9781118041581

A useful way to arrive at the BCB distribution is to begin by assessing the dis-

tribution for a typicalcompeting bid. For instance, let G(b) denote the cumula-

tive distribution function for the bid of a singlecompetitor. In other words, if the

firm submits bid b, the chance that it will better the bid of this single competitor

is simply G(b). What if the firm faces two competitors? For the firm to win, its bid

must be greater than bothcompeting bids. Under the assumption that the com-

peting bids are independent of one another, this occurs with probability [G(b)]^2.

More generally, suppose the firm faces (n 1) competitors whose bids are inde-

pendent and come from the common distribution G(b). Then the probability

that bid b is greater than all the others is H(b) [G(b)]n ^1.

To illustrate this result, consider the sealed-bid competition for the office

building, again taking firm 1’s point of view. Suppose the firm’s best assessment

680 Chapter 16 Auctions and Competitive Bidding

FIGURE 16.1

Probability Distribution

of the Best Competing

Bid

The curve’s height

shows the probability

that the best

competing bid is

smaller than the value

shown on the

horizontal axis. The

buyer’s optimal sealed

bid is $328,000.

1.0

.8

.6

.49

.4

.2

0

Probability

H curve

Probability

b = 328 wins

Firm 1’s profit

v – b

⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

300 310 320 328 342

Firm 1’s

optimal bid

Firm 1’s

reservation price

Range of Competing Bids (Thousands of Dollars)

c16AuctionsandCompetitiveBidding.qxd 9/26/11 1:09 PM Page 680