This document discusses decision analysis and different methods for making decisions under uncertainty. It introduces decision analysis concepts including alternative actions, states of nature, and outcomes. It then covers three decision making conditions: certainty, ignorance, and risk. Under risk, it discusses expected return/value and how to calculate it using probabilities and payoffs. It also introduces the concepts of maximax, maximin, expected value of perfect information, and utility functions to incorporate risk attitudes into decisions.

1. Chapter 8
Decision Analysis
Slides 8a: Introduction

2. Decision Analysis
 A set of alternative actions
 We may chose whichever we please
 A set of possible states of nature
 Only one will be correct, but we don’t know in
advance
 A set of outcomes and a value for each
 Each is a combination of an alternative action and a
state of nature
 Value can be monetary or otherwise

3. Decision Analysis
 Certainty
 Decision Maker knows with certainty what the state of
nature will be - only one possible state of nature
 Ignorance
 Decision Maker knows all possible states of nature,
but does not know probability of occurrence
 Risk
 Decision Maker knows all possible states of nature,
and can assign probability of occurrence for each
state

4. Decision Making Under Certainty
Decision Variable
Units to build 150
Parameter Estimates
Cost to build (/unit) 6,000
$
Revenue (/unit) 14,000
$
Demand (units) 250
Consequence Variables
Total Revenue 2,100,000
$
Total Cost 900,000
$
Performance Measure
Net Revenue 1,200,000
$

5. Decision Making Under Ignorance
– Payoff Table
Kelly Construction Payoff Table (Prob. 8-17)
Low (50 units) Medium (100 units) High (150 units)
Build 50 400,000 400,000 400,000
Build 100 100,000 800,000 800,000
Build 150 (200,000) 500,000 1,200,000
State of Nature
Demand
Alternative
Actions

6. Decision Making Under Ignorance
 Maximax
Select the strategy with the highest possible
return
 Maximin
Select the strategy with the smallest possible
loss
 LaPlace-Bayes
All states of nature are equally likely to occur.
Select alternative with best average payoff

7. Maximax:
The Optimistic Point of View
 Select the “best of the best” strategy
 Evaluates each decision by the maximum possible
return associated with that decision (Note: if cost data
is used, the minimum return is “best”)
 The decision that yields the maximum of these
maximum returns (maximax) is then selected
 For “risk takers”
 Doesn’t consider the “down side” risk
 Ignores the possible losses from the selected
alternative

8. Maximax Example
Low (50 units) Medium (100 units) High (150 units) Max
Build 50 400,000 400,000 400,000 400,000
Build 100 100,000 800,000 800,000 800,000
Build 150 (200,000) 500,000 1,200,000 1,200,000
State of Nature
Maximax
Criterion
Demand
Alternative
Actions
Kelly Construction

9. Maximin:
The Pessimistic Point of View
 Select the “best of the worst” strategy
Evaluates each decision by the minimum
possible return associated with the decision
The decision that yields the maximum value
of the minimum returns (maximin) is selected
 For “risk averse” decision makers
A “protect” strategy
Worst case scenario the focus

10. Maximin
Low (50 units) Medium (100 units) High (150 units) Min
Build 50 400,000 400,000 400,000 400,000
Build 100 100,000 800,000 800,000 100,000
Build 150 (200,000) 500,000 1,200,000 (200,000)
State of Nature
Maximin
Criterion
Demand
Alternative
Actions
Kelly Construction

11. Decision Making Under Risk
 Expected Return (ER)*
 Select the alternative with the highest (long term)
expected return
 A weighted average of the possible returns for
each alternative, with the probabilities used as
weights
* Also referred to as Expected Value (EV) or Expected
Monetary Value (EMV)
**Note that this amount will not be obtained in the short
term, or if the decision is a one-time event!

12. Expected Return
Low (50 units) Medium (100 units) High (150 units) ER
Build 50 400,000 400,000 400,000 400,000
Build 100 100,000 800,000 800,000 660,000
Build 150 (200,000) 500,000 1,200,000 570,000
Probability 0.2 0.5 0.3 1.0
State of Nature
Expected
Return
Demand
Alternative
Actions

13. Expected Value of Perfect Information
 EVPI measures how much better you could do on
this decision if you could always know when each
state of nature would occur, where:
 EVUPI = Expected Value Under Perfect Information
(also called EVwPI, the EV with perfect information, or
EVC, the EV “under certainty”)
 EVUII = Expected Value of the best action with
imperfect information (also called EVBest )
 EVPI = EVUPI – EVUII
 EVPI tells you how much you are willing to pay for
perfect information (or is the upper limit for what you
would pay for additional “imperfect” information!)

