1. INFO 631 Prof. Glenn Booker
Week 9 – Chapters 24-26
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2. Decisions Under Risk
Ch. 24
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3. Decisions Under Risk Outline
Introducing decisions under risk
Different techniques
Expected value decision making
Expectation variance
Monte Carlo analysis
Decision trees
Expected value of perfect information
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4. Decisions Under Risk
When you know the probabilities of the different outcomes and will incorporate them
Expected value decision making
Expectation variance
Monte Carlo analysis
Decision trees
Expected value of perfect information
When you don’t know (or can’t say) the probabilities, use Decisions Under Uncertainty (next chapter)
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5. Expected Value Decision Making
The value of an alternative with multiple outcomes can be thought of as the average of the random individual outcomes that would occur if that alternative were repeated a large number of times
Can use PW(i), FW(i), or AE(i)
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6. Expected Value of a Single Alternative
Denali project at Mountain Systems
Imagine 1000 parallel universes where the Denali project could be run at the same time
Should expect most favorable outcome would happen in 15% or 150 of those universes
Fair outcome would happen in 650
Least favorable outcome would happen in 200
Least Most
favorable Fair favorable
PW(MARR) $ $ $9012
Probability
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7. Expected Value of a Single Alternative
Total PW(i) income generated
Average PW(i) income in each universe
Notice
200 * -$1234 = ,800
650 * $5678 = $3,690,700
150 * $9012 = $1,351,800
$4,795,700
$4,795,700 / 1000 = $
(0.20 * -$1234) + (0.65 * $5678) + (0.15 * $9012) = $
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8. Expected Value of a Single Alternative
General formula
Can be used to help decide between multiple alternatives
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9. Expected Value of Multiple Alternatives
Same probability
Several projects at Mountain Systems
Expected values
Choose Shasta, it has the highest expected value
Least Most
favorable Fair favorable
Alternative % % %
Denali $ $ $9012
Shasta
Washington
The concept can be extended to help when choosing from among several alternatives
Denali (0.20 * -$1234) + (0.65 * $5678) + (0.15 * $9012) = $
Shasta (0.20 * -$1201) + (0.65 * $6601) + (0.15 * $9282) = $
Washington (0.20 * -$3724) + (0.65 * $4104) + (0.15 * $9804) = $
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10. Expectation Variance
What if probabilities were different for each alternative?
Comparing projects
Lassen has higher expected value but win big-lose big
Moana Loa has lower expected value but more probability of profit
Outcome Probability AE(i)
Least favorable % $3494
Nominal %
Most favorable %
Expected value = $665
Least favorable % $200
Low nominal %
High nominal %
Most favorable %
Expected value = $466
Lassen
Moana Loa
Depending on how risk-averse you are. Lower risk tolerance would take Moana Loa project (better choice in lean times?), higher risk tolerance would likely take Lassen (better choice in boom times?)
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11. Monte Carlo Analysis
Randomly generate combinations of input values and look at distribution of outcomes
Named after gambling resort in Monaco
Use [a variant of] Zymurgenics project (different data)
Least favorable Fair Most favorable
estimate estimate estimate
Initial investment $500, $400, $360,000
Operating & maintenance $ $ $800
Development staff cost / month $49, $35, $24,500
Development project duration months months months
Income / month $24, $40, $56,000
Zymurgenics project was in last chapter but data here is actually wrong data from book. Need to rerun simulation…
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12. Monte Carlo Analysis Simulation run results
Income range Number of occurrences
-$75,000 to -$50,
-$50,000 to -$25,
-$25,000 to -$
$0 to $24,
$25,000 to $49,
$50,000 to $74,
$75,000 to $99,
$100,000 to $124,
$125,000 to $149,
$150,000 to $174,
$175,000 to $199,
$200,000 to $224,
$225,000 to $249,
$250,000 to $274,
Zymurgenics project was in last chapter but data here is actually wrong data from book. Need to rerun simulation…
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13. Monte Carlo Analysis INFO631 Week 9
Zymurgenics project was in last chapter but data here is actually wrong data from book. Need to rerun simulation…
Point out EV, probability of – outcome, …
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14. Decision Trees
Maps out possible results when there are sequences of decisions and future random events
Useful when decisions can be made in stages
Basic Elements
Decision nodes – points in time where a decision maker makes a decision (square)
Chance nodes – points in time where the outcome is outside the control of the decision maker (circles)
Node sequencing
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15. Sample Decision Tree
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16. Decision Tree Analysis, Part 1
Add the financial consequences for each arc (PW(i), FW(i), or AE(i))
Properly adjust for time periods as required
Sum financial consequences from the root node to all leaf nodes
