BU.520.601
Decision Models
Decision Analysis
DecisionAnalysis
Summer 2013
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 Let us flip a fair coin once (there is a fee).
If you win I give you $102. If I win, you give me $100.
How much fee will you pay me for playing the game: $5, $2,
$1, $0? You can select any other amount.
 Let us flip a fair coin 1000 (there is a fee).
If you win a toss, I give you $102. If I win, you give me $100.
How much fee will you pay for the playing the entire game?
Suppose you are getting ready to go the office in a crowded metro.
Carrying an umbrella is a hassle; you will carry it only when you feel
necessary. Forecast for today is 70% chance of rain and the sky is
overcast. Should you carry an umbrella - Yes or no?
My decision would be “Yes” and it is a good decision.
However, there are two possible outcomes - it will rain or not.
If it does not rain, it does not mean I have made a bad decision.
DecisionAnalysis
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Decision Analysis (DA)
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DA is a methodology applicable to analyze a wide variety of
problems.
Although DA was used in the 1950s (at Du Pont) and early
1960s (at Pillsbury), major DA development took place in mid
sixties. One of the earliest application (at GE) was to analyze
whether a super heater should be added to the current power
reactor.
DA has been considered as a technology to assist
(individuals and) organizations in decision making by
quantifying the considerations (even though they may be
subjective) to deduce logical actions.
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Decision Analysis (DA)
One can discuss many topics listed below; we will look at a few.
• Problem Formulation.
• Decision Making with / without Probabilities.
• Risk Analysis and Sensitivity Analysis.
• Decision Analysis with Sample / Perfect Information.
• Multistage decision making.
Tools and terminology
• Basic statistics and probability
• Influence diagram / payoff table /
decision tree
• EMV: Expected Monetary Value
• EVSI / EVPI : Expected Value of
Sample / Perfect Information
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Bayes’ rule
Decision vs. outcome
Risk management
Minimax / maximin /
Utility theory
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Decision analysis without probabilities
Concepts covered: Payoff table.
Different approaches: Maximax, maximin, minimax regret
Example: There are four projects; I can select only one. The
payoff table shows potential “payoff” depending upon likely
economic conditions.
Alternatives
Economic Condition
Recession
Normal
Boom
Project A
4075
5000
6100
Project B
0
5250
12080
Project C
2500
7000
10375
Project D
1500
6000
9500
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Since the payoff in
project C is higher than
the payoff for D for
every economic
condition, we say that
project C is dominant.
We can eliminate
project D from
consideration.
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Maximax
If you are an optimist, you will decide on the basis of Maximax.
Alternatives
Economic Condition
Recession Normal Boom
Project A
4075
Project B
0
5250 12080
Project C
2500
7000 10375
5000
Step 1: Pick the max value
for each alternative.
6100
6100
12080
10375
Step 2:Then pick the
alternative with max payoff.
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Maximin
If you are a conservative you will use Maximin.
Alternatives
Economic Condition
Recession
Normal
Boom
1: Pick the min value
for each alternative.
Project A
4075
5000
6100
4075
Project B
0
5250
12080
0
Project C
2500
7000
10375
2500
2: Then pick the alternative
with max payoff.
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Minimax Regret You are neither optimist nor conservative.
Alternatives
Economic Condition
Recession
Normal
Boom
Project A
4075
5000
6100
Project B
0
5250
12080
Project C
2500
7000
10375
4075|
7000|
12080
Alternatives
Regret Table
Recession
Normal
Boom
Project A
0
2000
5980
Project B
4075
1750
0
Project C
1575
0
1705
Step 1: Calculate
the maximum for
each outcome.
Stet 2: Prepare
“Regret Table” by
subtracting each
outcome cell
value from its
maximum.
At least one number for each regret table outcome is zero and there
are no negative numbers. Why?
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Minimax Regret..
Alternatives
Economic Condition
Recession Normal
Boom
Project A
4075
5000
6100
Project B
0
5250
12080
Project C
2500
7000
10375
4075|
7000|
12080
Alternatives
Regret Table
Recession Normal
Step 3: Pick the max value
Boom for each alternative.
Project A
0
2000
5980
5980
Project B
4075
1750
0
4075
Project C
1575
0
1705
1705
Step 4: Pick the alternative
with minimum regret.
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General comments Payoff table
Alternatives
Table columns show
outcomes (also called
state of nature).
Economic Condition
Recession Normal Boom
Project A
4075
Project B
0
5250 12080
Project C
2500
7000 10375
5000
6100
• The maximax payoff criterion seeks the largest of the
maximum payoffs among the actions.
• The maximin payoff criterion seeks the largest of the minimum
payoffs among the actions.
• The minimax regret criterion seeks the smallest of the
maximum regrets among the actions.
The above three approaches we used involved Decision
Making without Probabilities.
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Decision analysis with probabilities
Typically, we use a tree diagram for the decision analysis.
