9781118041581

Optimal Auctions 689

L12

n  1n  1

U

E(v2nd)E(vmax)

n  1

n  1

n

n  1

FIGURE 16.3

Expected Prices When

Buyer Values Are

Uniformly Distributed

The English and sealed-

bid auctions have

identical revenues.

the average price (corresponding to the second-highest buyer value) achieved

by an English auction.

Now let’s compare this to the result in a sealed-bid auction. Here the win-

ning buyer bases his bid on vmax, the highest outstanding value. The E(vmax) is

also shown in the figure. But the winning bidder shades his optimal bid below

his private value. In equilibrium, his bid corresponds to the expectation of the

next-highest value, E(v2nd), so the degree of shading exactly balances the

advantage of basing the bid on the highest value. In the uniform case, his value

is n/(n 1) toward U, but he applies the factor (n 1)/n in shading his bid

below his value (see Equation 16.4). Multiplying n/(n 1) by (n 1)/n

results in an expected winning sealed bid exactly (n 1)/(n 1) of the way

toward U. This is exactly the same price as in the English auction.

CHECK

STATION 5

Suppose there are four bidders with values independently and uniformly distributed

between 0 and 100. Compute the expected price at an English auction. Write down the

typical buyer’s equilibrium strategy in a sealed-bid auction and then compute E(vmax)

and E(bmax). Confirm that the auctions deliver the same expected price.

Revenue equivalence should be viewed as a benchmark, a result that holds

in a prescribed set of circumstances. As we shall see, modifying the basic auc-

tion setting can confer a revenue advantage on one type of auction or the other.

For instance, suppose the private-value model is replaced by a common-value

setting. One can show that in the common-value setting, the English auction produces

greater revenue, on average, than the sealed-bid auction.Although the proof of this

result requires advanced mathematical methods, the intuition is readily

grasped.^13 Recall that, in the common-value setting, buyers must discount their

bids below their private estimates to avoid the winner’s curse. Roughly speak-

ing, the greater the uncertainty about the item’s value, the greater is this bid

discount. (The discount also increases with the number of bidders.) The next

point to recognize is that a buyer faces less uncertainty in the English auction

than in the sealed-bid auction. The English auction conveys more information

(^13) The first formulation and proof of this result can be found in P. Milgrom and R. Weber, “A
Theory of Auctions and Competitive Bidding,” Econometrica(1982): 1089–1122.
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