![8
Decision States of Nature
Alternatives Large Rise Small Rise No Change Small Fall Large Fall
Gold -100 100 200 300 0
Bond 250 200 150 -100 -150
Stock 500 250 100 -200 -600
C/D account 60 60 60 60 60
Stock option 200 150 150 -200 -150
The Payoff Table
The Payoff Table
The states of nature are mutually
exclusive and collectively exhaustive.
Define the states of nature.
DJA is down more
than 800 points
DJA is down
[-300, -800]
DJA moves
within
[-300,+300]
DJA is up
[+300,+1000]
DJA is up more
than1000 points](https://image.slidesharecdn.com/ch06-240904052324-c63c751a/85/Decision-models-in-production-and-operations-ch06-ppt-8-320.jpg)
![98
Worksheet: [IGA.xls]Sheet1
Adjustable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$A$2 X1 0.388888889 0 0 4 6
$B$2 X2 0.5 0 0 4 2
$C$2 X3 0.111111111 0 0 1.5 2
$D$2 V -6.75062E-29 0 1 1E+30 1
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$E$4 -1.11022E-16 -0.333333333 0 0 1E+30
$E$5 6.75062E-29 0 0 0 1E+30
$E$6 3.88578E-16 -0.333333333 0 1E+30 0
$E$7 -2.77556E-16 -0.333333333 0 1E+30 0
$E$8 1 0 1 0.000199941 1E+30
IGA Optimal Solution - worksheet
IGA Optimal Solution - worksheet](https://image.slidesharecdn.com/ch06-240904052324-c63c751a/85/Decision-models-in-production-and-operations-ch06-ppt-98-320.jpg)
Decision models in production and operations ch06.ppt
- 1. 1 Decision Models Decision Models Chapter 6 Chapter 6
- 2. 2 6.1 6.1 Introduction to Decision Analysis Introduction to Decision Analysis • The field of decision analysis provides a 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 optimize the resulting payoff in terms of a decision criterion.
- 3. 3 • Maximizing the decision maker’s utility function is the mechanism used when risk is factored into the decision making process. • Maximizing expected profit is a common criterion when probabilities can be assessed. 6.1 6.1 Introduction to Decision Analysis Introduction to Decision Analysis
- 4. 4 6.2 6.2 Payoff Table Analysis 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 table entries are the payoffs.
- 5. 5 TOM BROWN INVESTMENT DECISION TOM BROWN INVESTMENT DECISION • Tom Brown has inherited $1000. • He has to decide how 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
- 6. 6 • The return on each investment depends on the (uncertain) market behavior during the year. • Tom would build a payoff table to help make the investment decision TOM BROWN TOM BROWN
- 7. 7 S1 S2 S3 S4 D1 p11 p12 p13 p14 D2 p21 p22 p23 P24 D3 p31 p32 p33 p34 • Select a decision making criterion, and apply it to the payoff table. TOM BROWN - Solution TOM BROWN - Solution S1 S2 S3 S4 D1 p11 p12 p13 p14 D2 p21 p22 p23 P24 D3 p31 p32 p33 p34 Criterion P1 P2 P3 • Construct a payoff table. • Identify the optimal decision. • Evaluate the solution.
- 8. 8 Decision States of Nature Alternatives Large Rise Small Rise No Change Small Fall Large Fall Gold -100 100 200 300 0 Bond 250 200 150 -100 -150 Stock 500 250 100 -200 -600 C/D account 60 60 60 60 60 Stock option 200 150 150 -200 -150 The Payoff Table The Payoff Table The states of nature are mutually exclusive and collectively exhaustive. Define the states of nature. DJA is down more than 800 points DJA is down [-300, -800] DJA moves within [-300,+300] DJA is up [+300,+1000] DJA is up more than1000 points
- 9. 9 Decision States of Nature Alternatives Large Rise Small Rise No Change Small Fall Large Fall Gold -100 100 200 300 0 Bond 250 200 150 -100 -150 Stock 500 250 100 -200 -600 C/D account 60 60 60 60 60 Stock option 200 150 150 -200 -150 The Payoff Table The Payoff Table Determine the set of possible decision alternatives.
- 10. 10 Decision States of Nature Alternatives Large Rise Small Rise No Change Small Fall Large Fall Gold -100 100 200 300 0 Bond 250 200 150 -100 -150 Stock 500 250 100 -200 -600 C/D account 60 60 60 60 60 Stock option 200 150 150 -200 -150 The stock option alternative is dominated by the bond alternative 250 200 150 -100 -150 -150 The Payoff Table The Payoff Table
- 11. 11 6.3 6.3 Decision Making Criteria 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.
