Introduction to Decision Analysis

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.

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)– discrete, continuous– finite, infinite

• Payoffs (returns or costs) associated with each decision/state of nature pair

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.

EXAMPLETOM 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.

• 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 marketIncrease 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

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:

Payoff Table

States of Nature

Decision AlternativesLarge Rise Small Rise No Change Small Fall Large Fall

Gold -100 100 200 300 0Bond 250 200 150 -100 -150Stock 500 250 100 -200 -600C/D Account 60 60 60 60 60Stock Option Hedge200 150 150 -200 -150

States of Nature



States of Nature



States of Nature



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.

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.

Decision Making Under Certainty

• Given the “certain” state of nature, select the best possible decision for that state of nature

Decisions for Decision Making Under Certainty

If we knew Large Small No Small Large

this ===> Rise Rise Change Fall Fall

Gold -100 100 200 300 0

Bond 250 200 150 -100 -150

Stock 500 250 100 -200 -600

C/D 60 60 60 60 60

Choose => Stock Stock Gold Gold C/D

• 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.

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

• 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)

TOM BROWN - continued

– 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.

The Maximax criterion MaximumDecision Large rise Small rise No changeSmall fall Large fall PayoffGold -100 100 200 300 0 300Bond 250 200 150 -100 -150 200Stock 500 250 100 -200 -600 500C/D 60 60 60 60 60 60

The Optimal decision

• 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.

TOM BROWN - Continued

– 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”.

The Maximin Criterion Minimum

Decisions LargeRrise Small Rise No change Small Fall Large Fall Payoff

Gold -100 100 200 300 0 -100Bond 250 200 150 -100 -150 -150Stock 500 250 100 -200 -600 -600C/D account 60 60 60 60 60 60

The Maximin Criterion Minimum

Decisions LargeRrise Small Rise No change Small Fall Large Fall Payoff

Gold -100 100 200 300 0 -100Bond 250 200 150 -100 -150 -150Stock 500 250 100 -200 -600 -600C/D account 60 60 60 60 60 60


• 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.

– 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”.


The Payoff TableDecision Large rise Small rise No change Small fall Large fallGold -100 100 200 300 0Bond 250 200 150 -100 -150Stock 500 250 100 -200 -600C/D 60 60 60 60 60

The Payoff TableDecision Large rise Small rise No change Small fall Large fallGold -100 100 200 300 0Bond 250 200 150 -100 -150Stock 500 250 100 -200 -600C/D 60 60 60 60 60

Let us build the Regret Table

The Regret Table MaximumDecision Large rise Small rise No change Small fall Large fall RegretGold 600 150 0 0 60 600Bond 250 50 50 400 210 400Stock 0 0 100 500 660 660C/D 440 190 140 240 0 440

The Regret Table MaximumDecision Large rise Small rise No change Small fall Large fall RegretGold 600 150 0 0 60 600Bond 250 50 50 400 210 400Stock 0 0 100 500 660 660C/D 440 190 140 240 0 440

Investing in Gold incurs a regretwhen the market exhibits

a large rise 500

500

500

500

500500

500

-100-100

-100-100

-100- (-100) = 600


• 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).


– 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.

Summary

• Maximax Decision -- Stock• Maximin Decision -- C/D• Minimax Regret -- Junk Bond• Insufficient Reason -- Gold

• Isn’t there a better way?

(Probability)(Payoff)Over States of Nature

• 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 =

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


The Expected Value Criterion ExpectedDecision Large rise Small rise No changeSmall fall Large fall ValueGold -100 100 200 300 0 100Bond 250 200 150 -100 -150 130Stock 500 250 100 -200 -600 125C/D 60 60 60 60 60 60Prior Probability0.2 0.3 0.3 0.1 0.1

(0.2)(250) + (0.3)(200) + (0.3)(150) + (0.1)(-100) + (0.1)(-150) = 130


• 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.

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)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


The Expected Value of Perfect Information Decision Large Rise Small Rise No change Small Fall Large FallGold -100 100 200 300 0Bond 250 200 150 -100 -150Stock 500 250 100 -200 -600C/D 60 60 60 60 60Probab. 0.2 0.3 0.3 0.1 0.1

If it were known with certainty that there will be a “Large Rise” in the market

Large rise

... the optimal decision would be to invest in...

