Measuring Preference:
A Brief Summary of
Utility Theory
Shannon Neeley
Jenny Zhang
Utility: Some Definitions
The desirability of preference that
individuals or societies have for a given
outcome
Quantitative measure of the attractiveness
of a potential outcome
Outlines
Bernoulli Family
Daniel Bernoulli’s Paper
Utility and Utility Function
Caius example
St. Petersburg Paradox
Extensions of Bernoulli theory
von Neumann and Morgenstern utility
Problems with Utility theory
Bernoulli Family
Daniel Bernoulli
Born: 8 Feb 1700 in Groningen, Netherlands
Died: 17 March 1782 in Basel, Switzerland
Simple Biography
The Victorian statistician Francis Galton describes him as “Physician,
botanist, and anatomist, writer on hydrodynamics; very precocious”.
He was also a powerful mathematician and statistician, with a special
interest in probability.
Exposition of a New Theory on the
Measurement of Risk (1954)
“Ever since mathematicians first began to study
the measurement of risk, there has been general
agreement on the following proposition:
Expected values are computed by multiplying
each possible gain by the number of ways in
which it can occur, and then dividing the sum of
these products by the total number of possible
cases.”
No characteristic of the persons themselves has
been taken into consideration.
Why Utility
Example: buying a lottery ticket
All men cannot use the same rule to evaluate the
gamble. There is no reason to assume that two
persons encountering identical risks.
“The determination of the value of an item must
not be based on its price, but rather on the
particular circumstances of the person making the
estimate.”
“It becomes evident that no valid measurement of
the value of a risk can be obtained without
consideration being given to its utility”
Utility and Utility Function
The utility is dependent on the particular circumstances of
the person making the estimate.
X
U  k
X
U(X) = a+b*X
U(X) = sqrt(X)
U(X) = log(X)
U is utility
K is a constant of proportionality
X is the base amount of wealth
Caius example
Caius was contemplating whether he should take
insurance on a shipment from Amsterdam to St.
Petersburg.
safe
10,000 rubles
storm
5% will be lost.
Shipment
The Amsterdam underwriters want a full covered policy
payment of 800 rubles, or 8% of the profit if no mishap
occurs.
Should Caius buy the policy?
Caius example
If we go by simple ruble values, then he
should self-insure.
If Caius follows Bernoulli’s advice, he
should buy the policy if his expected utility
for insuring is greater than that for not
insuring.
Caius example
Suppose Caius’s capital is X rubles
W.L.O.G. take k=1
Caius’s expected utility of not insuring minus that of
insuring is given by
X  10,000
X  9,200
f ( X )  .95 * log[
]  log[
]
X
X
f ( Xn)
Newton’s Method
X n 1  X n 
f ' ( Xn)
If Caius has less than 5,042 rubles, he should (according
to Bernoulli), buy the 800 ruble policy.
Caius example
How much the underwriter (insurer) should have
in hand in order to sell the 10,000 ruble policy for
the amount of 800 rubles?
Y  800
Y  9,200
g (Y )  .95 * log[
]  .05 * log[
]
Y
Y
If the underwriter has a stake of 14,242 rubles or
more, Bernoulli’s rule tells us that selling the
policy will improve the underwriter’s position.
“But no one, however rich, would be managing his
affairs properly if he individually undertook the
insurance for less than five hundred rubles”
St. Petersburg Paradox
From Nicolas Bernoulli’s letter
Consider the following game
Peter flips a coin and will give Paul:
• $2 if the first flip is a head
• $2k if the kth flip is the first head
How much should Paul pay to play this game?
A poor person might be unwilling to pay
more than two dollars, the minimum
possible pay-off of the game.
St. Petersburg Paradox (cont)
k
k 
1
E ( payoff )   2    1  1    
2
k 1
k
The expectation is infinitely great but no one
would be willing to purchase it at a moderately
high price.
Applying the theory
(  1) * 4 (  2) * 8 (  4) *16 (  8)   
where α is Paul’s fortune
A relationship between one’s wealth and the amount
one would pay to play this paradoxical game.
If Paul owned nothing, his opportunity would be
worth approximately two ducats
10 – 3 ducats
100 – 4 ducats
1000 – 6 ducats
“We can easily see what a tremendous fortune a man
must own for it to make sense for him to purchase
Paul’s opportunity for twenty ducats.”
Applying the theory (cont)
“The amount which the buyer ought to pay
for this proposition differs somewhat from
the amount it would be worth to him were it
already in his possession. Since, however,
this difference is exceedingly small if alpha
is great.”
Bernoulli’s Contribution
“His paper is one of the most profound documents
ever written, not just on the subject of risk but on
human behavior as well. Bernoulli’s emphasis on the
complex relationships between measurement and gut
touches on almost every aspect of life.”
“For the first time in history Bernoulli is applying
measurement to something that cannot be counted.
