In
general, a measure is required in inferring or explaining behavior or
phenomena. Inferring that one would turn right instead of left on a crossroads
can be due to it being faster, safer, or more comfortable, etc. These are all
measures. Not many options are required to measure, though two are needed at
the very least. It is a measure in saying that A is bigger than B. And if I say
under certain circumstances one will pick a big one instead of a small one,
this is inference.
To measure
is to rank: in order of size, in order of quantity, in order of weight, etc.
Supposing there are too many options for ranking, and if it is still not enough
after exhausting A, B, C, D …, then we will use numbers. Numbers are infinite.
The definition of measure is to use numbers to arbitrarily rank. Yet number
itself has no content. When I say 17, 29, you would have no idea what I am
talking about. But if I say 29 pounds you would know that I am referring to the
weight of an object, and that 29 pounds is heavier than 17 pounds.
To
describe a selfish person maximizing his self-interest, we can use numbers to
rank his options. If I say under certain circumstances, this person will choose
29 instead of 17, you will then ask: what does 29 or 17 stand for?
The
problem lies exactly here. I have to use numbers to rank your options, yet
number itself has no content. What can we do? I could say the numbers you have
chosen represent pounds, but “pound” also refers to weight which may cause
confusion. Somehow a name has to be given to these numerically-ranked options.
What can we do? I therefore close my eyes, casually open an English dictionary,
put my finger down and then open my eyes, the word is utility.
In the
mid-twentieth century, after the cultivation of numerous scholars over more
than a century, the only creditable definition of utility is this simple:
utility is a numerical assignment in ranking options. It represents neither
happiness, nor enjoyment, nor welfare. Utility represents options ranking, and
since there are innumerable options, we arbitrarily use numbers and proclaim
that bigger numbers are preferable to smaller ones, or smaller numbers are
preferable to bigger ones, but not that big or small numbers are equally
preferable.
“Utility”
is an arbitrary numerical assignment in ranking options. Whether the number is
big or small is not critical. Of essence is the sequence: if we say the utility
of a big number is preferable to that of a small one, we cannot reverse that
halfway and say a smaller number is more preferable. This is a requisite of
logic.
In
general, there are three usages of number, out of which two are for
measurement. The non-measure usage of number is for identification. For
instance, when you go horse racing, on each horse is a number like number 7,
number 3, etc. These numbers do not indicate big or small, fast or slow, but
are for identification purposes only. If you bet on horse number 7 and that
particular horse wins, you can go to collect your winning.
The other
two usages of number are in relation to measure. Two types of measure exist
since there are two kinds of rankings in numerical measure. One kind of ranking
that can be added together is called cardinal measure; the other kind of numbers
that can only be ranked but not added together is called ordinal measure.
A fish
weighs 2 pounds and a chicken weighs 3 pounds, the two added together is 5
pounds. If you are in vain looking for an 8-foot long rope, then adding up a
3-foot rope with a 5-foot one gives you 8 feet. Foot is therefore cardinal. All
cardinal measures can be linearly transformed. For instance, Fahrenheit is a
cardinal measure, so is Celsius. By knowing the degree in Fahrenheit, we can
safely use an equation to find out the exact degree in Celsius. Pound and
kilogram, yard and meter, can all be linearly transformed.
A problem
in measuring utility is that utilities cannot necessarily be summed. The
utility number of one pound of bread is 4 while the utility number of one ounce
of butter is also 4, when both of them are eaten, the utility number will be
greater than 8. The utility number of a cup of coffee is 4 while the utility
number of a cup of tea is also 4, but the utility number per cup, when both of
them are drunk, will be less than 4. When dealing with complements (like bread
and butter) or substitutes (like coffee and tea), there is no easy solution to
the so-called additive utility issue.
Nonetheless,
economists have spent substantial effort on devising certain solution to
cardinally measure utility. The most remarkable is the book “Theory of Games and Economic Behavior”
written by twentieth century’s mathematics guru, John von Neumann (1903 –
1957), and economist, Oskar Morgenstern (1902 – 1977). In the second edition (1946)
of this popular book, the authors pointed out that in the presence of risks,
utility could be cardinally measured. However, four assumptions are required
for such measurement, out of which two are problematic.
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