An Incomplete Information Static Game Interpretation of the 1^st Price Auction

We can analyze the first price auction using the machinery of the static game with incomplete information. We’ll do it from Kristy’s perspective. She knows her own type, but is uncertain of Sutheby’s type.

Kristy will never submit a bid below μ. If she were to do so then she would surely lose. The same goes for Sutheby. Since she will never submit a bid below μ, that is her minimum bid. We’ll also allow her to submit a higher bid in the amount b. The same goes for Sutheby. When Kristy submits a bid in the amount b_K against a bid by Sutheby in the amount b_S there is a probability p that her bid wins, and a probability q that his bid wins. The payoffs in the two games are expected payoffs.

		Sutheby
			bS		bS
Kristy
	bK

Sutheby can be either one of two types. For each type he has two possible actions. Therefore, from Kristy’s perspective he has four strategic plans. With this in mind we can write the normal form of the game as:

		Sutheby
					bS, bS
Kristy
	bK

Sutheby’s second strategy of is seen to weakly dominate his fourth strategy of b_S, b_S . His first strategy of is seen to weakly dominate his third strategy. Sutheby’s first strategy is not credible since he would never bid μ when he is a θ – type. Therefore, Kristy believes that he will play his second strategic plan. If μ and b_K are both to be best responses to a play of by Sutheby then the associated payoffs must be equal.

If you solve for p then you will get the earlier result for the probability with which Kristy will win when she plays the strategy b_K.