9781118041581

As an example, consider a typical sealed-bid auction—say, for a small sub-

urban office building expected to receive bids from three firms. Each bidding

firm plans to occupy the building (if it wins the auction) and places a greater

or lower value on the building, depending on its main features: location, office

space, amenities, and so on. Needless to say, an additional desired feature is a

low purchase price at auction. Let’s consider the bidding problem faced by a

typical firm, say firm 1. Its bidding strategy begins with an assessment of its

reservation price,that is, its monetary value for the building. For concreteness,

suppose its reservation price is v 1 $342 thousand. This value can be thought

of as a break-even price. The firm is just indifferent to the alternatives of acquir-

ing the building at this price or not acquiring it at all. It never would pay more

and would be happy to pay less.^5

Firm 1’s profit from winning the auction at bid b is $342 thousand b, the

difference between its value and its bid. If it does not win the auction, its profit

is, of course, zero. It follows that the firm’s expected profit is

where the second term denotes the probability that bid b wins (i.e., is the high-

est bid). The key to determining a profit-maximizing bid is to assess accurately

the way the firm’s winning chances depend on its bid. Recognizing that it faces

two other bidders for the building, the firm has thought carefully about its win-

ning chances and has made the probability assessments listed in the third col-

umn of Table 16.1. As we would expect, the firm’s winning chances increase

E()[342b][Pr(b wins)],

678 Chapter 16 Auctions and Competitive Bidding

(^5) In fact, the $342,000 estimate probably represents an expected value; that is, the company recog-
nizes that the value of the building is more or less uncertain. Given this uncertainty, a risk-neutral
buyer values the building at its expected value.
TABLE 16.1
Finding a Profit-
Maximizing Bid
(Thousands of Dollars)
Raising a sealed bid
increases the
probability, but
lowers the
profitability, of
winning. A bid of
$328,000 maximizes
the buyer’s expected
profit.
Winning Probability Expected
Bid Profit of Winning Profit
$300 $42 .00 $0.00
310 32 .06 1.92
320 22 .25 5.50
326 16 .42 6.76
328 14 .49 6.86
332 10 .64 6.40
336 6 .81 4.86
340 2 1.00 2.00
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