Chapter 3:DECISION ANALYSIS

Part 2

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Decision Making Under Risk Probabilistic decision situation States of nature have probabilities of occurrence. The probability estimate for the occurrence of each

state of nature( if available) can be incorporated in the search for the optimal decision.

For each decision calculate its expected payoff by

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(Probability)(Payoff)

Over States of NatureExpected Payoff =

Decision Making Under Risk (cont.) Select the decision with the best expected payoff

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TOM BROWN - continued

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

The Optimal decision

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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.

Decision Making Criteria (cont.)

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Expected Value of Perfect Information

The gain in Expected Return obtained from knowing with certainty the future state of nature is called:

Expected Value of Perfect Information Expected Value of Perfect Information

(EVPI)(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 6

Expected Value of Perfect Information (cont.) EVPI = ERPI - EREV

EREV: Expected Return of the EV criterion .

Expected Return with Perfect Information ERPI=

(best outcome of 1st state of nature)*(Probability of 1st state of nature) + ….. +(best outcome of last state of nature)*(Probability of last state of nature)

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TOM BROWN - continued

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 60Stock

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 = $1418

Expected Value of Perfect Information (cont.) Another way to determine EVPI as follows

If Tom knows the market will show a large rise, he should buy the “stock”, within profit $500, or a gain of $250 over what he would earn from the “bond” (optimal decision without the additional information).

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Expected Value of Perfect Information (cont.)

If Tom knows in advance the market would undergo

His optimal decision With gain of payoff

A large rise stock 500-250= $250

A small rise stock 250-200= $ 50

No change gold 200-150= $ 50

A small fall gold 300-(-100)=$400

A large fall C/D 60-(-150)= 210

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EVPI= 0.2(250) + 0.3(50) +0.3(50)+ 0.1(400)+ 0.1(210)= 141

Baysian Analysis - Decision Making with Imperfect Information

Baysian 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.

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TOM BROWN - continued

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.

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TOM BROWN - continued P(forecast predicts “positive” | small rise in

market) = 0.7 P(forecast predicts “ negative” | small rise in

market) = 0.3

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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.

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SOLUTION To find Expected payoff with forecast Tom

should determine what to do when: The forecast is “positive growth” The forecast is “negative growth”

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SOLUTION 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”)

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SOLUTION 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”)

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Bayes’ Theorem provides a procedure to calculate these probabilities

Bayes’ Theorem P(A|B) =

Proof: p(A|B)= P (A and B) / P(B) (1)

P(B|A)= P(A and B)/P(A)

P(A and B) = P(B|A)*P(A)

(1) P(A|B)=P(B|A)*P(A)/P(B)

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Bayes’ Theorem (cont.) Often we begin probability analysis with initial or

prior probabilities. Then, from a sample , special report, or product

test we obtain some additional information. Given this information, we calculate revised or

posterior probability.

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Prior probabilities

New information

Posterior probabilities

Bayes’ Theorem(cont.)

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Posterior probabilities

Probabilities determined after the additional info becomes available

Prior probabilities

Probabilities estimated Determined based on Current info, before New info becomes available

The tabular approach to calculating posterior probabilities for positive economical forecast

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States of Prior Conditnal JointNature Probab. Probab. Probab.Large Rise 0.2 0.8 0.16Small Rise 0.3 0.7 0.21No Change 0.3 0.5 0.15Small Fall 0.1 0.4 0.04Large Fall 0.1 0 0

X

Ai: large riseB: forecast positive

P(Bi |Ai )P(Ai)P(forecast= Positive| large rise)P( large rise)

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No 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 = 0.56X =

The probability that the stock market shows “Large Rise”

given that the forecast predicted “Positive”

0.16/ 0.56

Probability( forecast= positive) = 0.16+ 0.21+0.15+ 0.04+ 0.0 = 0.56

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

The tabular approach to calculating posterior probabilities for “negative” ecnomical forecast

Probability (forecast= negative) = 0.44

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Nature Probab. Probab. Probab. Probab.Large Rise 0.2 0.2 0.04 0.091Small Rise 0.3 0.3 0.09 0.205No Change 0.3 0.5 0.15 0.341Small Fall 0.1 0.6 0.06 0.136Large Fall 0.1 1 0.1 0.227

Sum = 0.44

X =

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WINQSB printout for the calculation of the Posterior probabilities

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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

$ 6536

Expected Value of Sample Information EVSI

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 ? 37

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

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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.

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Game Theory (cont.) 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)

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Game Theory (cont.) Sequence of Moves

Sequential - Each player gets a play in a given sequence.

Simultaneous - All players play simultaneously.

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