Thinking and Deciding

348 DECISION ANALYSIS AND VALUES
decreases.) To answer this, ask yourself at what price $x you would be indifferent
between 128K for $2,500 and 64K for $x. You can determine x by starting with
a value that is clearly too high and lowering it until you are indifferent, and then
starting with a value that is clearly too low and raising it. If you encounter a range
of indifference, you can choose the middle of the range as your best guess. If you
are indifferent between 128K for $2,500 and 64K for $x, then it would make sense
to assume that u($2, 500) + u(128K) =u($x)+u(64K),oru($x) − u($2, 500) =
u(128K)−u(64K). Hence, if u(128K)−u(64K) is 1 utile, then u($x)−u($2, 500)
must be 1 utile as well.
We could then mark off another utile on the price dimension by asking at what
price $y you would be indifferent between 128K for $x and 64K for $y. Forthenext
utile, we could ask at what price $z you would be indifferent between 128K for $y
and 64K for $z, and so on.
Once we have defined 1 utile on the price dimension, we can then use this unit to
mark off steps on the memory dimension in the same way. For example, we can ask
at what memory size A you would be indifferent between A at $2,500 and 128K at
$x. In theory, the method of conjoint measurement is like the method of differences,
except that the differences compared are on various attributes instead of a single
attribute. 1
Once we have gone this far, we ought to be able to check what we have done
by asking about the two points labeled T in Figure 14.1. We ought to be indifferent
between these two points: A for $x and 128K for $y. This condition is called the
Thomsen condition (Krantz, Luce, Suppes, and Tversky, 1971, ch. 6). In order for
this scaling method to work, it must be satisfied for all sets of points of this form.
The Thomsen condition serves as a check on this method, just as monotonicity does
on the difference method.
When there are three or more dimensions, we can replace the Thomsen condition
with a simpler condition, called independence 2 This means that the tradeoff between
any two dimensions does not depend on the level of a third. For example, if you are
indifferent between 128K for $2,500 and 64K for $2,000 when the computer has a
ten-megabyte hard disk, you will still be indifferent between 128K for $2,500 and
64K for $2,000 when you have a thirty-megabyte hard disk. The tradeoff between
money and memory is not affected by the size of the hard disk. This condition
ensures that the contribution of each dimension to overall utility will be the same,
regardless of the levels of other dimensions.
1 In practice, the method of conjoint measurement can be applied to data in which judges simply
express a large number of preferences among pairs of options spread throughout the space in Figure 14.1.
The analyst then infers the equivalent intervals from these preferences (Keeney and Raiffa, 1976/1993;
Tversky, 1967). This is the basis of conjoint analysis, discussed in Chapter 15.
2 Independence implies the Thomsen condition (Keeney and Raiffa, 1976/1993, sec. 3.5.3). The term
“independence” has been used for a variety of conditions, each of which may imply different types of
measurement (von Winterfeldt and Edwards, 1986, chs. 8–9). It is also analogous to the sure-thing principle,
if we look at states as dimensions of choices. The probability of a state corresponds exactly to the
weight of the dimension in MAUT.