1. Chapter 19: Decision Analysis

2. Learning Objectives
LO1 Make decisions under certainty by constructing a decision table.
LO2 Make decisions under uncertainty using the maximax criterion, the maximum criterion, the Hurwicz criterion, and the minimax regret.
LO3 Make decisions under risk by constructing decision trees, calculating expected monetary value and expected value of perfect information, and analyzing utility.
LO4 Revise probabilities in light of sample information by using Bayesian analysis and calculating the expected value of sample information.

3. Decision-Making Scenarios
Decision-making under certainty
Decision-making under uncertainty
Decision-making under risk
LO1

4. Three Variables in Decision Analysis Model
Many decision analysis problems can be viewed as having variables
Decision Alternatives are the various choices or options available to the decision maker in any given problem situation (actions or strategies)
States of nature are the occurrences of nature that can happen after a decision is made that can affect the outcome of the decision and over which the decision maker has little or no control.
States of nature can be environmental, business climate, political, or any condition or state of affairs.
Payoffs are the benefits or rewards (positive or negative) that result from selecting a particular decision alternative. They are often expressed in dollars, but may be stated in other units, such as market share.
LO1

5. Construction of the Decision or Payoff Table
The concepts of decision alternatives, states of nature, and payoffs can be examined jointly by using a decision table or payoff table.
The table a cross tabular table with “states of nature” and “decision alternatives” as classification variables. The associated outcomes in the cells are the payoffs or benefits resulting from making certain choices and a certain state of nature occurring.
Table 19.1 on the next slide illustrates the structure of the problem.
LO1

6. Table 19.1: The Decision or Payoff Table
States of Nature
Decision
Alternatives
where: sj = state of nature
dj = decision alternative
Pi,j = payoff for decision i under state j
LO1

7. Yearly Payoffs on an Investment of $10 000: Description of Problem
An investor is faced with the decision of where and how to invest $10,000 under several possible states of nature
States of Nature
A stagnant economy
A slow-growth economy
A rapid-growth economy
Decision Alternatives being considered
Invest in the stock market
Invest in the Bond market
Invest in GICs
Invest in a mixture of stocks and bonds
The payoffs are presented in Table 19.2 on the next slide.
LO1

8. Table 19.2: Decision Table for an Investor
Stagnant
Slow
Growth
Rapid
Stocks
(500) $
700
2,200
Bonds
(100)
600
900
GICs
300
500
750
Mixture
(200)
650
1,300
Annual payoffs for an investment of $10,000
LO1

9. Rule for Decision Making Under Certainty
In making decisions under certainty the states of nature are known.
The decision maker needs merely to examine the payoffs under different decision alternatives and select the alternative with the highest with the largest payoff
LO1

10. Decision Making Under Certainty
The states of nature are known.
The Greatest Possible Payoff
The economy
will grow
rapidly.
Invest in stocks.
Stagnant
Slow
Growth
Rapid
Stocks
(500) $
700
2,200
Bonds
(100)
600
900
GICs
300
500
750
Mixture
(200)
650
1,300
Annual payoffs for an investment of $10,000
LO1

11. Criteria for Decision Making Under Uncertainty
Maximax payoff: Choose the best of the best (An optimist’s choice)
Maximin payoff: Choose the best of the worst (A pessimist’s choice)
Hurwicz payoff: Use a weighted average of the extremes (optimist and pessimist)
Minimax regret: Minimize the maximum opportunity loss
LO2

12. Maximax Criterion Stagnant Slow Growth Rapid Maximum Stocks Bonds GICs
1. Identify the maximum payoff for each alternative.
2. Choose the alternative with the largest maximum.
Stagnant
Slow
Growth
Rapid
Maximum
Stocks
Bonds
GICs
Mixture
(500) $
700
2,200
(100)
600
900
300
500
750
(200)
650
1,300
LO2

13. Maximin Criterion Slow Rapid Stagnant Growth Growth Minimum Stocks $
1. Identify the minimum payoff for each alternative.
2. Choose the alternative with the largest minimum.
Slow
Rapid
Stagnant
Growth
Growth
Minimum
Stocks $
(500) $
700 $
2,200 $
(500)
Bonds $
(100) $
600 $
900 $
(100)
GICs $
300 $
500 $
750 $
300
Mixture $
(200) $
650 $
1,300 $
(200)
LO2