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17. Sample Decision Tree
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18. Decision Tree Analysis, Part 2
Write probabilities for each arc out of each chance node
Probabilities out of a chance node must = 1.0
Roll back values from leaf nodes to root
If node is chance node, calculate expected value at that node based on values on all nodes to its right
If node is decision node, select the maximum profit (or minimum cost) from nodes to its right
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19. Sample Decision Tree
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20. Expected Value of Perfect Information
Value at root node is expected value of decision tree based on current information
Current information is known to be imperfect
Reasonable follow-on question
Research, experimentation, prototyping, …
Might even be able to eliminate one or more paths through the tree because you may discover them to be impossible
Analyzed decision tree provides information that will help answer that question
“Would there be any value in taking actions that would reduce
the probability of ending up in an undesirable future state?”
Just introducing topic here—need to see the details in the book…
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21. Expected Value of Perfect Information
If we had a crystal ball and knew outcomes for chance nodes, we could find which path would be best
Finding best path can be repeated for all possible combinations of random variables
Probabilities for random variables are known
Can calculate probability for each combination of outcomes
For each combination of outcomes, multiply its best value by probability of that combination
Sum the results of (value * probability) for all combinations of outcomes
Sum is expected value given perfect information
Difference between sum and expected value given current information is expected value of perfect information
Just introducing topic here—need to see the details in the book…
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22. Expected Value of Perfect Information
EVPI is upper limit on how much to spend to gain further knowledge
Probably impossible to actually get perfect information, organization should plan on spending less
Just introducing topic here—need to see the details in the book…
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23. Key Points
Value of an alternative with multiple outcomes is the average of the random individual outcomes that would occur if that alternative were repeated a large number of times (expected value)
The alternative with the highest expected value is best
With expectation variance, differing probabilities could influence the decision
Alternative with lower expected value might be a better choice if it also has a much lower probability of a negative outcome
Monte Carlo analysis generates random combinations of the input variables and calculates results under those conditions
Repeated many times and statistical distribution of outcomes is analyzed
Decision trees map out possible results when there are sequences of decisions together with a set of future random events that have known probabilities
Useful with many possible future states and decisions can be made in stages
The Expected value of perfect information provides answer to, “Would there be any value in taking actions that would reduce the probability of ending up in an undesirable future state?”
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24. Decisions Under Uncertainty
Ch. 25
Slides adapted from Steve Tockey – Return on Software
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25. Decisions Under Uncertainty Outline
Introducing decisions under uncertainty
Different Techniques
Payoff matrix
Laplace Rule
Maximin Rule
Maximax Rule
Hurwicz Rule
Minimax Regret Rule
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26. Decisions Under Uncertainty
Used when impossible to assign probabilities to outcomes
Can also be used when you don’t want to put probabilities on outcomes, e.g., safety-critical software system where a failure could threaten human life
People may not react well to an assigned probability of fatality
If probabilities can be assigned, Decision Making under Risk should be used
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27. Payoff Matrix Shows all possible outcomes to consider
One axis lists mutually exclusive alternatives
Other axis lists different states of nature
Each state of nature is a future outcome the decision maker doesn’t have control over
Cells have PW(i), FW(i), AE(i), …
Alternative State1 State2 State3 A A A A A
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28. Reduced Payoff Matrix One alternative may be “dominated” by another
Another alternative has equal or better payoff under every state of nature
Reduced payoff matrix has no dominated alternatives
Less work if dominated alternatives are removed
Alternative State1 State2 State3 A A A A A
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29. Laplace Rule Assumes each state of nature is equally likely
Sometimes called “principle of insufficient reason”
Calculate average payoff for each alternative across all states of nature
Same as expected value analysis for multiple alternatives with equal probabilities
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30. Laplace Rule
Example
Alternative A4 is chosen; the highest payoff always wins!
