Question

Without using technical jargon or mathematics, briefly explain the predicted Nash equilibrium of a second price auction Are second price auctions simultaneous or sequential?

Answer 1

A Second-Price Auction is one in which bidders submit bids, the highest bidder wins and pays a price equal to the second highest bid.

Second price auctions will have the structure very similar to a complete information auction. In the auction, each bidder submits a sealed bid and has value Vn but no idea of other values . Dominant strategy equilibrium for this game is for every player to bid their true value. Since the winner pays the bid of the second highest bidder, everyone should bid their true value and will obtain the highest value of surplus possible. Let us also assume that if there is a tie,the object goes to each winning bidder with equal probability. With the reasoning with complete information, in a second-price auction, it is a weakly dominant strategy to bid truthfully. The winning bidders surplus is difference between the winner's valuation and the second highest valuation

There are no other optimal strategies and thus the (Bayesian) equilibrium will be unique, since the valuation of other players are not known. Therefore, we have established that in the second price auction, there exists a unique Bayesian Nash equilibrium.

In a second price auction, it is a weakly dominant strategy to bid one’s value, bi(si) = si. Suppose i’s value is si, and she considers bidding bi > si. Let ˆb denote the highest bid of the other bidders not equal to i (from i’s perspective this is a random variable). There are three possible outcomes from i’s perspective: (i) ˆb>bi, si; (ii) bi > ˆb>si; or (iii) bi, si > ˆb. In the event of the first or third outcome, i would have done equally well to bid si rather than bi > si. In (i) she won’t win regardless, and in (ii) she will win, and will pay ˆb regardless. However, in case (ii), i will win and pay more than her value if she bids ˆb, something that won’t happen if she bids si. Thus, i does better to bid si than bi > si. A similar argument shows that i also does better to bid si than to bid bi < si. Q.E.D. Since each bidder will bid their value, the seller’s revenue (the amount paid in equilibrium) will be equal to the second highest value. Let Si:n denote the ith highest of n draws from distribution F (so Si:n is a random variable with typical realization si:n). Then the seller’s expected revenue is E£ S2:n¤ . The truthful equilibrium described in Proposition 1 is the unique symmetric Bayesian Nash equilibrium of the second price auction. There are also asymmetric equilibria that involve players using weakly dominated strategies. One such equilibrium is for some player i to bid bi(si) = v and all the other players to bid bj (sj ) = 0.

The first price auction will have lower expected revenue than the second price auction because the winner’s payment in the first price auction is based only on her own signal, while in the second price auction it is based on her own signal and the second-highest signal.

Bids in a second price auction are submitted simultaneously.