The Value of Additional Alternatives 565of independently drawn alternatives. The last column of Table 13.3 shows the
expected maximum value when values come from a normal (bell-shaped) dis-
tribution. (The mean is 52, and the standard deviation is 8.)^6 Although the
uniform and normal distributions are very different in shape, they display a
qualitatively similar pattern of maximum values. Note that expected maximum
values are higher for the normal than for the uniform distribution. Roughly
speaking, this is because the uniform distribution has a fixed upper limit of
possible values, whereas the normal distribution does not.
Table 13.3 shows the expected benefit from pursuing additional buyers.
Clearly, if this pursuit is costless, the firm should seek out as many buyers as it
can possibly find. More realistically, suppose finding additional buyers is costly—
in fact, the total fee the firm can expect to pay its investment banker depends
on how wide and costly a search the banker makes on the firm’s behalf. For
concreteness, suppose the banker sets its fee (C) according to the rough for-
mula C 1,000,000n; that is, the average cost per found buyer is $1 million.
From the firm’s point of view, what is the optimal number of potential buyers?(^6) The mean and standard deviation have been set so that the normal distribution roughly matches
the uniform one above. Remember that two-thirds of the time a normally distributed variable falls
within one standard deviation of the mean. For the uniform distribution here, the probability of
a value within plus or minus 8 around the mean is also two-thirds.
TABLE 13.3
Expected Maximum
Prices When Choosing
from Different
Numbers of Buyers
The greater the num-
ber of buyers from
which to choose, the
higher the seller’s
expected price.
Expected Maximum Price (Millions of Dollars)
Number of Buyers Uniform Distribution Normal Distributiona
1 52.0 52.0 (0)
2 56.0 56.5 (.56)
3 58.0 58.8 (.85)
4 59.2 60.2 (1.03)
5 60.0 61.3 (1.16)
6 60.6 62.2 (1.27)
7 61.0 62.8 (1.35)
8 61.3 63.4 (1.42)
9 61.6 63.9 (1.49)
aNumbers in parentheses indicate the difference between the expected price and the mean of the normal distribution—measured in number of
standard deviations. For instance, on average, the highest of three prices drawn independently from a normal distribution lies .85 standard
deviations above the distribution mean. In our example, the distribution mean is 52, and the standard deviation is 8. Therefore, the expected
price is found to be 52(.85)(8)58.8, as shown.
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