Chapter 6
It is a truth very certain that when it is not
in our power to determine what is true we
ought to follow what is probable.—
Descartes
Decision Making and Risk:
Certainty Equivalents
and Utility
1
Decision Making Using
Certainty Equivalents
The certainty equivalent (CE) is the payoff
amount we would accept in lieu of undergoing the uncertain situation.
Shirley Smart would pay $25 to insure her 1983
Toyota against total theft loss. CE = – $25.
For $1,000, Willy B. Rich would sell his FarFetched Lottery rights. CE = $1,000.
2
Lucky Chance would pay somebody $100 to fill
her Far-Fetched Lottery contract. CE = -$100.
Risk Premiums
 A situation’s risk premium (RP) is the
difference its between expected payoff (EP)
and certainty equivalent (CE):
RP = EP - CE
 Shirley Smart’s car is worth $1,000 and
there is a 1% chance of its being stolen.
Thus, going without insurance has
EP = (– $1,000)(.01) + ($0)(.99) = – $10
RP = – $10 – (– $25) = $15
3
Risk Premiums
 Playing the Far-Fetched Lottery has EP =
$2,500. Thus,
 For Willy B. Rich,
RP = EP – CE = $2,5000 – ($1,000) = $1,500
 For Lucky Chance,
RP = EP – CE = $2,5000 – (– $100) = $2,600
 Different people will have different CEs and
RPs for the same circumstance.
 They have different attitudes toward risk.
4
Attitude Toward Risk
 People with positive RPs are risk averters.
 Lucky Chance has greater risk aversion than
Willy B. Rich, as reflected by her greater RP.
 We cannot compare Shirley’s risk aversion to
the others’ because circumstances differ.
 Risk averse persons have RPs that increase:
 When the downside amounts become greater.
 Or when the chance of downside increases.
 A risk seeker will have negative RP.
5  A risk neutral person has zero RP.
Maximizing Certainty Equivalent
 A plausible axiom:
Decision makers will prefer the act yielding
greatest certainty equivalent.
 A logical conclusion:
The ideal decision criterion is to maximize
certainty equivalent.
 Doing so guarantees taking the preferred action.
 But CEs are difficult to determine. One
approach is to discount the EPs.
 RP = EP – CE implies that CE = EP – RP.
6
Using Risk Premiums to Get
Certainty Equivalents
 Ponderosa Records president has the following
risk premiums, keyed to the downside.
 These were found by extrapolating from three
equivalencies (white boxes).
7
 Exact amounts are unknowable, but these values seem
to fit his risk profile.
Decision Tree Analysis with CEs
(Discounted Expected Payoffs)
8
How Good is the Analysis?
 This result is different from that of ordinary
back folding (Bayes decision rule).
 It specifically reflects underlying risk aversion.
 The result must be correct if CEs are right.
 The major weakness is the ad hoc manner
for getting the RPs, and hence the CEs.
 Many assumptions are made in extrapolating to
get the table of RPs.
 There is a cleaner way to achieve the same
thing using utilities.
9
Utility Theory
 Consider a set of outcomes, O1, O2, ..., On.
The following assumptions are made:
10
 Preference ranking can be done.
 Transitivity of preference: A is preferred to B
and B to C, then A must be preferred to C.
 Continuity: Consider Obetween. Take a gamble
between two more extreme outcomes; winning
yields Obest and losing Oworst. There is a win
probability q making you indifferent between
getting Obetween and gambling. Such a gamble is
called a reference lottery.
Utility Assumptions
 Continuity (continued):
 e.g., +$1,000 v. Far-Fetched Lottery, you pick q.
 For Willy B. Rich, q = .5. (His CE was = +$1,000.)
 For Lucky Chance, q = .9.
 If the win probability were .99, would you risk
+$1,000 to gamble? What is your q?
 Substitutability: In a decision structure, you
would willingly substitute for any outcome a
gamble equally preferred.
11
 One outcome on Lucky Chance’s tree is +$1,000;
she would accept substituting for it the Far-Fetched
Lottery gamble with .9 win probability.
Utility Assumptions and Values
 Increasing preference: Raising q makes any
reference lottery more preferred.
 Anybody would prefer the revised Far-Fetched
Lottery when two coins are tossed and just one head
will win the $10,000. (The win probability goes
from .5 to .75.) You still might not like that gamble!
 Outcomes can be assigned utility values
arbitrarily, so that the more preferred always
gets the greater value:
u(Obest) = 10
12
u(Oworst)=0
u(Obetween)=50
 Willy has u(+$10,000) = 500, u(-$5,000) = 0 and
u(+$1,000) = 250. These are his values only.
Utility Values
 Lucky has different values: u(+$10,000) = 50,
u(-$5,000) = -99, and u(+$1,000) = 35.1.
13
 Like temperature, where 0o and 100o are
different states on Celsius and Fahrenheit
scales, so utility scales may differ.
 The freezing point for water is 0o C and the
boiling point 100o C. In-between states will
have values in that range, and hotter days will
have greater temperature values than cooler.
 So, too, with utility values. They will fall into
the range defined by the extreme outcomes,
Oworst and Obest. More preferred outcomes will
have greater utilities
Utility Values
 A reference lottery can be used to find the
utility for an outcome Obetween by:
 First, establish an indifference win probability
qbetween making it equally preferred to the
gamble:
 Obest with probability qbetween and
Oworst with probability 1 - qbetween
 Second, compute the lottery’s expected utility:
 u(Obest)(qbetween) + u(Oworst)(1 - qbetween)
 The above is u(Obetween).