Alternative State1 State2 State Average payoff A A A A
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31. Maximin Rule Assumes worst state of nature will happen Formula
Most pessimistic technique
Pick alternative that has best payoff from all worst payoffs
Formula
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32. Maximin Rule Example Alternative A5 is chosen
Alternative State1 State2 State Worst payoff A A A A
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33. Maximax Rule Assumes best state of nature will happen Formula
Most optimistic technique
Pick alternative that has best payoff from all best payoffs
Formula
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34. Maximax Rule Example Alternative A3 is chosen
Alternative State1 State2 State Best payoff A A A A
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35. Hurwicz Rule
Assumes that without guidance people will tend to focus on extremes
Blends optimism and pessimism using a selected ratio
Index of optimism, a, between 0 and 1
a = 0.2 means more pessimism than optimism
a = 0.1 means more pessimism than a = 0.2
a = 0.85 means lots of optimism but a small amount of pessimism (15%) remains
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36. Hurwicz Rule Formula Example a = 0.2 Alternative A2 is chosen
Alternative State1 State2 State Blended payoff
A (0.2 * 4021) + (0.8 * 948) = 1563
A (0.2 * 6004) + (0.8 * -2005) = -403
A (0.2 * 5104) + (0.8 * 0) = 1021
A (0.2 * 3014) + (0.8 * 1005) = 1407
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37. Hurwicz Rule .25 A3 6000 A4 A2 4000 A5 2000 0.5 -2000 INFO631 Week 9
-2000
0.5 A2 A3 A4 A5
.25
Blended payoffs for each alternative as a function of alpha
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38. Minimax Regret Rule
Minimize regret you would have if you chose wrong alternative under each state of nature
If you selected A1 and state of nature happened where A1 had the best payoff then you would have no regrets
If you selected A1 and state of nature happened where another alternative was better, you can quantify regret as difference between payoff you chose and best payoff under that state of nature
Regret matrix
Need to calculate
Difference between payoff you chose and best payoff under that state of nature
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39. Minimax Regret Rule – Calculate Regret matrix
Difference between payoff you chose and best payoff under that state of nature
For State 1 – A2
1005 – 948 = 57
For State 1 – A3
1005 – (-2005) = 3010
Etc.
NOTE: use numbers from original matrix
Alternative State1 State2 State3 A A A A
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40. Minimax Regret Rule Choose alternative with smallest maximum regret
Alternative A4 is chosen
Alternative State1 State2 State Maximum regret A A A A
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41. Summary of Uncertainty Rules
Decision rule Alternative selected Optimism or pessimism
Laplace A Neither
Maximin A Pessimism
Maximax A Optimism
Hurwicz (a=0.2) A Blend
Minimax regret A Pessimism
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42. Key Points
Uncertainty techniques used when impossible, or impractical, to assign probabilities to outcomes
Payoff matrix shows all possible outcomes to consider
Laplace rule assumes each state of nature is equally likely
Essentially expected value with equal probabilities
Maximin rule is most pessimistic
Pick alternative with best payoff from all worst payoffs
Maximax rule is most optimistic
Pick alternative with best payoff from all best payoffs
Hurwicz Rule assumes that without guidance people will tend to focus on the extremes
Blend optimism and pessimism using selected ratio
Minimax Regret rule minimizes regret you would have if you chose the wrong alternative under each state of nature
Choose alternative with smallest maximum regret
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43. Multiple Attribute Decisions
Ch. 26
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44. Multiple Attribute Decisions Outline
Introducing multiple attribute decisions
Case study: Fly-by-Night Air
Different kinds of “value”
Choosing attributes
Measurement scales
Non-compensatory techniques
Compensatory techniques
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45. Introducing Multiple Attribute Decisions
Previous chapters explained how to make decisions using a single criterion, money
Alternative with best PW(i), AE(i), incremental IRR, incremental benefit-cost ratio, etc. is selected
Aside from technical feasibility, money is almost always the most important decision criterion
But not the only one
Often, other criteria (“attributes”) must be considered and can’t be cast in terms of money
If you were responsible for buying a new laptop computer for use on a development project, price would probably be the most important decision criterion. But other attributes like processor speed, memory capacity, disk capacity, size of the screen, and reliability would probably also be factored into your decision. This chapter presents several techniques for making decisions when there is more than one attribute to consider.