1. A decision point is shown by a rectangle
2. Alternatives available at a decision point
DB
are shown as decision branches (DB).
3. At the end of each DB, there
CB
can be two or more chance
20%
events shown by a node and
55%
chance branches (CB).
Decision
25%
Chance events must be mutually
point
exclusive and exhaustive (total
probability = 1).
4. At the end of each branch is an endpoint shown as a triangle
where a payoff will be identified.
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Decision analysis with probabilities
Decision point:
Chance event :
End point:
DB: Decision Branch
CB: Chance Branch
At the chance node, we calculate the average
(i.e. expected) payoff. The terminology used is
DB
Expected Monetary Value (EMV)
If there is no chance event for a
CB
particular decision branch, it’s
20%
EMV is equal to the payoff.
55%
25%
We select the decision with the
highest EMV .
What if we are dealing with costs?
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A larger tree diagram
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Example 1 You bought 500 units of X @$10 each.
A dealer has offered to buy these from you @$14 each ( you can
make $4/unit profit).
You can sell these yourself for $16
Demand: X 300 400 500 600
each ($6/unit profit) but the
0.30 0.45 0.20 0.05
demand is uncertain. The demand Pr(X)
distribution is shown in the table.
Obviously, if demand exceeds 500, you will sell all 500. On the
other hand, if demand is under 500, you will have leftover units.
These leftover items can disposed off for $7 each ($3 loss, the
dealer will no longer buy these leftover units from you).
What’s your decision?
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Example 1 ..
Suppose you have 500 units of X in
Demand: X 300 400 500 600 stock, purchased for $10 each. Dealer
Pr(X)
0.30 0.45 0.20 0.05 sales price:$14, self sale price:$16
with salvage value:$7.
Start with the tree having 2 branches (DB) at the decision point.
There are no chance events in the dealer sale branch,
For the self sale, there are 4
mutually exclusive possibilities.
Dealer
Sale
Self sale
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500, 20%
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Example 1 ...
Suppose you have 500 units of X in
Demand: X 300 400 500 600 stock, purchased for $10 each. Dealer
Pr(X)
0.30 0.45 0.20 0.05 sales price:$14, self sale price:$16
with salvage value:$7.
EMV = 2000
Payoff = 500*4 = 2000
Dealer
Sale
Payoff = 300*6 – 200*3 = 1200
Payoff = 400*6 – 100*3 = 2100
Self sale
500, 20%
Payoff = 500*6 = 3000
Payoff = 500*6 = 3000
EMV = 0.3*1200 +
0.45*2100 + 0.2* 3000
+ 0.05*3000 = 2055
DecisionAnalysis
Your decision?
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Risk profile is the probability distribution for
the payoff associated with a particular action.
Risk Profile
30% 300
Self Sale
Payoff = 1200
400
45%
Payoff = 2100
20%
500
5%
600
Payoff = 3000
Payoff = 3000
The risk profile shows all the possible economic outcomes and
provides the probability of each: it is a probability distribution for
the principal output of the model.
DecisionAnalysis
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Example 3
We have received RFP (Request For Proposal).
• We may not want to bid at all (our cost: 0)
• If we bid, we will have to spend $5k for proposal preparation.
Based on the information provided in the RFP, a quick decision
is to bid either $115k or $120k or $125k.
We must select among 4 alternatives (including no bid).
• A quick estimate of the cost of the project (in addition to the
preparation cost) is $95k.
• Looks like we may have a competitor.
• If we bid the same amount as the competitor, we will get the
project because of our reputation with the client.
• We have gathered some probabilities based on past
experience.
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Example 3..
All numbers in thousand dollars
Our bid (OB) must be 0 (no bid), 115, 120 or 125.
Competitor’s bid (CB): 0, under 115, 115 to under 120, 120 to under
125, 125 and over.
Assumption: If bids are equal, we get the contract.
Information : Preparation cost: $5 + Cost of work : $95 = $100 total
Profit for our bid
 Competitor’s bid
1. No bid
2a. Under $115
2b. $115 to under $120
2c. $120 to under $125
2d. Over $125
DecisionAnalysis
0 115 120 125
0 15 20 25
0 -5
-5
-5
0 15
-5
-5
0 15 20
-5
0 15 20 25
BU.520.601
Use mini-max,
maxi-max, etc?
There are
probabilities
involved.
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Example 3…
1. There is a 30% probability that the competitor will not bid.
2. If the competitor does bid, there is
(a) 20% probability of bid under $115.
(b) 40% probability of bid $115 to under $120.
(c) 30% probability of bid under $120 to under $125.
(d) 10% probability of bid over $125.