14
Utility Values
 Using the Far-Fetched Lottery as reference:
Lottery
Willy
Lucky
Outcomes
Prob. Utility Prob. Utility
Obest (+$10,000) q=.5 500 q=.9
50
0
Oworst (-$5,000) 1 -.5
1 -.9 -99
Expected Utility: 250
35.1
Obetween (+$1,000):
15
250
35.1
 The indifference q plays a role analogous to the
thermometer, reading attitude towards the
outcome similarly to measuring temperature.
The Utility Function
 Utility values assigned to monetary
outcomes constitute a utility function.
 From a few points we may graph the utility
function and apply it over a monetary range.
 Those points may be obtained from an
interview posing hypothetical gambles.
 Using u(+$10,000)=100 and u(-$5,000)=0
Shirley Smart gave the following equivalencies:
16
 A: +$10,000 @ qA v -$5,000 ≡ +$1,000 if qA =.70
 B: +$10,000 @ qB v +$1,000 ≡ +$5,000 if qB =.75
 C1: +$1,000 @ qC1 v -$5,000 ≡ -$500 if qC1 =.70
 C2: +$1,000 @ qC2 v -$5,000 ≡ -$2,000 if qC2 =.30
Shirley’s Utility Function
 Shirley’s utilities for the equivalent amounts are
equal to the respective expected utilities:
 u(+$1,000) = u(+$10,000)(.70) + u(-$1,000)(1-.70)
= 100(.70) + 0(1-.70) = 70
 u(+$5,000) = u(+$10,000)(.75) + u(+$1,000)(1-.75)
= 100(.75) + 70(1 - .75) = 92.5
 u(-$500) = u(+$1,000)(.70) + u(-$5,000)(1-.70)
= 70(.70) + 0(1 - .70) = 49
 u(-$2,000) = u(+$1,000)(.30) + u(-$5,000)(1-.30)
= 70(.30) + 0(1 - .30) = 21
17
 Altogether, Shirley gave 6 points, plotted on the
following graph. The smoothed curve fitting
through them defines her utility function.
Shirley’s Utility Function
18
Using the Utility Function
 This utility function applies to the
Ponderosa decision.
19
Using the Utility Function
 Read the utility payoffs corresponding to
the net monetary payoffs.
 Apply the Bayes decision rule, with either:
 A utility payoff table, computing the expected
payoff each act.
 Or a decision tree, folding it back.
 The certainty equivalent amount for any act
or node may be found from the expected
utility by reading the curve in reverse.
 The following Ponderosa Records decision
tree was folded back using utility payoffs.
20
Decision Tree Analysis
with Utilities
21
Shape of Utility Curve and
Attitude Toward Risk
 The following shapes generally apply.
22
 The risk averter has decreasing marginal utilility for money. He will buy casualty insurance
and losses weigh more heavily than like gains.
 Risk seekers like some unfavorable gambles.
 Risk neutrality values money at its face amount.
Important Utility Ramifications
 Hybrid shapes (like Shirley’s) imply shifting
attitudes as monetary ranges change.
 Regardless of shape, maximizing expected
utility also maximizes certainty equivalent.
 Therefore, applying Bayes decision rule with
utility payoffs discloses the preferred action.
 Primary impediments to implementation:
 Clumsiness of the interview process.
 Multiple decision makers.
 Attitudes change with circumstances and time.
23
Ratification of
Bayes Decision Rule
 Over narrow monetary ranges, utility curves
resemble straight lines.
 For a straight line, expected utility equals
the utility of the expected monetary payoff.
 Maximizing expected monetary payoff then
also maximizes expected utility. Thus:
 The Bayes decision rule discloses the preferred
action as long as the outcomes are not extreme.
 Managers can then delegate decision
making without having to find utilities.
Preferred actions will be found by the staff.
24
Using Utility Functions with
PrecisionTree
PrecisionTree can be used to evaluate
decision trees with with exponential and
logarithmic utility functions.
To get started, click on the name box of a
decision tree and the Tree Setting dialog box
appears, as shown next.
A
1
2
25
tree #1
B
1
0
Tree Settings Dialog Box
(Figure 6-14)
1. Check the
Use Utility
Function box.
3. Select the
risk coefficient,
R, in the R value
line. Here
10,000 is used.
4. Select
Expected Utility
in the Display
line. Other
options are
Certainty
Equivalent and
Expected Value.
5. Click OK.
26
2. Select the
type of utility
function in the
Function line.
Here exponential
is chosen.
Decision Tree with Exponential Utility
Function for R = 10,000 (Figure 6-15)
A
1
The2 optimal strategy is:
3
4
1. Not
test market and to
5
abort.
6
7
8
2. The
corresponding
9
expected
utility is 0.
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Ponderosa Record Company
24
25
26
27
28
29
30
31
32
33
27
34
B
C
D
E
80.0%
Success
$90,000
Market nationally
FALSE
-$50,000
-48
20.0%
$0
0.0
-244
20.0%
$90,000
0.0
1
Failure
50.0%
$10,000
Favorable
-1
TRUE
$0
Abort
Test market
FALSE
-$15,000
0
-1
-2
Success
Market nationally
FALSE
-$50,000
-531
Failure
Unfavorable
50.0%
$0
-3
Abort
TRUE
$0
0
-3
50.0%
$100,000
0
1
0
Success
Market nationally
FALSE
-$60,000
-201
50.0%
$0
Failure
Don't test market
F
0.0
1
TRUE
$0
0
Abort
TRUE
$0
1
0
0
-402
80.0%
$0
0.0
-664