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46. Case Study: Fly-by-Night (FBN) Airlines
10-year old regional airline with above average growth
Moving into nationwide market as no-frills carrier
As part of strategic planning, IT department charged with examining airline reservations systems
10 year planning horizon, effective income tax rate=37%, after-tax MARR=15%
Research has identified five technically-viable alternatives
Keep existing software
Buy Jupiter commercial system
Buy Sword commercial system
Buy Guppy commercial system
Develop new software in-house
Develop new software offshore
Assumed to be service alternatives—income is expected to be essentially same on all
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47. Different Kinds of “Value”
Decision process is all about maximizing value
Choose from available alternatives the one that maximizes value
When value is expressed as money, decision process may be complex but is straightforward
Money isn’t the only kind of value
Money is really only a way to quantify value
Two kinds of value
Use-value - the ability to get things done, the properties of the object that cause it to perform
Esteem value - the properties that make it desirable
Two kinds of value
Use-value—the ability to get things done, the properties of the object that cause it to perform
Esteem value—the properties that make it desirable
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48. Choosing Attributes
Decisions should be based on appropriate attributes
Each attribute should capture a unique dimension of decision
Set of attributes should cover important aspects of decision
Differences in attribute values should be meaningful in distinguishing among alternatives
Each attribute should distinguish at least two alternatives
Selection of attributes may be subjective
Too many attributes is unwieldy
Too few attributes gives poor differentiation
Potential for better decisions needs to be balanced with extra effort of more attributes
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49. FBN Air: Decision Attributes
Total cost of ownership
In-service availability
Liffey performance index
From Liffey Consultancy, Ltd in Dublin, Ireland
Alignment with existing business processes
Liffey index is entirely mythical—made up to support an interval scale
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50. Measurement Scales
Each alternative will be evaluated on each attribute
Many ways to measure things
In fact, different “classes” of measurements
Within a class, some manipulations make sense and others don’t
So it’s important for you to know what the different classes of measurements are, how to recognize them, and what can and can’t be done with them.
A given multiple attribute decision, such as the case study, often includes attributes that are in different classes. So it’s important for you to know what the different classes of measurements are, how to recognize them, and what can and can’t be done with them.
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51. Measurement Scales Scale type Description Example Operations Nominal
Two things are assigned the same symbol if they have the same value
House style (Colonial,
Contemporary, Ranch,
Craftsman, Bungalow, …)
=, <>
Ordinal
The order of the symbols reflects
an order defined on the attribute
Letter grades in school
(A, B, C, ...)
=, <>,
<, >, <=, =>
Interval
Differences between the numbers
reflect differences in the attribute
Temperature in degrees
Fahrenheit or Celsius,
Calendar date
<, >, <=, =>,
+, -
Ratio
Differences and ratios between the
numbers reflect differences and
ratios of the attribute
Length in centimeters,
Duration in seconds,
Temperature in Kelvin
+, -, *, /
Here are four different scale types that can be used in measuring things.
Nominal -> is existence. Put things in buckets.
Ordinal -> One item is higher, lower, or equal to another. Buckets have order.
Interval -> Same as ordinal but the space between them is a fixed amount. Consistent gap between buckets.
Ratio -> Same as interval but zero is meaningful.
Note the types of comparisons allowed.
