Question

Question 6. There are two bidders in a sealed-bid, second-price auction. The object for sale has a common value. Each bidder, i = 1,2, receives a signal i that is independently and uniformly distributed on the interval [0, 1]. The true value of the object, v, is the average of the two signals,

v = (σ1 + σ2) / 2

(a) If bidder 1 gets the signal σ = 0.7, how much does he think the object is worth?

(b) Suppose that each bidder submits a bid equal to the expected value of the object (conditional on his signal). If bidder 1 submits the winning bid, what is the expected value of the object?

(c) What is the expected surplus for bidder 1 under the assumptions in (b)?

Answer 1

a) In a second-price sealed bid auction, each bidder submits a sealed bid to the seller. The high bidder wins and pays the second-highest bid for the good. The second-price auction is a special case of a Vickrey auction. As in other Vickrey auctions, it is a dominant strategy in a second-price auction for a bidder to bid their true value.

Hence we can safely conclude that the bidder 1 thinks that the object is worth equal to its signal that is 0.7.

b) It can be shown that in a second-price sealed bid auction, it is a dominant strategy for a bidder to bid her true value. The argment is as follows: say there is a bidder Susie, and Susie's valuation for the item is v. Suppose the highest competing bid is p. If v is greater than p, then Susie wins the item and earns v-p if she bids her own value v. No other bids can give Susie better earning. If Susie's value v is less than p, then the best Susie can do is to bid below p and earn zero dollar. Susie earns exactly zero dollar by bidding v. Over all, to bid v is the best choice for Susie. Therefore, sometimes the second-price auction is also called demand revealing mechanism.

On thebasis of above reasoning we know that bidders bid their expected value in other words conditional on its signal. If bidder 1 submits the winning bid, then the bid of bidder 1 will be the expected value of object.

c) The expected surplus in this case will be the difference between the competitibg bid ( Player 2's bid) and the winner ( player 1) bid. Since the winner has to pay second highest bid and his valuation is higher than that, the difference is surplus.