Engineering Risk Benefit Analysis
1.155, 2.943, 3.577, 6.938, 10.816, 13.621, 16.862, 22.82,
ESD.72, ESD.721
DA 4. Introduction to Utility
George E. Apostolakis
Massachusetts Institute of Technology
Spring 2007
DA 4. Introduction to Utility 1

The Concept of utility
• Utility of a consequence: A quantification of a person's relative preference for that consequence
• A simple extension of the indifference probability
(or "preference value") concept (Axiom 2a)
• Utility function: Expresses a person's relative preferences among a set of consequences (often defined over a continuous range)
DA 4. Introduction to Utility 2

Axiom 2a: Quantification of preferences
• For each C i
, the DM can specify a number
π
(C i
), with 0
≤ π
(C i
)
≤
1, such that the DM is indifferent between
¾ possessing C i with certainty and
¾ possessing the lottery L (C*, C
*
;
π
(C i
), 1 -
π
(C i
))
DA 4. Introduction to Utility 3

Schematic representation p
1 p
2
C
1
C
2 p
1 p
2
π
( C
1
)
1−π
( C
1
)
π
( C
2
)
1−π
( C
2
)
C *
C
*
C
*
C
*
~ p m
C m p m
π
( C m
)
1−π
( C m
)
C
*
C
*
DA 4. Introduction to Utility 4

Preference measures for lotteries
• Having established the relative preferences, i.e., the
π
(C i we can derive the relative preferences for lotteries.
), i = 1,…,m,
Recall Axiom 1: Given
L
1
= L (C * , C
*
; p
1
, 1 - p
1
) and L
2
= L (C * , C
*
; p
2
, 1 - p
2
)
Then: L
1 f
L
2 if p
1
> p
2
L
1 p
L
2 if p
1
< p
2
; L
2
~ L
1 if p
1
= p
2
⇒
The probabilities p
1 lotteries.
and p
2 are the preference measures of the two
DA 4. Introduction to Utility 5

Schematic representation (cont’d) p m p
1 p
2
π
( C
1
)
1−π
( C
1
)
π
( C
2
)
1−π
( C
2
)
C *
C
*
C
*
C
*
P = p
1
.
π
(C
1
) + ... + p m
.
π
(C m
)
L
1 – P
π
( C m
)
1−π
( C m
)
C
*
C
*
P = m
Σ p i
.
π
(C i
) i=1 expected value of preference values of possible outcomes of L
DA 4. Introduction to Utility
C
*
C
*
6

Preference value of a lottery
The DM is indifferent between the original lottery on slide 4 and the last lottery on slide 6. The preference value of the latter is
P
= i m
=
1 p i
π
( C i
)
Conclusion:The preference value for a lottery is the expectation of the preference values of the possible outcomes of the lottery.
DA 4. Introduction to Utility 7

Utility
• The indifference probability or any positive linear transformation of the form
U(x) = a
π
(x) + b a>0 is said to be a utility function over the set of consequences.
Convention: x = 0 represents the present assets of the DM.
DA 4. Introduction to Utility 8

Certainty Equivalent of a Lottery
• The certainty equivalent (CE) of a lottery L is a consequence such that the DM is indifferent between the certainty equivalent and the lottery.
[Note: U(CE) = U(L)]
DA 4. Introduction to Utility 9

Risk Premium for a Lottery
• The risk premium (RP) for a single-attribute lottery is given by
RP = EMV - CE
• For risk-averse individuals, RP is a positive quantity.
Often, in practice, RP also decreases as additional wealth is acquired.
Example: If a DM has a CE of $40 for the lottery
L($100, $0; 0.5, 0.5), then his RP is $50 - $40 = $10.
DA 4. Introduction to Utility 10

Utility of outcomes
Let the utility of payoffs be
U(x) = 1.18 ln(x+5) - 1.29
-2
≤ x
≤
2 (x in $M)
[U(2) = 1, U(-2)= 0]
For L(2, -2; 0.5, 0.5) EMV = 0
U(L) = 0.5x1 +0.5x0 = 0.5 = U(CE)
CE = -0.442M
DA 4. Introduction to Utility 11

DA 4. Introduction to Utility 12

Marketing a new product
N
O s, 0.3
m, 0.5
L
2
$300K
L
3
$100K w, 0.2
L
4
-$100K
1.0
DA 4. Introduction to Utility
L
1
$150K
13

0.855
N
Decision tree with utilities
L
2
1.0
0.30
0.50
0.20
L
3
0.87
L
4
0.60
O
0.92
1.00
L
1
0.92
DA 4. Introduction to Utility 14

Choice of decision option
• If the DM chooses to market the new product,
(s)he is choosing a lottery with preference value
0.855.
• If the DM chooses to market the old product, (s)he is choosing a lottery with preference value 0.920.
• Since 0.920
>
0.855, the DM should market the old product.
DA 4. Introduction to Utility 15

Decision Rule: Maximization of expected utility
• Each possible consequence is replaced by its utility.
• Each decision option is a lottery whose preference value is the expected utility of its outcomes.
• Choose the option with the highest preference value.
• This is the result of accepting the axioms (unlike the maximization of the EMV).
DA 4. Introduction to Utility 16

D
The survey problem
.34
D
.900
O
.949
N
C
C
1
.706
.294
0
L
1
L
2
L
3
L
4
Π (C i
)
0.90
0.99
0.85
0.55
.917
S
C .42
D
.900
O
.827
N
C
C
1
.143
.714
.143
L
1
L
2
L
3
L
4
0.90
0.99
0.85
0.55
.24
.920
S
C 1
.900
w
D
.920
D
.900
O
.675
N
C
C
.920
O
.855
N
C
C
1
0
.417
.583
1
.3
.5
.2
L
1
L
2
L
3
L
4
0.90
0.99
0.85
0.55
L
1
L
2
L
3
L
4
0.92
1.00
0.87
0.60
DA 4. Introduction to Utility
Figure by MIT OCW.
17

Best Decision
• Do not buy the survey and keep marketing the old product.
• The best decision using EMV was to buy the survey and act according to its results, i.e., if the result is “strong,” then market the new product, otherwise market the old product.
DA 4. Introduction to Utility 18