Utility is the Name of Numbers

Chapter 4, Section 2: Utility is the Name of Numbers

Generally speaking, to predict or explain behaviors/phenomena requires a measurement. To predict you'd take a right turn at a crossroad instead of a left one, is because turning right is faster, safer, or more comfortable, etc., which are all measurements. A measurement doesn't have to have many options, but at least two. To say A is bigger than B is a measurement, and that I assert under certain circumstance you'd choose the bigger over the smaller, is a prediction.

A measurement is ranking: ranked by big and small, by more and less, by heavy and light, etc. If the options for ranking are too plenty, A, B, C, D ... all used up but are still not enough, we then need numbers. Numbers are unlimited in amount. A measurement is thus defined as ranking with numbers. Yet these numbers have no content. When I say 17 and 29, you don't know what I am talking about. But if I say 29 pounds, you'd know it is the weight of some item, and also know 29 pounds is heavier than 17.

Because a selfish man maximizes his interest, we can use numbers to rank his options. If I assert under some circumstance this man would choose 29 over 17, you will certainly ask, what are exactly the 29 and 17?

Here right is the problem. I use numbers to rank your options, but the numbers don't have content, how does that happen? I can say the numbers you'd pick are in pounds, but "pound" means weight and that'd bring confusion. Anyway, I have to give a name to these ranking numbers, what shall I do? I therefore close my eyes, open a random page in dictionary, put down my finger on a word, reopen my eyes again and read it: utility. (Translator: Utility here corresponds to weight instead of pound.)

After more than one hundred years of nurture by countless scholars, at the time of mid-twentieth century, feasible definition of utility was simple: utility is the name of the numbers used in option ranking. It neither stands for happiness, nor enjoyment, nor welfare. Utility stands for option ranks, and as numbers are unlimited in amount, we let them come into play, and assert the option corresponding to a larger number is preferable, or vice versa, but never equally preferable.

"Utility" is an arbitrary name for the numbers of option ranking. It doesn't matter how big a number is, but what rank it takes: if the utility of a bigger number is set preferable over that of a smaller one, it cannot be reversed in the midway of analysis. This is a requirement by logic.

Roughly speaking, numbers have three applications, two of which are measurable. First, the non-measurable application is identification. If you go to a horse race and make a bet, there'd be a number for each horse, like 7, 3, etc. These numbers mean neither size nor speed, but are used only for identification. If you bet on the 7 and it wins, you can then have your prize.

The other two applications are about measurement. There are two types of measures because numbers can have two types of ranking. By one ranking, numbers can be added up, and that's called cardinal measure; while by the other, numbers can only be ranked but not added up, it's called ordinal measure.

A fish is 2 pounds and a chicken 3, with the two added up it's 5 pounds. So pound is cardinal. If you can't find an 8 feet rope, you can just add up a 3 feet and a 5. Foot is also cardinal. All cardinal measures can go through linear transformation. For instance: Fahrenheit degree of temperature is cardinal, and so is Celsius, thus with the Fahrenheit number at hand we can obtain the Celsius degree through a formula, securely. Pound and kilogram, or yard and meter, are all linearly transformable.

One difficulty for measuring utility is, utility is not always additive. The utility number for a pound of bread is 4, and that for an ounce of butter is 4 either, but when the two are eaten together, the utility number would be greater than 8. The utility for a cup of coffee is 4, and that of a cup of tea is 4 either, but when drunk together, the number of each cup would be less than 4. So when dealing with compliments like bread and butter, or substitutes like coffee and tea, additive utility would have insurmountable difficulties.

That said, economists once had devoted great energy to find a way so that utility can be cardinally measured. The most fabulous was the book cooperated by the twentieth century master of mathematics J. von Neumann (1903-1957) and economist O. Morgenstern (1902-1977), Theory of Games and Economic Behavior. In its second edition (1946), the authors pointed out, when there is risk, utility can be cardinally measured. Yet this cardinal measurement requires four assumptions, while two of them are problematic.