because each buyer can observe roughly the price levels at which rivals drop out
of the bidding, and this provides indirect information about their estimates. All
in all, the English auction’s informational advantage translates into a revenue
advantage for the seller.
Next, consider the effect of bidder risk aversionon bidding strategies and
auction performance. (Thus far, the presumption has been that bidders are
risk neutral, always seeking to maximize expected profit.) Interestingly, risk
aversion has no effect on bidding behavior in an English auction. Bidding up
to full value (if necessary) is a dominant strategy regardless of the bidder’s atti-
tude toward risk. Things are different, however, in a sealed-bid auction. Here,
risk aversion implies higher bids by buyers—the more risk averse the buyer, the
higher is the bid. As always, the optimal bid depends on the trade-off between
the probability and profitability of winning. The upshot is that a risk-averse
buyer raises its bid, settling for a smaller but more certainprofit.^14 Thus, risk
aversion on the part of bidders tends to confer a revenue advantage on the
sealed-bid auction relative to the English auction. (One other effect can be
mentioned. If a bidder’s value for an item is uncertain, an increase in risk aver-
sion implies a reduction in the bidder’s certainty equivalent value, which in
turn lowers bids. However, this value reduction is the same regardless of the
auction method being used.)
Finally, the revenue equivalence result presumes that buyers’ private values
are drawn independently from a common probability distribution. Allowing for
value asymmetry(buyer values drawn from different distributions) upsets rev-
enue equivalence. Roughly speaking, the English auction tends to have a rev-
enue advantage when two or more bidders have similar distributions of values
(and therefore tend to bid up the price close to their reservation values).
Conversely, the sealed-bid auction does better when the value ranges are quite dif-
ferent. As an extreme example, suppose there are two bidders with v 1 uniformly
distributed between 100 and 150 and v 2 uniformly distributed between 0 and 50.
In the English auction, buyer 2 always drops out before buyer 1 and the expected
price is only E[PE] E[v2nd] 25. In the sealed-bid auction, buyer 1’s optimal
bid is b 1 50, just enough to guarantee winning the auction. Thus, the sealed-
bid auction yields twice the expected revenue of the English auction.^15RESERVE PRICES A common feature of auctions is the setting of a reserve
price. If the high bid in the auction does not exceed the seller’s minimum or690 Chapter 16 Auctions and Competitive Bidding(^14) Suppose the firm that bids for the office building has utility function , where y is the
firm’s bidding profit. Using Table 16.1, we can confirm that the firm’s utility-maximizing bid is
$332 thousand (whereas its profit-maximizing bid was $328 thousand).
(^15) For less dramatic asymmetric distributions, the sealed-bid auction continues to hold a revenue
advantage, though a narrower one. Let one buyer’s values be uniformly distributed between 0 and
60 and the other’s between 30 and 90. Here expected English revenue is calculated to be 28.75.
In the sealed-bid auction, expected revenue is 34.4 (about 20 percent) higher.
U 101 y
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