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Decision AnalysisDecision-Making Under Uncertainty

updated 10.24.01

Systems Analysis MethodsDr. Jerrell T. Stracener, SAE Fellow

NTUSY-521-N

SMUEMIS 5300/7300

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Decision Making Under Uncertainty

• The decision-maker knows for sure which stateof nature will occur, and he or she makes thedecision based on the optimal payoff availableunder that state.

• It is unknown which states of nature will occurand the probability of the likelihood of a stateoccurring is also unknown

• The decision-maker has virtually no informationregarding which state of nature will occur, andhe or she attempts to develop a strategy basedon payoffs

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Approaches to Decision Making Under Uncertainty

• Maximax criterion

• Maximin criterion

• Hurwicz criterion

• Minimax regret

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Maximax Criterion

• An optimistic approach in which the decision-maker acts based on a notion that thebest things will happen

• The decision-maker isolates the maximumpayoff under each decision alternative and thenselects the decision alternative that produces thehighest of these maximum payoffs

• The name maximax means selecting the maximum overall payoff from the maximum payoffs of each decision alternative

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Decision Table

State of the Economy

Stagnant Slow Rapid Growth Growth Maximum

Stocks -$500 $700 $2200 $2200

Investment Bonds -$100 $600 $900 $900 DecisionAlternative CD’s $300 $500 $750 $750

Mixture -$200 $650 $1300 $1300

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Decision Table

• The maximax criterion approach requires that the decision-maker select the maximum payoff of these four

Maximum of {$2200, $900, $750, $1300} = $2200

• Since maximax criterion results in $2200 as the optimal payoff, the decision alternative selectedis the stock alternative, which is associated withthe $2200

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Maximin Criterion

• A pessimistic approach to decision-making under uncertainty

• This approach assumes that the worst will happen and attempts to minimize the damage

• Using the maximin criterion approach, the decision-maker starts by examining the payoffsunder each decision alternative and selects theworst, or minimum, payoff that can occur underthat decision

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Decision Table

State of the Economy

Stagnant Slow Rapid Growth Growth Minimum

Stocks -$500 $700 $2200 -$500

Investment Bonds -$100 $600 $900 -$100 DecisionAlternative CD’s $300 $500 $750 $300

Mixture -$200 $650 $1300 -$200

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Decision Table

• With the maximin criterion, the decision-makerexamines the minimum payoffs for each decisionalternative given in the last column and selectsthe maximum of these values

Maximum of {-$500, -$100, $300, -$200} = $300

• The decision is to invest in CD’s because thisinvestment alternative yields the highest, ormaximum, payoff under the worst-case scenario

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Hurwicz Criterion

• An approach that lies somewhere in between the Maximax and the Maximin approaches

• Selects the maximum and the minimum payoff from each decision alternative

• A value called (not the same as the probabilityof a Type I error), which lies between 0 and 1, isselected as a weight of optimism

• The nearer is to 1, the more optimistic is thedecision-maker

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Hurwicz Criterion

• The maximum payoff under each decisionalternative is multiplied by and the minimumpayoff under each decision alternative is multiplied by 1 -

• These weighted products are summed for eachdecision alternative, resulting in a weighted valuefor each decision alternative

• The maximum weighted value is selected, and the corresponding decision alternative is chosen

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Decision Table

State of the Economy

Stagnant Slow Rapid Growth Growth Maximum Minimum

Stocks -$500 $700 $2200 $2200 -$500

Bonds -$100 $600 $900 $900 -$100

CD’s $300 $500 $750 $750 $300

Mixture -$200 $650 $1300 $1300 -$200

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Decision Table

Suppose we are more optimistic than pessimisticand select = 0.7 for the weight of optimism. Thecalculations of weighted values for each decisionalternative are as follows:

stocks ($2200)(.7) + (-$500)(.3) = $1390bonds ($900)(.7) + (-$100)(.3) = $600CD’s ($750)(.7) + ($300)(.3) = $615mixture ($1300)(.7) + (-$200)(.3) = $850

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Decision Table

• The Hurwicz criterion leads the decision-makerto choose the maximum of these values, $1390

• The result under the Hurwicz criterion with = 0.7 is to choose stocks as the decisionalternative

• An advantage of the Hurwicz criterion is thatit allows the decision-maker the latitude to explorevarious weights of optimism

