Chapter 19:Decision Analysis

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.

• Decision-making under certainty

• Decision-making under uncertainty

• Decision-making under risk

Decision-Making Scenarios

LO1

• 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.

Three Variables in Decision Analysis Model

LO1

• 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.

Construction of the Decision or Payoff Table

LO1

Table 19.1: The Decision or Payoff Table

1 2 3

1 1 1 1 2 1 3 1

2 2 1 2 2 2 3 2

3 3 1 3 2 3 3 3

1 2 3

s s s sd P P P Pd P P P Pd P P P P

d P P P P

n

n

n

n

m m m m m n

, , , ,

, , , ,

, , , ,

, , , ,

States of Nature

DecisionAlternatives

where: sj = state of naturedj = decision alternativePi,j = payoff for decision i under state j

LO1

• 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.

Yearly Payoffs on an Investment of $10 000: Description of Problem

LO1

Table 19.2: Decision Table for an Investor

StagnantSlow Growth

Rapid Growth

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

• 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

Rule for Decision Making Under Certainty

LO1

Decision Making Under Certainty

The states of nature are known.

StagnantSlow Growth

Rapid Growth

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

The Greatest Possible PayoffThe economy

will growrapidly.

Invest in stocks.

The Greatest Possible PayoffThe economy

will growrapidly.

Invest in stocks.

LO1

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

Maximax Criterion

1. Identify the maximum payoff for each alternative.2. Choose the alternative with the largest maximum.

StagnantSlow

GrowthRapid Growth Maximum

StocksBondsGICs

Mixture

(500)$ 700$ 2,200$ 2,200$ (100)$ 600$ 900$ 900$

300$ 500$ 750$ 750$ (200)$ 650$ 1,300$ 1,300$

LO2

Maximin Criterion

1. Identify the minimum payoff for each alternative.2. Choose the alternative with the largest minimum.

StagnantSlow Growth

Rapid 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

Hurwicz Criterion

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.

StagnantSlow Growth

Rapid Growth Maximum Minimum

Weighted Average

Stocks (500)$ 700$ 2,200$ 2,200$ (500)$ 1,390$ Bonds (100)$ 600$ 900$ 900$ (100)$ 600$ GICs 300$ 500$ 750$ 750$ 300$ 615$ Mixture (200)$ 650$ 1,300$ 1,300$ (200)$ 850$

=.7 =.3

LO2

Decision Alternatives for Various Values of

Stocks Bonds GICs MixtureMax Min Max Min Max Min Max Min

1- 2,200 -500 900 -100 750 300 1,300 -2000.0 1.0 -500 -100 300 -2000.1 0.9 -230 0 345 -500.2 0.8 40 100 390 1000.3 0.7 310 200 435 2500.4 0.6 580 300 480 4000.5 0.5 850 400 525 5500.6 0.4 1120 500 570 7000.7 0.3 1390 600 615 8500.8 0.2 1660 700 660 10000.9 0.1 1930 800 705 11501.0 0.0 2200 900 750 1300

LO2

Graph of Hurwicz Criterion Selections for Various Values of

LO2

Investment Example: Selected Regrets

StagnantSlow Growth

Rapid Growth

Stocks (500)$ 700$ 2,200$ Bonds (100)$ 600$ 900$ GICs 300$ 500$ 750$ Mixture (200)$ 650$ 1,300$

I invested in GICs.Then the economygrew rapidly. I am

out $1,450.

I invested in stocks.Then the economy

stagnated. I regret notinvesting in GICs. I am$800 down from where

I could have been.

I invested in stocks, andthe economy grew slowly.

I have no regrets.

LO2

Investment Example:Opportunity Loss Table

StagnantSlow

GrowthRapid

GrowthStocks 800 0 0Bonds 400 100 1,300GICs 0 200 1,450Mixture 500 50 900

LO2

Investment Example:Calculating Opportunity Loss

StagnantSlow Growth

Rapid Growth

Stocks (500)$ 700$ 2,200$ Bonds (100)$ 600$ 900$ GICs 300$ 500$ 750$ Mixture (200)$ 650$ 1,300$

Payoff Table

StagnantSlow

GrowthRapid

GrowthStocks 800 0 0Bonds 400 100 1,300GICs 0 200 1,450Mixture 500 50 900

Opportunity Loss Table

OLi,j = Max(column j) - Pi,j

LO2

Minimax Regret

1. Identify the maximum regret for each alternative.2. Choose the alternative with the least maximum regret.

StagnantSlow

GrowthRapid

Growth MaximumStocks 800 0 0 800Bonds 400 100 1,300 1,300GICs 0 200 1,450 1,450Mixture 500 50 900 900

LO2

• 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

Decision Making under Risk

LO3

Decision Table with States of Nature Probabilities for Investment Example

LO3

Decision Tree for the Investment Example

Stocks

Bonds

GICs

Mixture

Slow growth (.45)

Slow growth (.45)

Slow growth (.45)

Slow growth (.45)

Stagnant (.25)

Stagnant (.25)

Stagnant (.25)

Stagnant (.25)

Rapid Growth (.30)

Rapid Growth (.30)

Rapid Growth (.30)

Rapid Growth (.30)

-$500

$700

$2,200

-$100

$600

$900$300

$500

$750

-$200

$650

$1,300

DecisionNode

ChanceNode

LO3

Expected Monetary Value Criterion

LO3

EMV Calculations for the Investment Example

LO3

Decision Tree with Expected Monetary Values for the Investment Example

Stocks

Bonds

GICs

Mixture

Slow growth (.45)

Slow growth (.45)

Slow growth (.45)

Slow growth (.45)

Stagnant (.25)

Stagnant (.25)

Stagnant (.25)

Stagnant (.25)

Rapid Growth (.30)

Rapid Growth (.30)

Rapid Growth (.30)

Rapid Growth (.30)

-$500

$700

$2,200

-$100

$600

$900$300

$500

$750

-$200

$650

$1,300

$850

$515

$525

$623.50

LO3

Decision Tree with Expected Monetary Values for the Investment Example

LO3

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

• 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.

