Bidder Strategies 681is that each competitor’s bid will be somewhere between the limits $300 thousand
and $340 thousand, with all values in between equally likely. This assessment
implies^7 G(b) (b 300)/40. Therefore, it follows that H(b) [(b 300)/40]^2 ,
because the firm faces two competitors. In fact, this is exactly the probability func-
tion depicted in Figure 16.1 and listed in Table 16.1. As we saw earlier, the firm’s
optimal bid against this distribution is $328 thousand. To sum up, based on an
assessment of possible bids by a typical competitor, the firm should compute the
distribution of BCB and then fashion an optimal bid against this distribution.
Now consider how the BCB distribution is affected if the number of bidders
increases, say, to five firms. Facing four competitors, firm 1 would assess H(b)
[(b 300)/40]^4. Notice that the firm’s win probability for any given bid goes
down drastically with the number of bidders. For example, a bid of $330 thou-
sand has a .75 chance of beating any single bidder, a .56 chance of beating two
other bids, but only a .32 chance of beating four others. Not surprisingly, the
increase in the number of competitors causes firm 1 to increase its optimal bid.
In fact, there is a second effect reinforcing firm 1’s raise in bid: With the increase
in competition, firm 1 would expect other firms to raise their bids as well. Thus,
the bid distribution of the typical firm, G(b), will be shifted toward higher bids.
In short, firm 1 faces an increase not only in the number of competing bids but
also in their levels. For both reasons, its optimal bid increases.(^7) To check this formula, note that G is .5 at the distribution median, $320,000. At a bid of $330,000,
G is .75 as expected. For a uniformly distributed value (as here), G (b L)/(U L), where L
and U denote the lower and upper limits of the distribution, respectively.
(^8) The material in this section is advanced and can be skipped without loss of continuity.
CHECK
STATION 2
Your firm is competing in a sealed-bid auction for a piece of computer equipment valued
(by the firm) at $30,000. You are contemplating one of four bids: $18,000, $20,000, $24,000,
and $27,000. Given the bid distribution of a typical buyer, these bids would win against a sin-
gle competitor with respective probabilities .4, .6, .8, and .9. What is the firm’s optimal bid
against one competitor? Against two competitors? Against three competitors?
EQUILIBRIUM BIDDING STRATEGIES^8 Thus far, we have taken the point of
view of a single firm whose task is to formulate an optimal bid, given a predic-
tion about the distribution of competing bids. Although this approach has cer-
tain advantages, it also suffers from two problems. First, firms often are severely
limited in their information about their competitors’ bidding behavior; that is,
there may be little empirical basis (i.e., a history of past bidding tendencies) for
assessing H(b) or G(b). An “optimal” bid that assumes the “wrong” competing
bid distribution will show a poor profit performance. Second, a purely empiri-
cal approach involves only one-sided optimization. It ignores the important fact
that competitors are profit maximizers—that they themselves are attempting to
set optimal bids. (Rather, it simply takes the bid distribution as given.)
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