Decision Theory Part-2

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Graduate Program in Business Information Systems
Decision Analysis - Part 2
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Ex: Decision to buy insurance
ACT (choice)
Event
Fire
No fire
Probability Buy insurance
Do not buy
insurance
0.002
-$100
-$40,000
0.998
-$100
0
-$100
-$80
Expected Payoff
Best act
IS IT SURPRISING?
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Is Bayes decision rule invalid?
 No, actually the true worth of outcomes is
not completely reflected by the payoffs!
Two approaches:
 Certainty Equivalents
 Utility Function
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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.
Game: win $5000 with probability 50%
win 0
with probability 50%
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Risk Premiums
 A situation’s risk premium (RP) is the
difference between its 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
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Risk Premiums
 Playing the Far-Fetched Lottery has EP = $2,500.
Thus,
 For Willy B. Rich,
RP = EP – CE = $2,500 – ($1,000) = $1,500
 For Lucky Chance,
RP = EP – CE = $2,500 – (– $100) = $2,600
 Different people will have different CEs and RPs
for the same circumstance.
 They have different attitudes toward risk.
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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.
 A risk neutral person has zero RP.
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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.
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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).
 Exact amounts are unknowable, but these values seem to fit his
risk profile.
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Calculation of Risk Premiums for
Panderosa
The president of panderosa Record Co. is asked the
following:
 How much would you be willing to pay for insuring
$100.000 recording equipment if there is 1% chance of
losing them due to external occasions. Note that, here
Expected payoff=$0(0.99)+(-$100.000)(0.01)=-$1000
 He is willing to pay $2500 to get rid of this danger.
Certainty equivalent= -$2500
RP=-1000-(-2500)=$1500
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Decision Tree Analysis with CEs
(Discounted Expected Payoffs)
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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.
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Decision Making with Utility
Expected monitary value may not accurately
reflect the DM’s preference when significant
risks are involved!
It is also hard to evaluate risk premiums to
calculate certainty equivalents.
An alternative aproach is to replace payoffs with
utilities.
Max. certainty equivalent
Max. utility

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Utility Assumptions
 Consider a set of outcomes, O1, O2, ..., On. The
following assumptions are made:
 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.
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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.
 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.
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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
u(Oworst)=0
u(Obetween)=5
 Willy has u(+$10,000) = 500, u(-$5,000) = 0 and u(+$1,000) =
250. These are his values only.
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Utility Values
 Lucky has different values: u(+$10,000) = 50,
u(-$5,000) = -99, and u(+$1,000) = 35.1.
 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
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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(Obetween)=u(Obest)(qbetween) + u(Oworst)(1 - qbetween)
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Utility Values
 Using the Far-Fetched Lottery as reference:
Lottery
Outcomes
Willy
Lucky
Prob.
Utility
Prob.
Utility
Obest (+$10,000)
q=.5
500
q=.9
50
Oworst (-$5,000)
1 -.5
0
1 -.9
-99
Expected Utility:
250
35.1
Obetween (+$1,000):
250
35.1
 The indifference q plays a role analogous to the
thermometer, reading attitude towards the outcome
similarly to measuring temperature.
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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:




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
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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
 Altogether, Shirley gave 6 points, plotted on the following graph.
The smoothed curve fitting through them defines her utility function.
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Shirley’s Utility Function
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Using the Utility Function
 This utility function applies to the Ponderosa
decision.
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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.
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Decision Tree Analysis
with Utilities
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Shape of Utility Curve and
Attitude Toward Risk
 The following shapes generally apply.
 The risk averter has decreasing marginal util- ility 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.
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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.
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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.
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Preferred actions will be found by the staff.
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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
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tree #1
B
1
0
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Tree Settings Dialog Box
(Figure 6-14)
1. Check the
Use Utility
Function box.
2. Select the
type of utility
function in the
Function line.
Here exponential
is chosen.
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.
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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
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25
26
27
28
29
30
31
32
33
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
F
0.0
1
0
-1
-2
Success
Market nationally
FALSE
-$50,000
-531
Failure
Unfavorable
50.0%
$0
80.0%
$0
0.0
-664
-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
TRUE
$0
0
-402
0
Abort
TRUE
$0
1
0
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Utility Functions
R: The risk tolerence
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