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Decision Analysis Decision-Making Under Uncertainty updated 10.24.01

SMU EMIS 5300/7300. NTU SY-521-N. Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow. Decision Analysis Decision-Making Under Uncertainty updated 10.24.01. Decision Making Under Uncertainty The decision-maker knows for sure which state

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Decision Analysis Decision-Making Under Uncertainty updated 10.24.01

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  1. SMU EMIS 5300/7300 NTU SY-521-N Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow Decision Analysis Decision-Making Under Uncertainty updated 10.24.01

  2. Decision Making Under Uncertainty • The decision-maker knows for sure which state • of nature will occur, and he or she makes the • decision based on the optimal payoff available • under that state. • It is unknown which states of nature will occur • and the probability of the likelihood of a state • occurring is also unknown • The decision-maker has virtually no information • regarding which state of nature will occur, and • he or she attempts to develop a strategy based • on payoffs

  3. Approaches to Decision Making Under Uncertainty • Maximax criterion • Maximin criterion • Hurwicz criterion • Minimax regret

  4. Maximax Criterion • An optimistic approach in which the • decision-maker acts based on a notion that the • best things will happen • The decision-maker isolates the maximum • payoff under each decision alternative and then • selects the decision alternative that produces the • highest of these maximum payoffs • The name maximax means selecting the • maximum overall payoff from the maximum • payoffs of each decision alternative

  5. Decision Table State of the Economy Stagnant Slow Rapid Growth Growth Maximum Stocks -$500 $700 $2200 $2200 Investment Bonds -$100 $600 $900 $900 Decision Alternative CD’s $300 $500 $750 $750 Mixture-$200 $650 $1300 $1300

  6. 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 selected • is the stock alternative, which is associated with • the $2200

  7. 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 payoffs • under each decision alternative and selects the • worst, or minimum, payoff that can occur under • that decision

  8. Decision Table State of the Economy Stagnant Slow Rapid Growth Growth Minimum Stocks -$500 $700 $2200 -$500 Investment Bonds -$100 $600 $900 -$100 Decision Alternative CD’s $300 $500 $750 $300 Mixture-$200 $650 $1300 -$200

  9. Decision Table • With the maximin criterion, the decision-maker • examines the minimum payoffs for each decision • alternative given in the last column and selects • the maximum of these values • Maximum of {-$500, -$100, $300, -$200} = $300 • The decision is to invest in CD’s because this • investment alternative yields the highest, or • maximum, payoff under the worst-case scenario

  10. 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 probability • of a Type I error), which lies between 0 and 1, is • selected as a weight of optimism • The nearer  is to 1, the more optimistic is the • decision-maker

  11. Hurwicz Criterion • The maximum payoff under each decision • alternative is multiplied by  and the minimum • payoff under each decision alternative is • multiplied by 1 -  • These weighted products are summed for each • decision alternative, resulting in a weighted value • for each decision alternative • The maximum weighted value is selected, and • the corresponding decision alternative is chosen

  12. 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

  13. Decision Table Suppose we are more optimistic than pessimistic and select  = 0.7 for the weight of optimism. The calculations of weighted values for each decision alternative are as follows: stocks ($2200)(.7) + (-$500)(.3) = $1390 bonds ($900)(.7) + (-$100)(.3) = $600 CD’s ($750)(.7) + ($300)(.3) = $615 mixture ($1300)(.7) + (-$200)(.3) = $850

  14. Decision Table • The Hurwicz criterion leads the decision-maker • to choose the maximum of these values, $1390 • The result under the Hurwicz criterion with •  = 0.7 is to choose stocks as the decision • alternative • An advantage of the Hurwicz criterion is that • it allows the decision-maker the latitude to explore • various weights of optimism • A decision-maker’s outlook might change from • scenario to scenario and from day to day

  15. Decision Table • In this case, if we had been fairly pessimistic • and chosen an  of 0.2, the result would have been • stocks ($2200)(.2) + (-$500)(.8) = $40 • bonds ($900)(.2) + (-$100)(.8) = $100 • CD’s ($750)(.2) + ($300)(.8) = $390 • mixture ($1300)(.2) + (-$200)(.8) = $100 • Under this scenario, the decision-maker would • choose the CD option because it yielded the • highest weighted payoff ($390) with  = 0.2

  16. Decision Alternatives for Various Values of  stocks bonds CD’s mixture  1-  max 2200 max 900 max 750 max 1300 min -500 min -100 min 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 1150 1 0 2200 900 750 1300 Bold indicates the choice given for the value 

  17. 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

  18. Decision Table stocks weighted payoff = CD’s weighted payoff $2200() + (-$500)(1 - ) = 750() + 300(1 - ) 2200 - 500 + 500 = 750 + 300 - 300 2250 = 800  = 0.3555 At  = 0.3555, both stocks and CD’s yield the same payoff under the Hurwicz criterion. For values less than  = 0.3555, CD’s are the chosen investment. Neither bonds nor mixture produce the optimum payoff under the Hurwicz criterion for any value of .

  19. 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) = $800 Bonds $300-(-$100) = $400 CDs $300-($300) = $0 Mixture $300-(-$200) = $500

  20. 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) = $0 Bonds $700-($600) = $100 CDs $700-($500) = $200 Mixture $700-($650) = $50

  21. Third State of Economy The opportunity losses for a rapid-growth state of economy are Stocks $2200-($2200) = $0 Bonds $2200-($900) = $1300 CDs $2200-($750) = $1450 Mixture $2200-($1300) = $900

  22. Minimax Regret • The strategy of minimax regret is based on lost • opportunity • Lost opportunity occurs when a decision-maker • loses out on some payoff or portion of a payoff • because he or she chose the wrong decision • alternative. • In analyzing decision-making situations under • uncertainty, an analyst can transform a decision • table (payoff table) into an opportunity loss table, • which can be used to apply the minimax regret • criterion.

  23. Opportunity Loss Table State of the Economy Stagnant Slow Rapid Growth Growth Stocks $800 $0 $0 Investment Bonds $400 $100 $1300 Decision Alternative CD’s $0 $200 $1450 Mixture$500 $50 $900

  24. Opportunity Loss Table • In summary, the maximum regrets under each • decision alternative are • stocks $800 • bonds $1300 • CD’s $1450 • mixture $900 • In making a decision based on a minimax regret • decision-maker examines the maximum regret • under each decision alternative given and selects • the minimum of these

  25. Opportunity Loss Table • The result is the stocks option, which has the • minimum regret of $800 • An investor who wants to minimize the maximum • regret under the various states of the economy • will choose to invest in stocks under the minimax • regret strategy

  26. Summary of Example Decision - Analysis Under Uncertainty Investment Decision Alternative Stocks CDs Stocks Stocks Decision Criteria Maximin Maximin Hurqicz(a = 0.7) Minimax Regret Payoff $2200 $300 $1390 $800

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