Actual
Prob. Prob. Prob.  Competitor’s bid
30% 30%
- 1. No bid
14%
20% 2a. Under $115
28%
40% 2b. $115 to under $120
70%
21%
30% 2c. $120 to under $125
7%
10% 2d. Over $125
DecisionAnalysis
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Profit for our bid
0 115 120 125
0
15
20
25
0
-5
-5
-5
0
15
-5
-5
0
15
20
-5
0
15
20
25
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Example 3:
Actual
Prob.  Competitor
No
bid
$115
0 115 120 125
30% 1. No bid
14% 2a. < $115
0
15
20
25
0
-5
-5
-5
28% 2b. $115 to < $120
21% 2c. $120 to < $125
0
15
-5
-5
0
15
20
-5
0
15
20
25
7% 2d. > $125
$0
Profit for our bid
Lose
Payoff = (-5), Probability 14%
Win
Payoff = 15, Probability 86%
bid
(-5)*(0.14) + 15 * (0.86) = $12.2
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Example 3:
No
bid
$12.2
Bid
$115
0
15
20
25
0
-5
-5
-5
28% 2b. $115 to < $120
21% 2c. $120 to < $125
0
15
-5
-5
0
15
20
-5
0
15
20
25
L
-5, 14%
W
15, 86%
Our decision?
$9.5
Bid
$120
Bid=
$125 $6.1
DecisionAnalysis
0 115 120 125
30% 1. No bid
14% 2a. < $115
7% 2d. > $125
$0
Profit for our bid
Actual
Prob.  Competitor
L
-5, 42%
W
20, 58%
L
W
-5, 63%
We will now use
Excel to solve the
problem.
25, 37%
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Ex. 3: Excel
=SUMPRODUCT(Profit_bid_115,Probabilities)
=MAX(D9:G9)
INDEX+MATCH
HLOOKUP ?
Value we are looking (12.2) is not in the ascending order in the table.
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Example 3: Sensitivity analysis
What if 30% probability of no bid from competitor is incorrect?
We can build a one variable data table.
Variable: Competitor’s no bid probability.
We select two outputs: bid and
(corresponding maximum) profit.
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Ex. 3: DA and value of information
Our decision was to bid $115 and EMV was $12.2. Suppose we
get competitor’s bid information. Can we improve our profit?
Profit for our bid
 Competitor’s bid
0 115 120 125
1. No bid
0
15
20
25
2a. Under $115
0
-5
-5
-5
2b. $115 to under $120
0
15
-5
-5
2c. $120 to under $125
0
15
20
-5
2d. Over $125
0
15
20
25
What is the probability?
0.30
0.7 * 0.2 = 0.14
0.7 * 0.4 = 0.28
0.7 * 0.3 = 0.21
0.7 * 0.1 = 0.07
EMV = 0.3*25+0.14*0+0.28*15+0.21*20+0.07*25 = 17.65
Earned Value of Perfect Information (EVPI) = $17.65 – $12.2 = $5.45
Sometimes we may have partial information.
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Example 3: Alternate method
CB=0
15(.3)+11(.7) = $12.2
$0
No
bid
CB
OB=
$115
OB=
$120
70%
$9.5
OB=
$125
$20
$5
-5(.2)+15(.4+.3+.1) = $11
30%
$6.1
DecisionAnalysis
30%
20%
<115 -$5
115 to <120 $15
40%
30% 120 to < 125
$15
10%
>125 $15
70%
bid
Our $12.2
decision
30% $15
70%
$25
-$2
EMV
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Payoff
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Example 3…..
$12.2
$0
No
bid
OB=
$115
This line indicates the decision
made.
$9.5
OB=
$120
bid
Values 12.2, 9.5 and 6.1
represent Expected Monetary
Values (EMV).
This is called folding back the
decision tree.
$12.2
OB=
$125
$6.1
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Utility theory
Consider the gambling
problems again.
– Let us flip a fair coin once.
– If you win I give you $102
– If I win, you give me $100
– How much will you pay me
to play this game: $5, $2,
$1, $0 ?
Consider another gamble
– Let us flip the same coin
(500 times) with the same
payoffs
– How much will you pay me
to play this game?
DecisionAnalysis
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Different people will pay different
amounts to play the first game
Expected payoff in the first game is $1
but most people do not want to play the
game at all.
Why? Losing $100 is a bigger event
than winning $102
Most people will play the second game.
Still differ in how much they will pay.
For most people a gain that is twice as
big is not twice as good.
A loss of twice as much is more than
twice as bad.
People’s attitude towards risk can be
categorized as: risk averse, risk seeker
and risk neutral.
A common way to express it is through
the decision-maker’s utility function.
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Utility is a measure of relative satisfaction. We can plot a graph of amount
of money spent vs. “utility” on a 0 to 100 scale. Typical shapes for different
types of risk takers generally follow the patterns shown below.
U(100)
U(100)
U(0)
U(0)
0
100
Risk seeker
U(100)
0
100
Risk averse
U(0)
0
100
Risk neutral
U(100)
U(100)
U(100)
U(0)
U(0)
U(0)
0
100
0
100
0
100
Graphs above show that to achieve 50% utility, risk seekers will pay maximum,
risk averse will pay minimum and risk neutral will pay an average amount.
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