14. Hurwicz Criterion Stagnant Slow Growth Rapid Maximum Minimum Weighted
1. Identify the maximum payoff for each alternative.
2. Identify the minimum payoff for each alternative.
3. Calculate a weighted average of the maximum and the minimum using 
and (1 - ) for weights.
4. The size of α is between 0 and 1 and will depend on how optimistic or
pessimistic the decision-maker is.
4. Choose the alternative with the largest weighted average.
Stagnant
Slow
Growth
Rapid
Maximum
Minimum
Weighted
Average
Stocks
(500) $
700
2,200
1,390
Bonds
(100)
600
900
GICs
300
500
750
615
Mixture
(200)
650
1,300
850
 =.7
 =.3
LO2

15. Decision Alternatives for Various Values of 
Stocks
Bonds
GICs
Mixture
Max
Min 
1-
2,200
-500
900
-100
750
300
1,300
-200
0.0
1.0
0.1
0.9
-230
345
-50
0.2
0.8 40
100
390
0.3
0.7
310
200
435
250
0.4
0.6
580
480
400
0.5
850
525
550
1120
500
570
700
1390
600
615
1660
660
1000
1930
800
705
1150
2200
1300
LO2

16. Graph of Hurwicz Criterion Selections for Various Values of 
LO2

17. Investment Example: Selected Regrets
I invested in stocks, and
the economy grew slowly.
I have no regrets.
Stagnant
Slow
Growth
Rapid
Stocks
(500) $
700
2,200
Bonds
(100)
600
900
GICs
300
500
750
Mixture
(200)
650
1,300
I invested in stocks.
Then the economy
stagnated. I regret not
investing in GICs. I am
$800 down from where
I could have been.
I invested in GICs.
Then the economy
grew rapidly. I am
out $1,450.
LO2

18. Investment Example: Opportunity Loss Table
Stagnant
Slow
Growth
Rapid
Stocks
800
Bonds
400
100
1,300
GICs
200
1,450
Mixture
500 50
900
LO2

19. Investment Example: Calculating Opportunity Loss
Stagnant
Slow
Growth
Rapid
Stocks
(500) $
700
2,200
Bonds
(100)
600
900
GICs
300
500
750
Mixture
(200)
650
1,300
Payoff Table
800
400
100
200
1,450 50
Opportunity Loss Table
OLi,j = Max(column j) - Pi,j
LO2

20. Minimax Regret Stagnant Slow Growth Rapid Maximum Stocks 800 Bonds 400
1. Identify the maximum regret for each alternative.
2. Choose the alternative with the least maximum regret.
Stagnant
Slow
Growth
Rapid
Maximum
Stocks
800
Bonds
400
100
1,300
GICs
200
1,450
Mixture
500 50
900
LO2

21. Decision Making under Risk
Probabilities of the states of nature have been determined
Decision making under uncertainty: probabilities of the states of nature are unknown
Decision making under risk: probabilities of the states of nature are known (have been estimated)
Decision Trees
Expected Monetary Value of Alternatives
LO3

22. Decision Table with States of Nature Probabilities for Investment Example
LO3

23. Decision Tree for the Investment Example
Stocks
Bonds
GICs
Mixture
Slow growth (.45)
Stagnant (.25)
Rapid Growth (.30)
-$500
$700
$2,200
-$100
$600
$900
$300
$500
$750
-$200
$650
$1,300
Decision
Node
Chance
LO3

24. Expected Monetary Value Criterion
LO3

25. EMV Calculations for the Investment Example
LO3

26. Decision Tree with Expected Monetary Values for the Investment Example
Stocks
Bonds
GICs
Mixture
Slow growth (.45)
Stagnant (.25)
Rapid Growth (.30)
-$500
$700
$2,200
-$100
$600
$900
$300
$500
$750
-$200
$650
$1,300
$850
$515
$525
$623.50
LO3

27. Decision Tree with Expected Monetary Values for the Investment Example
LO3

28. EMV Criterion for the Investment Example
1. Calculate the expected monetary value of each alternative.
2. Choose the alternative with the largest EMV: $850
LO3