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52. FBN Air: Evaluation and Attribute Scales
Cost Availability Liffey index Alignment
Alternative PW(i) Months [ ] [Ex, Vg,Ok,Pr, Vpr]
Existing $1.8M Excellent
Jupiter $15.4M Poor
Sword $21.6M Ok
Guppy $16.7M Very poor
New in-house $30.3M Excellent
New off-shore $17.5M Very good
Attribute Scale
Cost Ratio
Availability Ratio
Liffey index Interval
Alignment Ordinal
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53. Dimensionality of Decision Techniques
Two families of decision techniques
Differ in how attributes used
Non-compensatory, or fully dimensioned, techniques
Each attribute treated as separate entity
No tradeoffs among attributes
Compensatory, or single-dimensioned, techniques
Collapse attributes onto single figure of merit
Lower score in one attribute can be compensated by—or traded off against—higher score in others
The different families of techniques are not mutually exclusive. You could, for example, use non-compensatory techniques to reduce the number of alternatives then use compensatory techniques to make the final selection.
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54. Non-compensatory Decision Techniques
Three will be described
Dominance
Satisficing
Lexicography
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55. Dominance
Compare each pair of alternatives on attribute-by-attribute basis
Look for one alternative to be at least as good in every attribute and better in one or more
When found, no problem deciding
One alternative is clearly superior to the other, inferior can be discarded
May not lead to selecting one single alternative
Good for filtering alternatives and reducing work using other techniques
In FBN Air, Jupiter dominates Guppy
Dominance technique is unlikely to select a single alternative but in some situations it could. If UFO’s choice were only between A3 and A5, the dominance technique would be sufficient. At the very least, dominated alternatives can be dropped from further consideration to reduce the amount of work to be done using the other techniques.
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56. Satisficing Sometimes called “method of feasible ranges”
Establish acceptable ranges of attribute values
Alternatives with any attributes outside acceptable range are discarded
May not lead to selecting one single alternative
Good for filtering alternatives and reducing work using other techniques
If FBN Air were to define a maximum cost at $25M and Alignment better than Poor, would eliminate alternatives Guppy and In-house
A decision that considers several alternatives is almost certain to yield a more optimal choice than a decision that stops at the first minimally acceptable one
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57. Satisficing
Can lead to selecting one alternative when used with an iterative propose-then-evaluate process
Iterative version is appropriate when satisfactory performance, rather than optimal performance, is good enough
If optimal performance needed, always identify several alternatives that meet satisficing criteria then do further decision analysis with one of other techniques
Repeat
Propose a new solution
Evaluate that solution against the decision attributes
Until the solution is within the acceptable range for all decision attributes
Note: Stops when 1st acceptable solution is proposed
If FBN Air were to define a maximum cost at $25M and Alignment better than Poor, would eliminate alternatives Guppy and In-house
A decision that considers several alternatives is almost certain to yield a more optimal choice than a decision that stops at the first minimally acceptable one
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58. Lexicography
Two previous techniques assume attributes have equal importance
If one attribute is far more important than others, final choice could be made on that one attribute alone
If alternatives have identical values for most-important attribute, use next-most-important attribute to break tie
If still tied, compare next most important attribute, …
Continue until a single alternative chosen or all alternatives evaluated
FBN Air
Alignment might be #1, eliminates all but Existing and In-house
Cost might be #2, eliminates in-house
What if the process ends without selecting a single alternative? On the assumption that every important attribute has been included in the decision analysis, then it doesn’t matter which of the remaining alternatives is chosen. Any one of them is, by definition, as good as the other remaining alternatives. In this case you can choose one at random, go with your gut feel, etc. Otherwise, you’ll need to introduce new attributes into the decision analysis.
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59. Compensatory Decision Techniques
Attribute values converted into common “figure of merit”
Units for common scale are usually arbitrary
If common scale is at least interval scale then scores can be compared meaningfully
Two will be presented
Nondimensional scaling
Additive Weighting
Analytical Hierarchy Process (see text)
Remember, worse performance in one attribute can be compensated by—traded off against—better performance in another
Analytic Hierarchy Process (AHP) described in book—read on your own
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60. Non-Dimensional Scaling
Convert attribute values into common scale so they can be added together to make composite score for each alternative
Alternative with best composite score is selected
All attributes are defined to have equal importance
Common scale needs same range for all attributes
Must also follow same trend on desirability; most-preferred value needs to always be biggest or always be smallest common scale value
Formula for converting attributes, as long as interval or ratio-scaled, into the common scale
Let’s entirely arbitrarily chose the common scale to be
In FBN’s case, lower cost is better so lowest cost alternative would need to be given, say, the highest common rating for the cost attribute. The cost attribute of the Existing System alternative could be given 50 as its common scale rating. But higher Liffey Index is better so the highest Liffey Index alternative would be given the highest common rating for the L-I attribute. The L-I attribute of the Sword would be given 50 as its common scale rating.
