bid as the price.  If there is a tie then the winner is decided by the toss of a fair coin.

Strategy for Sutheby

Remember, Sutheby doesn't know Kristy's valuation for sure. Sutheby will never bid more than his valution .  If he does so then his consumer surplus is negative.   If he bids less than then he will surely lose and his surplus will be zero.  By bidding Sutheby has an expected payment of ( (probability of Kristy being a type) x (probability of winning the coin toss) x () =) /4.

Strategy for Kristy

Remember that Kristy knows she has a high valuation, but doesn't know Sutheby's valuation.

Let Kristy's low bid be .   Whenever she submits this bid against someone that values the painting at she will win the painting 50% of the time.  If she submits this bid against a -type person she will lose with certainty. Her expected surplus is (-)/2 for the low bid of .

Now consider any other bid, b, by Kristy.  Since every bid that is a best response must have the same surplus we have

 ½(1)(-b)  + ½(p)(-b) =  (-)/2 Payoff against a person Payoff against a person

where p is the probability that her bid exceeds her opponent.   On rearranging

(1+p)(-b) = -

1+p =

or p = . What value of b makes it certain that she will win?

1 = implies her high bid will be .

This is equivalent to Kristy playing half the time and half the time.  Such a strategy would make Sutheby indifferent between playing a pure strategy of and a pure strategy of .

Let's summarize the arguments

 Probability of bidding against a type person 1/2 Probability of bidding against a type person 1/2 Probability of winning against a type person Probability of winning against a type person 1 Payoff upon winning (-b) Payoff upon winning (-b)

Using all of this information we find that with her mixed strategy Kristy will have a net surplus of

This is exactly the same net surplus that Kristy obtained in the the 2nd price auction! As in the 2nd price auction Sutheby, the bidder with a low valuation, gets a net surplus of 0 here as well.

Finally, the expected revenue to Monsieur Garde is the same as the expected revenue to the seller in the 2nd price auction. A type player can only win against another type player, and then only with probability 1/2, so P() = 1/4.  Therefore P() = 3/4, as in the 2nd price auction. Since the net surplus to the buyer is P() minus their payment, and is the same in both types of auction, it must be the case that the expected revenue to the seller is the same in both auctions.

The first price auction can also be analyzed using the framework of a static game with incomplete information.