1. Introduction to Decision Analysis
The field of decision analysis provides framework for making important decisions.
Decision analysis allows us to select a decision from a set of possible decision alternatives when uncertainties regarding the future exist.
The goal is to optimized the resulting payoff in terms of a decision criterion.

2. Components of a Decision Model
A set of possible decisions
discrete, continuous
finite, infinite
A set of possible events that could occur (states of nature)
Payoffs (returns or costs) associated with each decision/state of nature pair

3. Payoff Table Analysis Payoff Tables
Payoff Table analysis can be applied when -
There is a finite set of discrete decision alternatives.
The outcome of a decision is a function of a single future event.
In a Payoff Table -
The rows correspond to the possible decision alternatives.
The columns correspond to the possible future events.
Events (States of Nature) are mutually exclusive and collectively exhaustive.
The body of the table contains the payoffs.

4. EXAMPLE TOM BROWN INVESTMENT DECISION
Tom Brown has inherited $1000.
He has decided to invest the money for one year.
A broker has suggested five potential investments.
Gold.
(Junk) Bond.
Growth Stock.
Certificate of Deposit.
Stock Option Hedge.
Tom has to decide how much to invest in each investment.

5. Constructing a Payoff Table
Determine the set of possible decision alternatives.
for Tom this is the set of five investment opportunities.
Defined the states of nature.
Tom considers several stock market states (expressed by changes in the DJA)
State of Nature DJA Correspondence
S.1: A large rise in the stock market Increase over 1000 points
S.2: A small rise in the stock market Increase between 300 and 1000
S.3: No change in the stock market Change between -300 and +300
S.4: A small fall in stock market Decrease between 300 and 800
S5: A large fall in the stock market Decrease of more than 800
The States of Nature are Mutually Exclusive
and Collective Exhaustive

6. Determining Payoffs Our broker reasons:
Stocks and bonds generally move in the same direction of the market
Gold is an investment hedge that seems to move opposite to the market direction
A C/D account pays the same interest irrespective of market conditions
Specifically our broker’s analysis leads to the following payoff table:

7. Payoff Table

8. Dominated Decision Alternatives
The Stock Option hedge is dominated by the Bond Alternative, because the payoff for each state of nature for the stock option  the payoff for the bond option. Thus the stock option hedge can be eliminated from any consideration.

9. Decision Making Criteria
Classifying Decision Making Criteria
Decision making under certainty
The future state of nature is assumed known
Decision making under risk
There is some knowledge of the probability of the states of nature occurring.
Decision making under uncertainty.
There is no knowledge about the probability of the states of nature occurring.

10. Decision Making Under Certainty
Given the “certain” state of nature, select the best possible decision for that state of nature

11. Decisions for Decision Making Under Certainty
If we knew Large Small No Small Large
this ===> Rise Rise Change Fall Fall
Gold
Bond
Stock
C/D
Choose => Stock Stock Gold Gold C/D

12. Decision Making Under Uncertainty
The decision criteria are based on the decision maker’s
attitude toward life.
These include an individual being pessimistic or optimistic, conservative or aggressive.
Criteria
Maximin Criterion - pessimistic or conservative approach.
Minimax Regret Criterion - pessimistic or conservative approach.
Maximax criterion - optimistic or aggressive approach.
Principle of Insufficient Reasoning.

13. Decision Making Under Uncertainty
The decision criteria are based on the decision maker’s attitude toward life.
These include an individual being pessimistic or optimistic, conservative or aggressive.
Criteria
Maximax (Minimin), Maximin (Minimax), Minimax Regret, Principle of Insufficient Reasoning

14. The Maximax Criterion
This criterion is based on the best possible scenario.
It fits both an optimistic and an aggressive decision maker.
An optimistic decision maker believes that the best possible outcome will always take place regardless of the decision made.
An aggressive decision maker looks for the decision with the highest payoff (when payoff is profit)

15. To find an optimal decision.
Find the maximum payoff for each decision alternative.
Select the decision alternative that has the maximum of the “maximum” payoff.
TOM BROWN - continued
The Optimal decision

16. The Maximin Criterion
This criterion is based on the worst-case scenario.
It fits both a pessimistic and a conservative decision maker.
A pessimistic decision maker believes that the worst possible result will always occur.
A conservative decision maker wishes to ensure a guaranteed minimum possible payoff.

17. To find an optimal decision
Record the minimum payoff across all states of nature for each decision.
Identify the decision with the maximum “minimum payoff”.
TOM BROWN - Continued
The Optimal decision

18. The Minimax Regret Criterion (“I shoulda”)
This criterion fits both a pessimistic and a conservative decision maker.
The payoff table is based on “lost opportunity,” or “regret”.
The decision maker incurs regret by failing to choose the “best” decision.