Bernoulli defines the motivations of the person who
does the choosing. This is an entirely new area of
study and body of theory. Bernoulli laid the
intellectual groundwork for much of what was to
follow, not just in economics, but in theories about
how people make decisions and choices in every
aspect of life.” (Bernstein, P.L. (1996). Against the Gods)
Jeremy Bentham
(1789) The Principles of Morals and Legislation
Suggested measurement of pleasure and pain
Utility: “…that property in any object, whereby it
tends to produce benefit, advantage, pleasure,
good, or happiness…when the tendency it has to
augment the happiness of the community is
greater than any it has to diminish it.”
Utility became a tool for discovering how prices
result from interactions between buyers and sellers
Law of supply and demand
William Stanley Jevons
(1871) The Theory of Political Economy
“Value depends entirely on utility”
Like Bernoulli, thought that the utility
varies with the amount of commodity in
one’s possession
Reflected era’s zeal for measurement
“Pleasure, pain, labor, utility, value, wealth, etc.
are notions admitting of quantity”
Game Theory
“The true source of uncertainty lies in the
intentions of others”
Invented by John von Neumann
(1944) Theory of Games and Economic Behavior
With Oskar Morgenstern
Rigidly mathematical with an emphasis on
numerical quantities
Claim that this matches reality in the same way
measurements have been attached to heat and light
Quantifying Preference
Assume a man prefers milk to coffee and prefers
coffee to tea
Do you prefer a cup of coffee to a cup that has a 50-50
chance of being milk or tea?
What about having $1 or $2
Would you rather have $1 or a 50-50 chance of either
winning $2 or nothing?
Vary the probability to find the point that the man
is indifferent between the sure thing and the
gamble.
Expected Utility
If the man is indifferent between $1 certain
and the gamble:
50% preference means his preference for $1
over 0 is half as great as his preference for $2
over 0
The utility of $2 is double the utility of $1
Maximize expected utility based on rational
behavior
von Neumann-Morgenstern
Axioms of Rational Behavior
Transitivity: If the subject is indifferent
between outcomes A and B and between B
and C, he must be indifferent between A and
C.
Continuity of preferences: If A is preferred
to B and B is preferred to no change, the
there is a probability α such that the subject
is indifferent between αA and B
von Neumann-Morgenstern
Axioms of Rational Behavior
Independence: If one is indifferent between
A and B then for any probability α, one is
indifferent between αA and αB
Desire for high probability of success: If A
is preferred to no change and if α1 > α2, then
α1A is preferred to α2A
Compound probabilities: If one is
indifferent between αA and B, and if α=
α1α2, then one is indifferent between α1α2A
and B
Problems with Utility Theory
Entire risk profile cannot be captured with a
single number (expected utility)
Utile has no meaning to most people
No natural utility function (i.e., when
should we use log or square root?)
People violate axioms
Prospect Theory from Daniel Kahneman and
Amos Tversky
0.0
0.2
0.4
0.6
0.8
1.0
Probability Profile for St. Petersburg Game
2
4
6
Dollars
8
10
Reference point
Bernoulli: “The utility resulting from any small
increase in wealth will be inversely proportionate
to the quantity of goods previously possessed”
The assessment of a risky opportunity is more
dependent on the reference point from which the
possible gain or loss will occur than on the final
value of the assets that would result.
“Our preferences can be manipulated by changes
in the reference points” (Tversky)
Reference Point: Example
Option 1: Everyone just won $30
Flip a coin vs. no coin flip
• Heads: win $9
• Tails: lose $9
70% chose the coin flip
Option 2: Everyone starts with $0
Flip a coin vs. $30 for sure
• Heads: win $39
• Tails: win $21
43% chose the coin flip
Students base their choice on the reference point
Failure of Invariance
Invariance: If A is preferred to B and B preferred
the C, then A is preferred to C
Suppose you go to the theater with a ticket
You lost your $40 ticket
Do you buy another ticket?
Suppose you plan to buy the ticket there
You lost the $40
Do you use another $40 to buy the ticket?
Most people won’t replace the lost ticket, but will
pay another $40
Conclusion
Although does not work in practice, forms
the foundation of
Economic theory, supply and demand
Theory of Utilitarianism
Decision Making
Game theory
References
Bernoulli,D., “Exposition of a new theory on the
measurement of risk”, Econometrica, Vol. 22, No. 1. (Jan.,
1954), pp. 23-36.
Bernstein, P. (1996). Against the Gods: The Remarkable
Story of Risk. John Wiley & Sons: New York
Stigler, G.J. (1968). “The Development of Utility Theory.”
in Utility Theory: A Book of Readings. John Wiley &
Sons: New York.
Thompson,J.R. (2002). Models for Investors in Real world
Markets. John Wiley & Sons: New York. pp.29-59
Von Neumann, J. and Morgenstern, O. (1947) Theory of
Games and Economic Behavior. Princeton University
Press: Princeton.