• A decision-maker’s outlook might change fromscenario to scenario and from day to day

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Decision Table

• In this case, if we had been fairly pessimisticand chosen an of 0.2, the result would have been

stocks ($2200)(.2) + (-$500)(.8) = $40bonds ($900)(.2) + (-$100)(.8) = $100CD’s ($750)(.2) + ($300)(.8) = $390mixture ($1300)(.2) + (-$200)(.8) = $100

• Under this scenario, the decision-maker wouldchoose the CD option because it yielded thehighest weighted payoff ($390) with = 0.2

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Decision Alternatives for Various Values of

stocks bonds CD’s mixture 1- max 2200 max 900 max 750 max 1300

min -500min -100min 300 min -200

.0 1 -500 -100 300 -200

.1 .9 -230 0 345 -50

.2 .8 40 100 390 100

.3 .7 310 200 435 250

.4 .6 580 300 480 400

.5 .5 850 400 525 550

.6 .4 1120 500 570 700

.7 .3 1390 600 615 850

.8 .2 1660 700 660 1000

.9 .1 1930 800 705 11501 0 2200 900 750 1300

Bold indicates the choice given for the value

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Graph of Hurwicz Criterion for Selected Values of

$2300 2100 1900 1700 1500 1300 1100 900 700 500 300 100 -100 -300 -500

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1

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Decision Table

stocks weighted payoff = CD’s weighted payoff$2200() + (-$500)(1 - ) = 750() + 300(1 - )

2200 - 500 + 500 = 750 + 300 - 3002250 = 800 = 0.3555

At = 0.3555, both stocks and CD’s yield the samepayoff under the Hurwicz criterion. For values lessthan = 0.3555, CD’s are the chosen investment.Neither bonds nor mixture produce the optimumpayoff under the Hurwicz criterion for any value of.

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First State of Economy

Suppose the state of economy turns out to be stagnant. The optimal decision choice would be CDs, which pay off $300. Any other decision would lead to an opportunity loss. The opportunity loss for each decision alternative other than CDs can be calculated by subtracting he decision alternative payoff from $300.

Stocks $300-(-$500) = $800Bonds $300-(-$100) = $400CDs $300-($300) = $0Mixture $300-(-$200) = $500

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Second State of Economy

The opportunity losses for the slow-growth state of economy are calculated by subtracting each payoff from $700, because $700 is the maximum payoff that can be obtained under this state; any other payoff is an opportunity loss. These opportunity losses are:

Stocks $700-($700) = $0Bonds $700-($600) = $100CDs $700-($500) = $200Mixture $700-($650) = $50

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Third State of Economy

The opportunity losses for a rapid-growth state of economy are

Stocks $2200-($2200) = $0Bonds $2200-($900) = $1300CDs $2200-($750) = $1450Mixture $2200-($1300) = $900

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Minimax Regret

• The strategy of minimax regret is based on lostopportunity

• Lost opportunity occurs when a decision-makerloses out on some payoff or portion of a payoffbecause he or she chose the wrong decision alternative.

• In analyzing decision-making situations underuncertainty, an analyst can transform a decisiontable (payoff table) into an opportunity loss table,which can be used to apply the minimax regretcriterion.

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Opportunity Loss Table

State of the Economy

Stagnant Slow Rapid Growth Growth

Stocks $800 $0 $0

Investment Bonds $400 $100 $1300 DecisionAlternative CD’s $0 $200 $1450

Mixture $500 $50 $900

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Opportunity Loss Table

• In summary, the maximum regrets under eachdecision alternative are

stocks $800bonds $1300CD’s $1450mixture $900

• In making a decision based on a minimax regretdecision-maker examines the maximum regret under each decision alternative given and selectsthe minimum of these

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Opportunity Loss Table

• The result is the stocks option, which has theminimum regret of $800

• An investor who wants to minimize the maximumregret under the various states of the economywill choose to invest in stocks under the minimaxregret strategy

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Summary of Example Decision - Analysis Under Uncertainty

Decision Criteria

Maximin

Maximin

Hurqicz( = 0.7)

Minimax Regret

Investment Decision Alternative

Stocks

CDs

Stocks

Stocks

Payoff

$2200

$300

$1390

$800