29. Definition of Expected Value of Perfect Information
What is the value of knowing which state of nature will occur and when? What is the value of sampling information or undertaking the prediction of an event?
The concept of the expected value of perfect information answers these questions and provide some insight into how much the decision maker should pay for market research.
LO3

30. Definition of Expected Value of Perfect Information
The expected value of perfect information
the difference between the payoff that would occur if the decision maker knew which state of nature would occur and the expected monetary payoff from the best decision alternative when there is no information about the occurrence about the states of nature
Expected Value of Perfect Information = Expected Monetary Payoff with Perfect Information – Expected Monetary Payoff with Information

31. Choice Criterion Under Perfect Information: Choose the Maximum Payoff for any Given State of Nature
LO3

32. Expected Monetary Payoff with Perfect Information for the Investment Example
The investment of stocks was selected under the EMV strategy because it resulted in the maximum payoff of $850. This decision was made with no information about the states of nature (Refer to slide above Perfect Information)
Maximum Payoffs for each state of nature under perfect information: Stagnant Economy = $300; Slow Growth = $700; Rapid Growth = $2,200 (refer to slide above: Perfect Information Criterion )
The expected Monetary Value with perfect information = (300)(0.25) + ($700)(0.45) + ($2,200)(0.30) = $1,050
LO3

33. Expected Value of Perfect Information for the Investment Example
= Expected Monetary Payoff with Perfect Information - Max(EMV[di])
= $ $850
= $200
It would not be economically wise to spend more than $200 to obtain perfect Information about these states of nature. The cost of collecting and processing the information is very high relative to the benefits.
LO3

34. Utility
Utility is the degree of pleasure or displeasure a decision maker has in being involved in the outcome selection process given the risks and opportunities available.
The degree of pleasure will depend on the individual tolerance of risk. An investor may be classified as
Risk-Avoider
Risk-Neutral
Risk-Taker
LO3

35. Measurement of Utility: Standard Gamble Method
A person has the chance to enter a contest with a chance of winning $100,000
If the person wins the contest, he or she wins $100,000.
If the person loses, he or she receives $0.
Cost of entering the game is zero dollars.
The Expected value of the game is :
($100,000)*(.5)+($0)*(.5) = $50,000. But the person betting will not get this unless he or she continues to bet indefinitely on the game.
Would a person take an offer of $30,000 for certain, in the condition that he or she drops out of the game. The answer to this depends on the person’s assets and whether the person is risk neutral, a risk avoider, or a risk taker.
LO3

36. Utility Curves for Three Types of Game Players
The straight line is where the expected value of the game is equal the payment offered to drop out of the game (Risk Neutral) rather than continue the gamble
For the risk avoider the expectation of winning must be higher than the long run probability that makes EMV = the equivalent certainty value: the utility curve is above the Risk Neutral line
The risk taker will bet on the gamble even if the chances of winning is below that required to make EMV = to the equivalent certainty value . The utility curve is below the Risk Neutral line
Chance of
Winning
the Contest
Monetary Payoff
Risk-Avoider
Risk
Neutral
Risk-Taker
LO3

37. Risk Neutral Game Player in a Standard Gamble Game: Indifferent to Owning “a” or “b”
The game player decides to take the $50,000 and not continue gambling. The amount is equal to the expected value of the game at probability of winning =0.5 a b $ - $0 .5
$50 000
LO3

38. Risk Avoider in a Standard Gamble Game: Indifferent to Owning “a” or “b”
Game player decides to take the $20,000 for certain, rather than continue to play, even though the expected value of the game is much higher ($50,000) a b $ - $0 .5
$20 000
LO3

39. Risk Taker in a Standard Gamble Game: Indifferent to Owning “a” or “b”
Game player decides not to take the offer of $70,000 to leave the game, despite the fact that the expected value of the gamble is much less ($50,000). a b $ - $0 .5
$70 000
LO3

40. Risk Curves For Three Game Players
LO3

41. Revising Probabilities in Light of Sample Information
Bayes’ Rule
Expected Value of Sample Information
LO4

42. Interpretation X represents the gamble responses of a risk-avoider
X makes decision based on a utility the segment of parabolic function above the risk-neutral line
Y represents the gamble responses of a risk-taker
Y makes decisions based on an exponential utility function below the risk –neutral.
LO4