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61. FBN Air: Scaled Attributes
Cost Availability Liffey index
Alternative [0..50] [0..50] [0..50] Total
Existing
Jupiter
Sword
Guppy
New in-house
New off-shore
Note: Let’s entirely arbitrarily chose the common scale to be In FBN’s case,
lower cost is better so lowest cost alternative highest common rating
higher Liffey Index (LI) is better so the highest LI alternative highest common rating.
Best = Sword
Best is Swords
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62. Non-Dimensional Scaling and Ordinal Attributes
When decision includes ordinal scaled attributes, you will need to:
Ignore ordinal-scaled attributes
Refine ordinal-scaled attributes to use interval or ratio scales and include them in nondimensional scaling
Do nondimensional scaling for all interval- and ratio-scaled attributes then finish using a non-compensatory technique
Alternative Total Alignment
Existing Excellent
Jupiter Poor
Sword Ok
Guppy Very poor
New in-house Excellent
New off-shore Very good
Let’s entirely arbitrarily chose the common scale to be
In FBN’s case, lower cost is better so lowest cost alternative would need to be given, say, the highest common rating for the cost attribute. The cost attribute of the Existing System alternative could be given 50 as its common scale rating. But higher Liffey Index is better so the highest Liffey Index alternative would be given the highest common rating for the L-I attribute. The L-I attribute of the Sword would be given 50 as its common scale rating.
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63. Additive Weighting
Identical to non-dimensional scaling except attributes have different “weights” or degrees of influence on the decision
An attribute that’s more important will have more influence on outcome
Most popular
Step 1: select common scale and convert all interval and ratio-scaled attribute values into that scale
Just like non-dimensional scaling
Step 2: assign weights based on relative importance
Many different approaches to this
Recommended approach is
Each attribute given “points” corresponding to importance
Weight for each attribute is its points divided by sum of points across all attributes
Most popular compensatory technique
This one separates dimensionalizing attribute values from weighting attributes & reduces decision complexity and allows for more precise analysis
Attribute with twice the importance is given twice as many points
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64. FBN Air: Weighting the Attributes
Suppose FBN Air gives point values as shown for ratio and interval-scaled attributes
Attribute Points Weight
Cost / ( ) = 0.588
Availability / ( ) = 0.118
Liffey index / ( ) = 0.294
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65. Additive Weighting
Step 3: calculate each alternative’s total weighted score
Example Existing = (0.588*50)+(0.118*50)+(0.294*0) = 35.3
Same as non-dimensional scaling, decision is made on total score if there are no relevant ordinal-scaled attributes
Cost Availability Liffey index
Alternative (0.588) (0.118) (0.294) Total
Existing
Jupiter
Sword
Guppy
New in-house
New off-shore
Ignoring the ordinal-scaled attributes in the FBN Air example, the existing system is the best under additive weighting. If there are important ordinal-scaled attributes they will need to be addressed as explained in the non-dimensional scaling section.
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66. Key Points
Aside from technical feasibility, money is almost always the most important decision criterion but it’s not always the only one
Use values can usually be quantified in terms of money
Esteem values can't be quantified in terms of money
Decisions involving more than one attribute are almost inevitable
Choose decision attributes to cover all relevant use values and esteem values
Several different classes of measurement
Nominal, Ordinal, Interval, and Ratio
Within each class, some comparisons will make sense and others won’t
Non-compensatory techniques treat each attribute as a separate entity
Dominance, Satisficing, Lexicography
Compensatory techniques allow better performance on one attribute to compensate for poorer performance in another
Nondimensional Scaling, Additive Weighting
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