20.2 Sealed-bid Auction

In a sealed-bid auctionAuction where bidders simultaneously submit sealed bids, and the highest bidder wins and pays the highest bid., each bidder submits a bid in an envelope. These are opened simultaneously, and the highest bidder wins the item and pays his or her bid. Sealed-bid auctions are used to sell offshore oil leases, and they are used by governments to purchase a wide variety of items. In a purchase situation, known often as a tender, the lowest bidder wins the amount he bids.

The analysis of the sealed-bid auction is more challenging because the bidders don’t have a dominant strategy. Indeed, the best bid depends on what the other bidders are bidding. The bidder with the highest value would like to bid a penny more than the next highest bidder’s bid, whatever that might be.

To pursue an analysis of the sealed-bid auction, we are going to make a variety of simplifying assumptions. These assumptions aren’t necessary to the analysis, but we make them to simplify the mathematical presentation.

We suppose there are n bidders, and we label the bidders 1, …, n. Bidder i has a private value v_i, which is a draw from the uniform distribution on the interval [0,1]. That is, if $0\le a\le b\le 1\text{,}$ the probability that bidder i’s value is in the interval [a, b] is b – a. An important attribute of this assumption is symmetry—the bidders all have the same distribution. In addition, the formulation has assumed independence—the value one bidder places on the object for sale is statistically independent from the value placed by others. Each bidder knows his own value but he doesn’t know the other bidders’ values. Each bidder is assumed to bid in such a way as to maximize his expected profit (we will look for a Nash equilibrium of the bidding game). Bidders are permitted to submit any bid equal to or greater than zero.

To find an equilibrium, it is helpful to restrict attention to linear strategies, in which a bidder bids a proportion of her value. Thus, we suppose that each bidder bids λv when her value is v and λ is a positive constant, usually between zero and one. With this set up we shall examine under what conditions these strategies comprise a Nash equilibrium. An equilibrium exists when all other bidders bid λv when their value is v, and the remaining bidders bid the same.

So fix a bidder and suppose that bidder’s value is v_i. What bid should the bidder choose? A bid of b wins the bidding if all other bidders bid less than b. Because the other bidders, by hypothesis, bid λv when their value is v, our bidder wins when $b\ge \lambda v_j$ for each other bidder j. This occurs when $\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.\ge v_j$ for each other bidder j, and this in turn occurs with probability $\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.\text{.}$ If $b>\lambda \text{,}$ then in fact the probability is 1. You can show that no bidder would ever bid more than λ. Thus, our bidder with value v_i who bids b wins with probability ${\left(\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.\right)}^{n-1}$ because the bidder must beat all n −1 other bidders. That creates expected profits for the bidder of $\pi =(v_i-b){\left(\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.\right)}^{n-1}\text{.}$

The bidder chooses b to maximize expected profits. The first-order condition requires $0=-{\left(\frac{b}{\lambda }\right)}^{n-1}+(v_i-b)(n-1)\frac{b^{n-2}}{{\lambda }^{n-1}}\text{.}$

The first-order condition solves for $b=\frac{n-1}{n}v\text{.}$

But this is a linear rule. Thus, if $\lambda =\frac{n-1}{n}\text{,}$ we have a Nash equilibrium.

The nature of this equilibrium is that each bidder bids a fraction $\lambda =\frac{n-1}{n}$ of his value, and the highest-value bidder wins at a price equal to that fraction of her value.

In some cases, the sealed-bid auction produces regret. Regret means that a bidder wishes she had bid differently. Recall our notation for values: v₍₁₎ is the highest value and v₍₂₎ is the second-highest value. Because the price in a sealed-bid auction is $\frac{n-1}{n}{v}_{(1)}\text{,}$ the second-highest bidder will regret her bid when $v_{(2)}>\frac{n-1}{n}{v}_{(1)}\text{.}$ In this case, the bidder with the second-highest value could have bid higher and won, if the bidder had known the winning bidder’s bid. In contrast, the English auction is regret-free: the price rises to the point that the bidder with the second-highest value won’t pay.

How do the two auctions compare in prices? It turns out that statistical independence of private values implies revenue equivalenceSituation in which two auctions produce the same price on average., which means the two auctions produce the same prices on average. Given the highest value v₍₁₎, the second-highest value has distribution ${\left(\frac{v_{(2)}}{v_{(1)}}\right)}^{n-1}$ because this is the probability that all n − 1 other bidders have values less than v₍₂₎. But this gives an expected value of v₍₂₎ of $E{v}_{(2)}={\displaystyle \underset{0}{\overset{v_{(1)}}{\int }}{v}_{(2)}(n-1)\frac{v_{(2)}^{n-2}}{v_{(1)}^{n-1}}}d{v}_{(2)}=\frac{n-1}{n}{v}_{(1)}\text{.}$

Thus, the average price paid in the sealed-bid auction is the same as the average price in the English auction.