Thinking and Deciding

322 UTILITY MEASUREMENT
I could continue in this manner, dividing the utility interval in half each time. If
I assigned a utility of 0 to $0 and a utility of 8 to $1 million, I would assign a utility
of 4 (halfway between 0 and 8) to $x, once I discovered what $x was. I could then
divide the interval between $0 and $x in half in the same way, and assign this wealth
a utility of 2, and so on. In my case, I might assign a utility of 2 to $50,000, since
this would be subjectively halfway between $0 and $150,000.
Notice that the bottom and top values of the scale could be chosen arbitrarily. To
use utilities in decision analysis, we need only relative values. For example, suppose
that I have $50,000 in savings and I want to know whether I ought to invest it in a
real estate deal that has a .5 probability of netting me $1 million and a .5 probability
of losing everything. The expected utility of the deal is .5·u($1 million)+.5·u($0).
The way I have scaled utility, this would be .5 · 8+.5 · 0, or 4. This is higher than
the utility of my current wealth, which is 2, so I ought to go for it.
Suppose I had used a different scale, though, so that I had assigned a utility of
100 to $0 and a utility of 200 to $1 million. When I cut this interval in half, I would
have discovered that the utility of $150,000 is 150, halfway between 100 and 200,
and the utility of $50,000 is 125. In this case, the expected utility of the real-estate
deal is 150, and the utility of my current wealth would be 125, still less than 150,
and by the same proportion of the total range as before.
In order for difference measurement to be used, the utilities of outcomes must
be related by a condition called monotonicity. Consider the following outcomes, laid
out in order of desirability. The letters A through F might stand for different amounts
of money, or different jobs.
A B C D E F
Suppose that the difference between A and B is subjectively equal to the difference
between D and E, and the difference between B and C is subjectively equal to the
difference between E and F. It must be the case, then, that the difference between A
and C is also equal to the difference between D and F (Krantz, Luce, Suppes, and
Tversky, 1971, ch. 4). If this monotonicity condition is not met, inconsistencies will
arise in the scale. To see how that might happen, imagine that F is directly above E,
rather than to the right of it. Then the distance between E and F could still be the
same as that between B and C, yet the distance between A and C would be greater
than that between D and F. The outcomes would not lie along the same line.
Note that the difference method can be used directly, as in the examples just
given, or it can be used as a check on the direct-rating method. If the rater understands
that the differences between ratings are supposed to be meaningful, this may
be enough to correct distortion of the scale. Either way, the monotonicity condition
can serve as a check. As yet, no one has asked experimentally whether monotonicity
applies either to direct ratings or to difference measurement. Perhaps this question
has not been asked because no one has thought of a reason why the monotonicity
condition would ever be violated systematically. Checks seem to be a good idea,
however.