Definition of Expected Value of Perfect Information

LO3

• 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

Definition of Expected Value of Perfect Information

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

MAXIMUMMAXIMUM

MAXIMUM

LO3

• 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

Expected Monetary Payoff with Perfect Information for the Investment Example

LO3

Expected Value of Perfect Information for the Investment Example

Expected Value of Perfect Information= Expected Monetary Payoff with Perfect Information -

Max(EMV[di])= $1050 - $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

• 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

Utility

LO3

• A person has the chance to enter a contest with a 50-50 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.

Measurement of Utility: Standard Gamble Method

LO3

Utility Curves for Three Types of Game Players

Chance ofWinningthe Contest

Monetary Payoff

Risk-Avoider

Risk Neutral

Risk-Taker

•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

LO3

• 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

Risk Neutral Game Player in a Standard Gamble Game: Indifferent to Owning “a” or “b”

a

b

$100 000

-$0

.5

.5

$50 000

LO3

• 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)

Risk Avoider in a Standard Gamble Game: Indifferent to Owning “a” or “b”

a

b

$100 000

-$0

.5

.5

$20 000

LO3

• 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).

Risk Taker in a Standard Gamble Game: Indifferent to Owning “a” or “b”

a

b

$100 000

-$0

.5

.5

$70 000

LO3

Risk Curves For Three Game Players

LO3

• Bayes’ Rule

• Expected Value of Sample Information

Revising Probabilities in Light of Sample Information

LO4

• 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.

Interpretation

LO4

• 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.

Interpretation

LO4

Decision Table for Investment Problem

No Growth

(.65)

Rapid Growth

(.35)Bonds 500$ 100$ Stocks (200)$ 1,100$

LO4

Expected Monetary Value Criterion for the Investment Example

No Growth

Rapid Growth

Expected Monetary Value

0.65 0.35Bonds 500$ 100$ 360.00$ Stocks (200)$ 1,100$ 255.00$

LO4

• 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.

Revising Probabilities in the Light of Sample Information

LO4

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

LO4

Decision Tree for the Investment Example: Figure 19.5

Stocks

Bonds

No Growth (.65)

No Growth (.65)

Rapid Growth (.35)

Rapid Growth (.35)

$500

$100

-$200

$1,100

EMV=$360

EMV=$255

($360)

LO4

The Success and Failure Rates of the Forecaster in Forecasting the Two States of the Economy

Actual State of Economy

No Growth

(s1)

Rapid Growth

(s2)Forecaster Predicts

No Growth (F )1 .80 .30

Forecaster Predicts

Rapid Growth (F2 ).20 .70

P(Fi|sj)

LO4

Bayes’ Rule

LO4

Revision Based on a Forecast of No Growth (F1)

State ofEconomy

PriorProbabilities

ConditionalProbabilities

JointProbabilities

RevisedProbabilities

NoGrowth(s 1)

P(s 1) = .65 P(F1| s 1) = .80 P(F1 s 1) = .520 .520/.625 = .832

RapidGrowth(s 2)

P(s 2) =.35 P(F1| s 2) = .30 P(F1 s 2) = .105 .105/.625 = .168

P(F1) = .625

P(sj|F1)

LO4

Revision Based on a Forecast of Rapid Growth (F2)

State ofEconomy

PriorProbabilities

ConditionalProbabilities

JointProbabilities

RevisedProbabilities

NoGrowth(s 1)

P(s 1) = .65 P(F2| s 1) = .20 P(F2 s 1) = .130 .130/.375 = .347

RapidGrowth(s 2)

P(s 2) =.35 P(F2| s 2) = .70 P(F2 s 2) = .245 .245/.375 = .653

P(F2) = .375

P(sj|F2)

LO4

Decision Tree for the Investment Example After Revision of Probabilities: Figure 19.6

Stocks

Bonds

No Growth (.832)

No Growth (.832)

Rapid Growth (.168)

Rapid Growth (.168)

$500

$100

-$200

$1,100

$432.80

$18.40

$432.80

Stocks

Bonds

No Growth (.347)

No Growth (.347)

Rapid Growth (.653)

Rapid Growth (.653)

$500

$100

-$200

$1,100

$238.80

$648.90

$648.90

ForecastNo Growth(.625)

ForecastRapid Growth(.375)

$513.84BuyForecast

LO4

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= $513.84 - $360= $153.84

But what if the decision maker had to pay $100 for theforecaster’s prediction?

This would reduce the value of getting perfect information from $513.84 shown in Figure 19.6 in the previous slide to $413.84.

Note that this is still superior to the $360 without sample information

LO4

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

Decision Tree Investment ExampleAll Options Included

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