19. To find an optimal decision
For each state of nature.
Determine the best payoff over all decisions.
Calculate the regret for each decision alternative as the difference between its payoff value and this best payoff value.
For each decision find the maximum regret over all states of nature.
Select the decision alternative that has the minimum of these “maximum regrets”.

20. TOM BROWN - continued 500 - (-100) = 600 500 -100 500 500 -100 -100
= 600
500
TOM BROWN - continued
-100
500
500
-100
-100
500
-100
-100
Investing in Gold incurs a regret
when the market exhibits
a large rise
500
500
The Optimal decision
Let us build the Regret Table

21. The Principle of Insufficient Reason
This criterion might appeal to a decision maker who is neither pessimistic nor optimistic.
It assumes all the states of nature are equally likely to occur.
The procedure to find an optimal decision.
For each decision add all the payoffs.
Select the decision with the largest sum (for profits).

22. TOM BROWN - continued Sum of Payoffs
Gold $600
Bond $350
Stock $50
C./D $300
Based on this criterion the optimal decision alternative is to invest in gold.

23. Summary Maximax Decision -- Stock Maximin Decision -- C/D
Minimax Regret -- Junk Bond
Insufficient Reason -- Gold
Isn’t there a better way?

24. Decision Making Under Risk
The Expected Value Criterion
If a probability estimate for the occurrence of each state of nature is available, payoff expected value can be calculated.
For each decision calculate its expected payoff by
Expected Payoff = S
Based on past market performance the analyst predicts:
P(large rise) = .2, P(small rise) = .3, P(No change) = .3,
P(small fall) = .1, P(large fall) = .1
(Probability)(Payoff)
Over States of Nature

25. TOM BROWN - continued The Optimal decision
(0.2)(250) + (0.3)(200) + (0.3)(150) + (0.1)(-100) + (0.1)(-150) = 130

26. When to Use the Expected Value Approach
The Expected Value Criterion is useful in cases where long run planning is appropriate, and decision situations repeat themselves.
One problem with this criterion is that it does not consider attitude toward possible losses.

27. Expected Value of Perfect Information (EVPI)
The Gain in Expected Return obtained from knowing with certainty the future state of nature is called:
Expected Value of Perfect Information (EVPI)
It is also the Smallest Expect Regret of any decision alternative.
Therefore, the EVPI is the expected regret
corresponding to the decision selected
using the expected value criterion

28. TOM BROWN - continued Stock If it were known with certainty
that there will be a “Large Rise” in the market
Large rise
-100
250
500 60
Stock
... the optimal decision would be to invest in...
Similarly,
Expected Return with Perfect information =
0.2(500)+0.3(250)+0.3(200)+0.1(300)+0.1(60) = $271
EVPI = ERPI - EV = $271 - $130 = $141

29. Bayesian Analysis - Decision Making with Imperfect Information
Bayesian Statistic play a role in assessing additional information obtained from various sources.
This additional information may assist in refining original probability estimates, and help improve decision making.

30. Should Tom purchase the Forecast ?
TOM BROWN - continued
Tom can purchase econometric forecast results for $50.
The forecast predicts “negative” or “positive” econometric growth.
Statistics regarding the forecast.
Should Tom purchase the Forecast ?
When the stock market showed a large rise the
forecast was “positive growth” 80% of the time.

31. SOLUTION
Tom should determine his optimal decisions when the forecast is “positive” and “negative”.
If his decisions change because of the forecast, he should compare the expected payoff with and without the forecast.
If the expected gain resulting from the decisions made with the forecast exceeds $50, he should purchase the forecast.

32. Tom needs to know the following probabilities
P(Large rise | The forecast predicted “Positive”)
P(Small rise | The forecast predicted “Positive”)
P(No change | The forecast predicted “Positive ”)
P(Small fall | The forecast predicted “Positive”)
P(Large Fall | The forecast predicted “Positive”)
P(Large rise | The forecast predicted “Negative ”)
P(Small rise | The forecast predicted “Negative”)
P(No change | The forecast predicted “Negative”)
P(Small fall | The forecast predicted “Negative”)
P(Large Fall) | The forecast predicted “Negative”)

33. [ P(B | A 1)P(A 1)+ P(B | A 2)P(A 2)+…+ P(B | A n)P(A n) ]
Bayes’ Theorem provides a procedure to calculate these probabilities
P(B |A i)P(A i)
[ P(B | A 1)P(A 1)+ P(B | A 2)P(A 2)+…+ P(B | A n)P(A n) ]
P(A i | B) =
Posterior Probabilities for “positive” economic forecast
0.16
0.56 X =
0.286
0.375
0.268
0.071
0.000
0.2
0.3
0.1
The probability that the stock market
shows “Large Rise” given that
the forecast predicted “Positive”
The Probability that the forecast is “positive” and the stock market shows “Large Rise”.
Observe the revision in
the prior probabilities