- 12. 12 • The decision criteria are based on the decision maker’s attitude toward life. • The criteria include the – Maximin Criterion - pessimistic or conservative approach. – Minimax Regret Criterion - pessimistic or conservative approach. – Maximax Criterion - optimistic or aggressive approach. – Principle of Insufficient Reasoning – no information about the likelihood of the various states of nature. Decision Making Under Uncertainty Decision Making Under Uncertainty
- 13. 13 Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Maximin Criterion The Maximin Criterion
- 14. 14 • This criterion is based on the worst-case scenario. – It fits both a pessimistic and a conservative decision maker’s styles. – 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. Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Maximin Criterion The Maximin Criterion
- 15. 15 TOM BROWN - The Maximin Criterion TOM BROWN - The Maximin Criterion • 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 Large Rise Small rise No Change Small Fall Large Fall Payoff Gold -100 100 200 300 0 -100 Bond 250 200 150 -100 -150 -150 Stock 500 250 100 -200 -600 -600 C/D account 60 60 60 60 60 60 The Maximin Criterion Minimum Decisions Large Rise Small rise No Change Small Fall Large Fall Payoff Gold -100 100 200 300 0 -100 Bond 250 200 150 -100 -150 -150 Stock 500 250 100 -200 -600 -600 C/D account 60 60 60 60 60 60 The optimal decision
- 16. 16 =MAX(H4:H7) * FALSE is the range lookup argument in the VLOOKUP function in cell B11 since the values in column H are not in ascending order =VLOOKUP(MAX(H4:H7),H4:I7,2,FALSE ) =MIN(B4:F4) Drag to H7 The Maximin Criterion - spreadsheet The Maximin Criterion - spreadsheet
- 17. 17 To enable the spreadsheet to correctly identify the optimal maximin decision in cell B11, the labels for cells A4 through A7 are copied into cells I4 through I7 (note that column I in the spreadsheet is hidden). I4 Cell I4 (hidden)=A4 Drag to I7 The Maximin Criterion - spreadsheet The Maximin Criterion - spreadsheet
- 18. 18 Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Minimax Regret Criterion The Minimax Regret Criterion
- 19. 19 • The Minimax Regret Criterion – This criterion fits both a pessimistic and a conservative decision maker approach. – The payoff table is based on “lost opportunity,” or “regret.” – The decision maker incurs regret by failing to choose the “best” decision. Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Minimax Regret Criterion The Minimax Regret Criterion
- 20. 20 • The Minimax Regret Criterion – 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.” Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Minimax Regret Criterion The Minimax Regret Criterion
- 21. 21 The Payoff Table Decision Large rise Small rise No change Small fall Large 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 The Payoff Table Decision Large rise Small rise No change Small fall Large 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 TOM BROWN – Regret Table TOM BROWN – Regret Table Let us build the Regret Table The Regret Table Decision Large rise Small rise No change Small fall Large fall Gold 600 150 0 0 60 Bond 250 50 50 400 210 Stock 0 0 100 500 660 C/D 440 190 140 240 0 Investing in Stock generates no regret when the market exhibits a large rise
- 22. 22 The Payoff Table Decision Large rise Small rise No change Small fall Large 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 The Payoff Table Decision Large rise Small rise No change Small fall Large 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 The Regret Table Maximum Decision Large rise Small rise No change Small fall Large fall Regret Gold 600 150 0 0 60 600 Bond 250 50 50 400 210 400 Stock 0 0 100 500 660 660 C/D 440 190 140 240 0 440 The Regret Table Maximum Decision Large rise Small rise No change Small fall Large fall Regret Gold 600 150 0 0 60 600 Bond 250 50 50 400 210 400 Stock 0 0 100 500 660 660 C/D 440 190 140 240 0 440 Investing in gold generates a regret of 600 when the market exhibits a large rise The optimal decision 500 – (-100) = 600 TOM BROWN – Regret Table TOM BROWN – Regret Table
- 23. 23 The Minimax Regret - spreadsheet The Minimax Regret - spreadsheet =MAX(B$4:B$7)-B4 Drag to F16 =VLOOKUP(MIN(H13:H16),H13:I16,2,FALSE) =MIN(H13:H16) =MAX(B14:F14) Drag to H18 Cell I13 (hidden) =A13 Drag to I16
- 24. 24 • 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). Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Maximax Criterion The Maximax Criterion
- 25. 25 • 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. Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Maximax Criterion The Maximax Criterion
- 26. 26 TOM BROWN - TOM BROWN - The Maximax Criterion The Maximax Criterion The Maximax Criterion Maximum Decision Large rise Small rise No change Small fall Large fall Payoff Gold -100 100 200 300 0 300 Bond 250 200 150 -100 -150 200 Stock 500 250 100 -200 -600 500 C/D 60 60 60 60 60 60 The optimal decision
- 27. 27 • 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). Decision Making Under Uncertainty - Decision Making Under Uncertainty - The Principle of Insufficient Reason The Principle of Insufficient Reason
- 28. 28 TOM BROWN TOM BROWN - - Insufficient Reason Insufficient Reason • Sum of Payoffs – Gold 600 Dollars – Bond 350 Dollars – Stock 50 Dollars – C/D 300 Dollars • Based on this criterion the optimal decision alternative is to invest in gold.