14. Expected Value of Perfect
Information
Low (50 units) Medium (100 units) High (150 units) ER
Build 50 400,000 400,000 400,000 400,000
Build 100 100,000 800,000 800,000 660,000
Build 150 (200,000) 500,000 1,200,000 570,000
Probability 0.2 0.5 0.3 1.0
Best Decision 400,000 800,000 1,200,000 840,000
EVPI 180,000
State of Nature
Expected
Return
Demand
Alternative
Actions

15. Using Excel to Calculate EVPI:
Formulas View
A B C D E
1
2
3 Payoffs States of Nature Expected Return
4 Alternatives Low (50 units) Medium (100 units) High (150 units) ER
5 Build 50 400000 400000 400000 =SUMPRODUCT(B5:D5,B$8:D$8)
6 Build 100 100000 800000 800000 =SUMPRODUCT(B6:D6,B$8:D$8)
7 Build 150 -200000 500000 1200000 =SUMPRODUCT(B7:D7,B$8:D$8)
8 Probability 0.2 0.5 0.3
9 Best Decision =MAX(B5:B7) =MAX(C5:C7) =MAX(D5:D7)
10
11 EVwPI = =SUMPRODUCT(B9:D9,B8:D8)
12 EVBest = =MAX(E5:E7)
13 EVPI = =E11-E12
14
Kelly Construction

16. A newsvendor can buy the Wall Street Journal
newspapers for 40 cents each and sell them for 75
cents.
However, he must buy the papers before he knows
how many he can actually sell. If he buys more
papers than he can sell, he disposes of the excess at
no additional cost. If he does not buy enough
papers, he loses potential sales now and possibly in
the future.
Suppose that the loss of future sales is captured by a
loss of goodwill cost of 50 cents per unsatisfied
customer.
The Newsvendor Model

17. The demand distribution is as follows:
P0 = Prob{demand = 0} = 0.1
P1 = Prob{demand = 1} = 0.3
P2 = Prob{demand = 2} = 0.4
P3 = Prob{demand = 3} = 0.2
Each of these four values represent the states of
nature. The number of papers ordered is the decision.
The returns or payoffs are as follows:

18. State of Nature (Demand)
0 1 2 3
Decision
0 0 -50 -100 -150
1 -40 35 -15 -65
2 -80 -5 70 20
3 -120 -45 30 105
Payoff = 75(# papers sold) –
40(# papers ordered) – 50(unmet demand)
Where 75¢ = selling price
40¢ = cost of buying a paper
50¢ = cost of loss of goodwill

19. Now, the ER is calculated for each decision:
State of Nature (Demand)
0 1 2 3
Decision
0 0 -50 -100 -150 -85
1 -40 35 -15 -65 -12.5
2 -80 -5 70 20 22.5
3 -120 -45 30 105 7.5
ER
Prob. 0.1 0.3 0.4 0.2
ER1 = -40(0.1) + 35(0.3) – 15(0.4) – 65(0.2) = -12.5
ER2 = -80(0.1) – 5(0.3) + 70(0.4) + 20(0.2) = 22.5
ER3 = -120(0.1) – 45(0.3) + 30(0.4) – 105(0.2) = 7.5
ER0 = 0(0.1) – 50(0.3) – 100(0.4) – 150(0.2) = -85
Of these four ER’s,
choose the maximum,
and order 2 papers

20. ER(new) = 0(0.1) + 35(0.3) + 70(0.4) + 105(0.2)
State of Nature
0 1 2 3
Decision
0 0 -50 -100 -150
1 -40 35 -15 -65
2 -80 -5 70 20
3 -120 -45 30 105
Prob. 0.1 0.3 0.4 0.2
= 59.5
ER(current) = 22.5
EVPI = 59.5 – 22.5 = 37.0 cents

21. The decision that yields the maximum of these maximum
returns (maximax) is then selected.
This method evaluates each decision by the maximum
possible return associated with that decision.
Maximax Criterion: The Maximax criterion is an
optimistic decision making criterion.

22. Then, the decision that yields the maximum value of the
minimum returns (maximin) is selected.
Maximin Criterion: The Maximin criterion is an
extremely conservative, or pessimistic, approach to
making decisions.
Maximin evaluates each decision by the minimum possible
return associated with the decision.

23. So, using the 3 criteria, we made the following
decisions regarding the newsvendor data:
Criteria Decision
Maximin Cash Flow Order 1 paper
Expected Return Order 2 papers
Maximax Cash Flow Order 3 papers

24. Most people are risk-averse, which means they
would feel that the loss of a certain amount of
money would be more painful than the gain of
the same amount of money.
Utility functions in decision analysis measure the
“attractiveness” of money.
Utility can be thought of as a measure of
“satisfaction.”
THE RATIONALE FOR UTILITY

25. Utility
1.0
0.910
0.850
0.775
0.680
0.524
100 200 300 400 500 600 Dollars
Typical risk-averse utility function:
Go from $400 to
$500 results in
A gain
in
utility
of 0.06

26. To illustrate, first suppose you have $100 and someone
gives you an additional $100. Note that your utility
increases by
U(200) – U(100) = 0.680 – 0.524 = 0.156
Now suppose you start with $400 and someone gives you
an additional $100. Now your utility increases by
U(500) – U(400) = 0.910 – 0.850 = 0.060
This illustrates that an additional $100 is less attractive if
you have $400 on hand than it is if you start with $100.

27. Utilities and Decisions under Risk
Summary:
Utility is a way to incorporate risk aversion into the
expected return calculation.
Calculating a utility function is out of the scope of
this course, but it can be calculated by a series of
lottery questions (e.g., Would you prefer one million
dollars or a 50% chance of earning five million?).