-100 250 500 60

Stock

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

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.


• Tom can purchase econometric forecast results for $50.

• The forecast predicts “negative” or “positive” econometric growth.

• Statistics regarding the forecast. The Forecast When the stock market showed a... predicted Large Rise Small Rise No Change Small Fall Large Fall

Positive econ. growth 80% 70% 50% 40% 0%Negative econ. growth 20% 30% 50% 60% 100%

When the stock market showed a large rise the forecast was “positive growth” 80% of the time.

Should Tom purchase the Forecast ?

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.

• 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”)

• 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 forecastStates of Prior Conditnal Joint PosteriorNature Probab. Probab. Probab. Probab.Large Rise 0.2 0.8 0.16 0.286Small Rise 0.3 0.7 0.21 0.375No Change 0.3 0.5 0.15 0.268Small Fall 0.1 0.4 0.04 0.071Large Fall 0.1 0 0 0.000

Sum = 0.56

X =

0.16 0.56

The Probability that the forecast is “positive” and the stock market shows “Large Rise”.

The probability that the stock market shows “Large Rise” given that the forecast predicted “Positive”

Observe the revision in the prior probabilities

0.2860.3750.2680.0710.000

0.20.30.30.10.1

Bayesian Prob. for “Negative” Forecast

s P(s) P(Neg|s) P(Neg and s) P(s|Neg)Large Rise .20 .20 .04 .04/.44= .091

Small Rise .30 .30 .09 .09/.44= .205

No Change .30 .50 .15 .15/.44= .341

Small Fall .10 .60 .06 .06/.44= .136

Large Fall .10 1 .10 .10/.44= .227

.44

• 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 forecastas follows.

• EV(Invest in……. |“Positive” forecast) ==.286( )+.375( )+.268( )+.071( )+0( ) =

• EV(Invest in ……. | “Negative” forecast) = =.091( )+.205( )+.341( )+.136( )+.227( ) =

Gold

Gold-100 100 200 300 0 $84-100

-100

-100

-100

100100

100

100

200200

200

200

300300

300

300

00

0

0 $120

Bond

Bond

250

250

200

200

150

150

-100

-100

-150

-150

$180

$ 65

– The rest of the revised EV s are calculated in a similar manner.

Prior Revised EVDecisions/large risesmall riseno changesmall falllarge fallEV Pos Neg

Gold -100 100 200 300 0 100 84 120Bond 250 200 150 -100 -150 130 180 65Stock 500 250 100 -200 -600 125 250 -37C/D Account60 60 60 60 60 60 60 60Prior Probability0.2 0.3 0.3 0.1 0.1Pos. Economic Forecast0.29 0.38 0.27 0.07 0 0.56Neg. Economic Forecast0.09 0.21 0.34 0.14 0.23 0.44

Invest in Stock when the Forecast is “Positive”

Invest in Gold when the forecast is “Negative”

ERSI = Expected Return with sample Information =

(0.56)(250) + (0.44)(120) = $193

ERSI = Expected Return with sample Information =

(0.56)(250) + (0.44)(120) = $193

EREV = Expected Value Without Sampling Information = 130EREV = Expected Value Without Sampling Information = 130

Expected Value of Sample Information - Excel

So,Should Tom purchase the Forecast ?

• EVSI = Expected Value of Sampling Information

= ERSI - EREV = 193 - 130 = $63.

Yes, Tom should purchase the Forecast.

His expected return is greater than the Forecast cost.

• Efficiency = EVSI / EVPI = 63 / 141 = 0.45

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.

• 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.

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

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.

