What is Utility?
Prepared by:
Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York
email:
malkevitch@york.cuny.edu
web page:
http://york.cuny.edu/~malk
1. Introduction
Although we are accustomed to measuring the value of things in monetary terms, people not only have an objective view of the value of things but also a “psychological view.” From one point of view a dollar is a dollar, but from a different view point, we see that the value of $100 bill to a millionaire is different from what it is to a poor person. While one person may lean over to pick up a dime he sees on the sidewalk, another person may pass by a quarter that she sees because she can not afford the extra two seconds of time to pick it up. What is the dollar value of a collection of your family’s photographs? To you it may have very high value but you might be unable to sell the photos to anyone. Thus, the usual dollar price approach to the value of something cannot always be used.
In negotiations that might take place between a company and its union there are monetary issues at stake but there are also subtle issues of power and worth at stake. In describing the consequences of various actions that might be taken, one knows which outcomes are preferable even though in many cases there are no dollar equivalents that make it possible to say that one outcome is worth more than another outcome. The concept of “utility” helps in understanding “psychological” value or worth.
The other major reason for studying utility lies in the area of analyzing the processes by which decisions are made between risky alternatives. If we have to make a decision as to whether to drill for oil in a variety of different locations where the probabilities of finding oil and/natural gas vary from site to site, and the possible “payoff” varies also, we need to have a framework for choosing which of the risky alternatives we should pursue. Utility theory offers a method for making computations which enable us to decide which risky alternative to pick.
2. Using Utilities
To deal with the fact that people can make value judgments about the relative merits of different outcomes or courses of action even though there is no explicit underlying scale that the outcomes can be measure on, it is convenient to invent a fictional quantity called “utility” that can be assigned to the different items one is comparing that can serve as the underlying “yardstick.” The “utility” of something is a number assigned to that thing and in comparing which of two things has more (psychological ) value, the assignment is to be made in such a way that if i is preferred to j then i has higher utility than j. The way to think of utilities is that they are a numerical scheme to “represent” the preferences that exist between a collection of objects. THE PREFERENCE STRUCTURE COMES FIRST AND THE UTILITIES USED TO REPRESENT THE PREFERENCE STRUCTURE COME LATER. The utilities that I assign to say 5 fruits enables an outsider to tell which of the fruits I like best or which ones I am indifferent between. (Allowing indifference adds more complications but when someone is indifferent between two things they will be assigned the same utility.)
Example (Nina’s utilities for four fruits):
Fruit Utility
empire apple 40
banana 25
orange 25
red delicious apple 10
From this table one could conclude that if offered the choice of a banana or an empire apple that Nina would chose the empire apple. On the other hand she is indifferent between having an orange or banana.
It is tempting in looking at the table to conclude that Nina likes empire apples 4 times as much as she like red delicious apples. To make this conclusion it would be necessary to assume that one can “do arithmetic” on utilities in the same way that one can do arithmetic with the weights of objects in kilograms or with time lengths in seconds. Forty kilograms is 4 times heavier than 10 kilograms and 100 seconds is 4 times as long as 25 seconds, however, is 80º Fahrenheit 4 times as hot as 20º Fahrenheit? As you probably realize one can not do arithmetic on Fahrenheit temperatures in the same way that you can do arithmetic with times and weights. On the other had we makes statements of the kind that yesterday was 10 degrees hotter than today. Does this mean that with respect to Nina’s preferences for fruits that we can say that since the difference between empire apple and banana is 15 which is equal to the difference between banana and red delicious apple that the amount that Nina likes empire apples more than bananas is the same amount that she likes banana more than red delicious apples?
In light of this discussion, can utilities be made to behave the way time and mass do or must we settle for something less?
3. Ordinal versus Cardinal Utilities
Since the utilities that we assign to a preference structure are numbers it is tempting to hope that these numbers have more significance than just their relative sizes. Consider the the person who ranked the three fruit, pears (P), apples (A), and kiwis (K) as shown in Figure 1. On the left, which represents June’s preferences, June has assigned each fruit a number which shows that the higher the number the more preferred the fruit. On the right, which represents May’s preferences, May also has assigned each fruit a number which shows that the higher the number the more preferred the fruit. The question is do the numbers themselves say anything about the strength of preference that the individual has. Does June like kiwis a lot more than pears or just a little? Does May like kiwis a lot more than pears or just a little? Does June like pears just a little more than an apple? Does May like pears much more than apples or might she like them just a little bit more?
Is there a way for a person meaningfully to assign “cardinal” rather than ordinal utilities to things? If cardinal utilities are possible, can they be compared between people? Can one multiply, add, and divide these utilities?
Workers in mathematics and behavioral science have investigated these questions at great length, leading to lots of new mathematics as well as attempts to produce practical guides for people who are trying to make thoughtful complex decisions.
References
1. Luce, R., and H. Raiffa, Games and Decisions, Wiley, New York, 1957. (Reprinted by Dover Press
2. Allingham, M., Choice Behavior, Oxford, New York, 2002.
3. Peterson, M., An Introduction to Decision Theory, Cambridge U. Press, New York, 2009.
4. Resnik, M., Choices: An Introduction to Decision Theory, U. Minnesota Press, Minneapolis, 1987.