34. Bayesian Prob. for “Negative” Forecast
s P(s) P(Neg|s) P(Neg and s) P(s|Neg)
Large Rise /.44= .091
Small Rise /.44= .205
No Change /.44= .341
Small Fall /.44= .136
Large Fall /.44= .227
.44

35. Expected Value of Sample Information
The expected gain from making decisions based on Sample Information.
With the forecast available, the Expected Value of Return is revised.
Calculate Revised Expected Values for a given forecast
as follows.
EV(Invest in……. |“Positive” forecast) =
=.286( )+.375( )+.268( )+.071( )+0( ) =
EV(Invest in ……. | “Negative” forecast) =
=.091( )+.205( )+.341( )+.136( )+.227( ) =
Gold
Bond
-100
250
-100
100
200
100
150
200
200
-100
300
300
-150
$84
$180
-100
100
200
300
-100
Bond
100
200
300
250
-100
100
200
150
200
-100
300
-150
$ 65
$120

36. Should Tom purchase the Forecast ?
The rest of the revised EV s are calculated in a
similar manner.
EREV = Expected Value Without Sampling Information = 130
Expected Value of Sample Information - Excel
So,
Should Tom purchase the Forecast ?
Invest in Stock when the Forecast is “Positive”
ERSI = Expected Return with sample Information =
(0.56)(250) + (0.44)(120) = $193
Invest in Gold when the forecast is “Negative”

37. EVSI = Expected Value of Sampling Information
= ERSI - EREV = = $63.
Yes, Tom should purchase the Forecast.
His expected return is greater than the Forecast cost.
Efficiency = EVSI / EVPI = 63 / 141 = 0.45

38. Decision Trees
The Payoff Table approach is useful for a single decision situation.
Many real-world decision problems consists of a sequence of dependent decisions.
Decision Trees are useful in analyzing multi-stage decision processes.

39. Characteristics of the Decision Tree
A Decision Tree is a chronological representation of the decision process
There are two types of nodes
Decision nodes (represented by squares)
State of nature nodes (representing by circles).
The root of the tree corresponds to the present time.
The tree is constructed outward into the future with branches emanating from the nodes.
A branch emanating from a decision node corresponds to a decision alternative. It includes a cost or benefit value.
A branch emanating from a state of nature node corresponds to a particular state of nature, and includes the probability of this state of nature.

40. BILL GALLEN DEVELOPMENT COMPANY
B. G. D. plans to do a commercial development on a property.
Relevant data
Asking price for the property is $300,000
Construction cost is $500,000
Selling price is approximated at $950,000
Variance application costs $30,000 in fees and expenses
There is only 40% chance that the variance will be approved.
If B. G. D. purchases the property and the variance is denied, the property can be sold for a net return of $260,000
A three month option on the property costs $20,000 which will allow B.G.D. to apply for the variance.
A consultant can be hired for $5000
P (Consultant predicts approval | approval granted) = 0.70
P (Consultant predicts denial | approval denied) = 0.90

41. What should BGD do? Construction of the Decision Tree
Initially the company faces a decision about hiring the consultant.
After this decision is made more decisions follow regarding
Application for the variance.
Purchasing the option.
Purchasing the property.

42. Let us consider the decision to not hire a consultant 113
Do nothing 2
Buy land
-300,000 4
Apply for variance
-30,000
Do not hire consultant
Hire consultant
-5000 1
Purchase option
-20,000
Let us consider the decision to
not hire a consultant 11
Apply for variance
-30,000

43. Apply for variance
Buy land and
-70,000 10
120,000 8 6 7
Build
Sell
950,000
-500,000
Approved
Denied
0.4
0.6 5
260,000
Sell 9
-300,000
-500,000
950,000 13 14 15
Buy land
Build
Sell 17
-50,000
100,000 16
Approved
Denied
0.4
0.6 12
Purchase option and
Apply for variance

44. Let us consider the decision to hire a consultant
This is where we are at this stage

45. Let us consider the decision to2
-5000 21 28 44 37 36 20
Apply for variance
-30,000
Do not hire consultant
Let us consider the decision to
hire a consultant 19 35
Do Nothing
Buy land
-300,000
Purchase option
-20,000 1
Hire consultant
-5000 18
Predict
Approval
Denial
0.4
0.6