Do not hire consultant

Hire consultant-5000

0

1

Let us consider the decision to

not hire a consultant

2

Do nothing

0Buy land-300,000Purchase option

-20,000

03

11

4

Apply for variance-30,000


5

Buy la

nd an

d

Apply

for va

rianc

e

Approved

Denied

0.4

0.6

6 7Build Sell

950,000-500,000

260,000Sell

9

-70,000

10

120,000

8

12

Purchase option and

Apply for variance

Approved

Denied

0.4

0.6

-300,000 -500,000 950,00013 14 15

Buy land Build Sell

17

-50,000

100,000

16

1

0

Buy land-300,000



3

4

11

18-50,000

-300,000 -500,000 950,000

260,000

120,000

-70,000

100,000

5

6

9

13

17

7

10

14 15 16

8

12

Build Sell

950,000-500,000

Sell

Buy land Build Sell

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

1

Do not hire consultant

0

2

Let us consider the decision to

hire a consultant

Hire consultant-5000

18

Predict

Approval

Predict

Denial

0.4

0.6

19

35

Do Nothing

Buy land-300,000

Purchase option-20,000

Do Nothing

Buy land-300,000Purchase option-20,000

-5000

21

28

44

37

36

20

Apply for variance

Apply for variance

Apply for variance

Apply for variance

-5000

-30,000

-30,000

-30,000

-30,000

22

Approved

Denied

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 theposterior probabilities for the approval and denial

of the variance application

?

?

Posterior Probability of approval | consultant predicts approval) = 0.8235Posterior Probability of denial | consultant predicts approval) = 0.1765

0.1765

0.8235

23 24Build Sell

950,000-500,000

260,000Sell

26

-75,000

27

115,000

25

• Determining the Optimal StrategyThe 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

-75,000

115,000115,000

-75,000

115,000

-75,000

115,000

-75,000

115,000

-75,00022

115,000

-75,000

Approved

Denied

(115,000)(.8235)=94702.5

(-75,000)(.1765)= -13237.5

-22500

8050080500

-22500

80500

-22500

80500

-?

?0.1765

0.8235

23 24Build Sell

950,000-500,000

260,000Sell

26 27

25

-75,000

115,000

•$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

•81483

•Predict approval

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

BEST COURSE OF ACTION

• 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

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.

• 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.

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.

• 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)

Sentry's EmphasisMeat Produce Grocery Bakery

IGA's Meat 2 2 -8 6Emphasis Produce -2 0 6 -4

Grocery 2 -7 1 -3

IGA needs to determine an advertising emphasis that IGA needs to determine an advertising emphasis that will maximize its expected change in market share will maximize its expected change in market share

regardless regardless of Sentry’s action.of Sentry’s action.

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.

– Constraints• IGA’s market share increase for any given advertising

focus selected by Sentry, must be at least V.– The Model

Maximize V

ST IGA expected change in

market share Sentry's Meat 2X1 - 2X2 + 2X3 V

Advertising Produce 2X1 - 7X3 VEmphasis Groceries (-8X1) +6X2 + X3 V

Bakery 6X1 - 4X2 - 3X3 VThe variables are probabilities X1 + X2 + X3 = 1

X1, X2, X3, are non negative: V is unrestricted

Maximize V

ST IGA expected change in

market share Sentry's Meat 2X1 - 2X2 + 2X3 V

Advertising Produce 2X1 - 7X3 VEmphasis Groceries (-8X1) +6X2 + X3 V

Bakery 6X1 - 4X2 - 3X3 VThe variables are probabilities X1 + X2 + X3 = 1

X1, X2, X3, are non negative: V is unrestricted

• 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.

– The ModelMinimize V

ST Sentry's expected change in

market share IGA's Meat 2Y1 + 2Y2 - 8Y3 + 6Y4 V

Advertising Produce -2Y1 + 6Y3 - 4Y4 VEmphasis Groceries 2Y1 - 7Y2 + Y3 - 3Y4 V

Y1 + Y2 + Y3 + Y4 = 1Y1, Y2, Y3, Y4 are non negative; V is unrestricted

Minimize V

ST Sentry's expected change in

market share IGA's Meat 2Y1 + 2Y2 - 8Y3 + 6Y4 V

Advertising Produce -2Y1 + 6Y3 - 4Y4 VEmphasis Groceries 2Y1 - 7Y2 + Y3 - 3Y4 V

Y1 + Y2 + Y3 + Y4 = 1Y1, Y2, Y3, Y4 are non negative; V is unrestricted

• Optimal solutions – For IGA

• X1 = 0.3889; 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).

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Introduction to Decision Analysis