1. Systems Analysis Methods
Dr. Jerrell T. Stracener, SAE Fellow
SMU
EMIS 5300/7300
NTU
SY-521-N
Utility Theory
Basic Concepts
updated

2. Terminology for Preferences
Suppose further that, out of this set the person
is given only two options, say A and B, and asked
to choose one. The choice taken reveals the
person’s preference between the options.
A > B denotes that A is preferred to B
A < B denotes that B is preferred to A

3. Terminology for Preferences
A ~ B
~ is read as “the person is indifferent between
A and B
A  B denotes that B is not preferred to A
A  B denotes that A is not preferred to B

4. Terminology for Preferences
The statement A is preferred to B is a strong
preference for A.
The statement B is not preferred to A is a weak
preference for B.
The strong preference for A indicates that A is
valued over B by some amount. This preference can be interpreted to mean that the cost of A
could be increased by at least some small
increment, and the person would still choose
A over B.

5. Terminology for Preferences
A preference ordering is rational if it is
transitive, that is, if it satisfies the following
condition:
if A > B > C then A > C
A person is rational if his or her preference
ordering is transitive.
We assume that we are dealing with rational
individuals

6. The Notion of Utility
We seek to assign to each option, i, a real
number, ui, such that
if A > B then uA > uB
ui is the utility of option i, and its units are utiles.
If this can be done, then a comparison of the real
numbers ui will reveal the decision maker’s set
of preferences.
Utilities that provide only preference orderings
are referred to as ordinal utilities. Ordinal utilities
cannot be added or subtracted, and they do not
measure strength of preference.

7. The Law of Diminishing Marginal Utility
It states that as the amount of a good
consumed increases, the marginal utility of the
good decreases.
The slope of the curve at any point is referred
to as the marginal utility.
saturation
point
utility
slope =
marginal
utility
marginal utility
saturation
point
quantity consumed
quantity consumed

8. The Law of Equal Marginal Utilities
Each good is demanded up to the point where
the marginal utility per dollar spent on it is exactly
equal to the marginal utility of a dollar spent on
any other good.This law says that the value of a
unit of currency lies in its purchasing power for
alternative goods.
The slope of the curve at any point is referred
to as the marginal utility.
 = PAqA + PBqB + PCqC
where P is price and q is quantity

9. The Law of Equal Marginal Utilities
marginal utility, utiles per $
quantity consumed qA qB qC A B C
budget constraint

10. Linearly Additive Utility
The additive utility function is perhaps the
multi-attribute utility form most commonly used
by engineers. For a utility-independent system
that may be described in terms of n attributes,
the system utility is given by
where ui is the utility of attribute i occurring in
quantity xi and ai is a weighting factor.

11. Multiplicative Utility
The multiplicative utility form is given by
where again the ui are the utilities of each
attribute, and  is an arbitrary positive constant.
 = 1 is perfectly acceptable. The multiplicative
utility form often enables a simpler form for each
of the component utilities than does the linearly
additive form.

12. Log-Linear Utility
The log-linear utility form is given by
This form bears resemblance to both the linearly
additive form and the multiplicative form. If we
allow the transformation we see that these forms
are actually identical, given the proper choice of
ui.

13. Log-Linear Utility
Also, we see that
Thus, the log-linear and the multiplicative forms
are identical, and hence all three forms are
equivalent, given the proper choice of the ui.