46. The consultant serves as a source for additional information
-75,000 27
115,000 25 23 24
Build
Sell
950,000
-500,000 22
Approved
Denied
0.1765
0.8235 ?
260,000
Sell 26
Posterior Probability of approval | consultant predicts approval) =
Posterior Probability of denial | consultant predicts approval) =
Consultant predicts approval
The consultant serves as a source for additional information
about denial or approval of the variance.
Therefore, at this point we need to calculate the
posterior probabilities for the approval and denial
of the variance application

47. Determining the Optimal Strategy
The rest of the Decision Tree is built in a similar manner.
Now,
Work backward from the end of each branch.
At a state of nature node, calculate the expected value of the node.
At a decision node, the branch that has the highest ending node value is the optimal decision.
The highest ending node value is the value for the decision node.
Let us illustrate by evaluating one branch --
Hire consultant and consultant predicts approval

48. Thus if consultant recommends approval -- BUY LAND
80500
-22500
115,000
-75,000
115,000
-75,000
115,000
-75,000
(115,000)(.8235)=
(-75,000)(.1765)=
115,000
-75,000
115,000
-75,000
-75,000
115,000
-22500
80500
-75,000
115,000
Build
Sell
80500
-22500 23 24 25
80500 -
-500,000
950,000
Predict approval
Approved ?
0.8235 22
Denied
81483
Sell
0.1765 26 27
260,000
$81,483 is the expected value if consultant predicts approval and land is bought
In a similar manner, the expected value if consultant predicts approval and the option is purchased is $68,525
The expected value if the consultant predicts approval and BGD does nothing is -$5,000 (the consultant’s fee).
Thus if consultant recommends approval -- BUY LAND

49. Evaluation of Other Branches
DO NOT HIRE CONSULTANT
Do nothing EV = $0
Buy land EV = $6,000
Purchase option EV = $10,000 <==BEST
HIRE CONSULTANT -- PREDICTS DENIAL
Do nothing EV = -$5,000 <===BEST
Buy land EV = -$40,458
Purchase option EV = -$27,730

50. BEST COURSE OF ACTION OPTIMAL STRATEGY --
EV(DO NOT HIRE CONSULTANT) = $10,000
EV(HIRE CONSULTANT) =
.4(81,483) + .6(-5000) = $29,593.20
OPTIMAL STRATEGY --
HIRE CONSULTANT
If consultant predicts approval -- buy the land
If consultant predicts denial -- do nothing

51. Game Theory
Game theory can be used to determine optimal decision in face of other decision making players.
All the players are seeking to maximize their return.
The payoff is based on the actions taken by all the decision making players.

52. Classification of Games
Number of Players
Two players - Chess
Multiplayer - More than two competitors (Poker)
Total return
Zero Sum - The amount won and amount lost by all competitors are equal (Poker among friends)
Nonzero Sum -The amount won and the amount lost by all competitors are not equal (Poker In A Casino)
Sequence of Moves
Sequential - Each player gets a play in a given sequence.
Simultaneous - All players play simultaneously.

53. IGA SUPERMARKET
The town of Gold Beach is served by two supermarkets: IGA and Sentry.
Market share can be influenced by their advertising policies.
The manager of each supermarket must decide weekly which area of operations to discount and emphasize in the store’s newspaper flyer.

54. Data
The weekly percentage gain in market share for IGA, as a function of advertising emphasis.
A gain in market share to IGA results in equivalent loss for Sentry, and vice versa (i.e. a zero sum game)

55. IGA needs to determine an advertising emphasis that
will maximize its expected change in market share
regardless of Sentry’s action.

56. IGA’s Perspective - A Linear Programming model
SOLUTION
IGA’s Perspective - A Linear Programming model
Decision variables
X1 = the probability IGA’s advertising focus is on meat.
X 2 = the probability IGA’s advertising focus is on produce.
X 3 = the probability IGA’s advertising focus is on groceries.
Objective Function For IGA
Maximize expected market change (in its own favor) regardless of Sentry’s advertising policy.
Define the actual change in market share as V.

57. Constraints
IGA’s market share increase for any given advertising focus selected by Sentry, must be at least V.
The Model

58. Sentry’s Perspective - A Linear Programming model
Decision variables
Y1 = the probability that Sentry’s advertising focus is on meat.
Y2 = the probability that Sentry’s advertising focus is on produce.
Y3 = the probability that Sentry’s advertising focus is on groceries.
Y4 = the probability that Sentry’s advertising focus is on bakery.
Objective function
Minimize changes in market share in favor of IGA
Constraints
Sentry’s market share decrease for any given advertising focus selected by IGA, must not exceed V.

59. The Model

60. Optimal solutions For IGA For Sentry
X1 = ; X2 = 0.5; X3 = 0.111
For Sentry
Y1 = 0.6; Y2 = 0.2; Y3 = 0.2; Y4 = 0
For both players V =0 (a fair game).