43. Interpretation
Let Z represent the responses of a risk-neutral game player
Z is indifferent between a certain guarantee amount, and gambling or not gambling. He remains on the EMV line
The gamble is $10,000. Probability of winning is p= 0.5, EMV = $50,000.
The risk curve shows that for a guarantee of $50,000 to drop gambling in the game , the risk avoider (X) will only gamble if the probability of winning is p=0.8. On the other hand the risk-taker will gamble even if the guarantee is just under$80,000, approximately $30,000 more than the EMV at p= 0.5.
LO4

44. Decision Table for Investment ProblemNo
Growth
(.65)
Rapid
(.35)
Bonds
500 $
100
Stocks
(200)
1,100
LO4

45. Expected Monetary Value Criterion for the Investment ExampleNo
Growth
Rapid
Expected
Monetary
Value
0.65
0.35
Bonds
500 $
100
360.00
Stocks
(200)
1,100
255.00
LO4

46. Revising Probabilities in the Light of Sample Information
In this section we address the revision of prior probabilities using Bayes’ rule with sampling information in the context of the $10,000 case discussed above.
The probabilities of the various states of nature are frequently not fixed or known in an exact way. Thus prior subjective probabilities (or probabilities based on our best guess) may be used initially to obtain the EMV. These probabilities can be updated by introducing information obtained from samples. The updated probabilities can be incorporated into the decision process to hopefully help make better decisions.
LO4

47. Simplified Version of the $10,000 Investment Decision Problem: Table 19.6
LO4

48. Decision Tree for the Investment Example: Figure 19.5
Stocks
Bonds
No Growth (.65)
Rapid Growth (.35)
$500
$100
-$200
$1,100
EMV=$360
EMV=$255
($360)
LO4

49. The Success and Failure Rates of the Forecaster in Forecasting the Two States of the Economy
Actual State of Economy
No Growth (s 1 )
Rapid Growth 2
Forecaster Predicts
No Growth (F )
.80
.30
Rapid Growth (F2 )
.20
.70
P(Fi|sj)
LO4

50. Bayes’ Rule
LO4

51. Revision Based on a Forecast of No Growth (F1)
State of
Economy
Prior
Probabilities
Conditional
Joint
Revised No
Growth (s 1 )
P(s
) = .65
P(F
| s
) = .80  s
) = .520
.520/.625 = .832
Rapid 2
) =.35
) = .30
) = .105
.105/.625 = .168
P(F1) = .625
P(sj|F1)
LO4

52. Revision Based on a Forecast of Rapid Growth (F2)
State of
Economy
Prior
Probabilities
Conditional
Joint
Revised No
Growth (s 1 )
P(s
) = .65
P(F 2
| s
) = .20  s
) = .130
.130/.375 = .347
Rapid
) =.35
) = .70
) = .245
.245/.375 = .653
P(F2) = .375
P(sj|F2)
LO4

53. Decision Tree for the Investment Example After Revision of Probabilities: Figure 19.6
Stocks
Bonds
No Growth (.832)
Rapid Growth (.168)
$500
$100
-$200
$1,100
$432.80
$18.40
No Growth (.347)
Rapid Growth (.653)
$238.80
$648.90
Forecast
No Growth
(.625)
Rapid Growth
(.375)
$513.84
Buy
LO4

54. Expected Value of Sample Information for the Investment Example
In general, the expected value of sample information
= expected monetary value with information
- expected monetary value without information
= $ $360
= $153.84
But what if the decision maker had to pay $100 for the
forecaster’s prediction?
This would reduce the value of getting perfect information from
$ shown in Figure 19.6 in the previous slide to $
Note that this is still superior to the $360 without sample information
LO4

55. Decision Tree Investment Example
All Options Included
Figure 19.7 is constructed by combining Figures 19.5 and 19.6.
This is the Investment Tree for the investment information with the options of buying the information or not buying the information included .
It includes a cost of buying Information ($100) and the EMV with this purchased information ($413.84)
LO4

56. COPYRIGHT
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