investment banker is hopeful that it can find as many as 8 to 10 potential buy-
ers for the division. Its best assessment is that the offer of a typical potential
buyer will be centered around $52 million, with a range of plus or minus $12
million. In fact, it assesses a uniform distributionfor the offer; that is, it regards
all values between $40 million and $64 million as equallylikely. The investment
banker also believes that buyer offers will, by and large, be independent of one
another. (Each buyer’s offer comes from the equally likely range just given,
regardless of others’ offers.) In looking for the best sale price, what strategy
should the firm pursue? What is the best price it can get, on average, from con-
tacting outside buyers?
Let’s consider the second question first. Suppose the firm contacts a single
buyer. Then the averageprice it can obtain is $52 million. In turn, what if the
investment banker can find two potential buyers, allowing the firm to choose
the higherprice of the two? How high will this “better” price be on average?
The answer is $56 million. For the moment, the exact number is less important
than understanding that the firm fares better on average from choosing the
higher of the two price offers than by being locked into a single price. Of
course, it does even better if it has the opportunity to pick the highest price
from among three potential buyers, better still with four buyers, and so on.
Table 13.3 lists the expected maximum price attainable as the number of
buyers varies up to nine. As we would expect, the “best” price rises steadily with
the number of buyers. In fact, there is a simple formula for computing the
expected maximum value among a number of variables (call this number n)
drawn independently from a uniform distribution. The expected maximum
value is[13.6]where L is the lowest possible value and U is the greatest possible value. In our
example, we have L 40 and U 64. For instance, if n 3, then E(Vmax)
(1/4)(40) (3/4)(64) 58. Observe that the expected maximum value is a
weightedaverage of the extreme values, L and U, the weights being 1/(n 1)
and n/(n 1). For a single buyer (n 1), the weights are .5, and the expected
price is a straight average of the minimum and maximum values (i.e., halfway
between them). As the number of buyers increases, the expected maximum
price approaches the upper end of the possible value range because the weight
on U approaches 1.^5
The preceding result generalizes as follows: For any distribution of values
(not only the uniform), the expected maximum value increases with the numberE(Vmax)a1
n 1bLan
n 1bU564 Chapter 13 The Value of Information(^5) We can rearrange Equation 13.6 in the form
In short, the expected value of Vmaxis n/(n 1) of the way between the lower and upper bounds.
E 1 Vmax 2 L 3 n/ 1 n 1243 UL 4.
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