- 29. 29 Decision Making Under Uncertainty – Decision Making Under Uncertainty – Spreadsheet template Spreadsheet template Payoff Table Large Rise Small Rise No Change Small Fall Large Fall Gold -100 100 200 300 0 Bond 250 200 150 -100 -150 Stock 500 250 100 -200 -600 C/D Account 60 60 60 60 60 d5 d6 d7 d8 Probability 0.2 0.3 0.3 0.1 0.1 Criteria Decision Payoff Maximin C/D Account 60 Minimax Regret Bond 400 Maximax Stock 500 Insufficient Reason Gold 100 EV Bond 130 EVPI 141 RESULTS
- 30. 30 Decision Making Under Risk Decision Making Under Risk • 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.
- 31. 31 Decision Making Under Risk – Decision Making Under Risk – The Expected Value Criterion The Expected Value Criterion Expected Payoff = (Probability)(Payoff) • For each decision calculate the expected payoff as follows: (The summation is calculated across all the states of nature) • Select the decision with the best expected payoff
- 32. 32 TOM BROWN - TOM BROWN - The Expected Value Criterion The Expected Value Criterion The Expected Value Criterion Expected Decision Large rise Small rise No change Small fall Large fall Value Gold -100 100 200 300 0 100 Bond 250 200 150 -100 -150 130 Stock 500 250 100 -200 -600 125 C/D 60 60 60 60 60 60 Prior Prob. 0.2 0.3 0.3 0.1 0.1 EV = (0.2)(250) + (0.3)(200) + (0.3)(150) + (0.1)(-100) + (0.1)(-150) = 130 The optimal decision
- 33. 33 • The expected value criterion is useful generally in two cases: – Long run planning is appropriate, and decision situations repeat themselves. – The decision maker is risk neutral. When to use the expected value When to use the expected value approach approach
- 34. 34 The Expected Value Criterion - The Expected Value Criterion - spreadsheet spreadsheet =SUMPRODUCT(B4:F4,$B$8:$F$8) Drag to G7 Cell H4 (hidden) = A4 Drag to H7 =MAX(G4:G7) =VLOOKUP(MAX(G4:G7),G4:H7,2,FALSE)
- 35. 35 6.4 6.4 Expected Value of Perfect Information 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)
- 36. 36 The Expected Value of Perfect Information Decision Large rise Small rise No change Small fall Large 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 Probab. 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,… TOM BROWN - TOM BROWN - EVPI EVPI
- 37. 37 The Expected Value of Perfect Information Decision Large rise Small rise No change Small fall Large 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 Probab. 0.2 0.3 0.3 0.1 0.1 -100 250 500 60 Expected Return with Perfect information = ERPI = 0.2(500)+0.3(250)+0.3(200)+0.1(300)+0.1(60) = $271 Expected Return without additional information = Expected Return of the EV criterion = $130 EVPI = ERPI - EREV = $271 - $130 = $141 TOM BROWN - TOM BROWN - EVPI EVPI
- 38. 38 6.5 6.5 Bayesian Analysis - Decision Making Bayesian Analysis - Decision Making with Imperfect Information with Imperfect Information • Bayesian Statistics 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.
- 39. 39 TOM BROWN – Using Sample Information TOM BROWN – Using Sample Information • Tom can purchase econometric forecast results for $50. • The forecast predicts “negative” or “positive” econometric growth. • Statistics regarding the forecast are: 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 predicted a “positive growth” 80% of the time. Should Tom purchase the Forecast ?
- 40. 40 • If the expected gain resulting from the decisions made with the forecast exceeds $50, Tom should purchase the forecast. The expected gain = Expected payoff with forecast – EREV • To find Expected payoff with forecast Tom should determine what to do when: – The forecast is “positive growth”, – The forecast is “negative growth”. TOM BROWN – Solution TOM BROWN – Solution Using Sample Information Using Sample Information
- 41. 41 • 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”) TOM BROWN – Solution TOM BROWN – Solution Using Sample Information Using Sample Information
- 42. 42 • Bayes’ Theorem provides a procedure to calculate these probabilities P(B|Ai)P(Ai) P(B|A1)P(A1)+ P(B|A2)P(A2)+…+ P(B|An)P(An) P(Ai|B) = Posterior Probabilities Probabilities determined after the additional info becomes available. TOM BROWN – Solution TOM BROWN – Solution Bayes’ Theorem Bayes’ Theorem Prior probabilities Probability estimates determined based on current info, before the new info becomes available.
- 43. 43 States of Prior Prob. Joint Posterior Nature Prob. (State|Positive) Prob. Prob. Large Rise 0.2 0.8 0.16 0.286 Small Rise 0.3 0.7 0.21 0.375 No Change 0.3 0.5 0.15 0.268 Small Fall 0.1 0.4 0.04 0.071 Large Fall 0.1 0 0 0.000 X = TOM BROWN – Solution TOM BROWN – Solution Bayes’ Theorem Bayes’ Theorem The Probability that the forecast is “positive” and the stock market shows “Large Rise”. • The tabular approach to calculating posterior probabilities for “positive” economical forecast
- 44. 44 States of Prior Prob. Joint Posterior Nature Prob. (State|Positive) Prob. Prob. Large Rise 0.2 0.8 0.16 0.286 Small Rise 0.3 0.7 0.21 0.375 No Change 0.3 0.5 0.15 0.268 Small Fall 0.1 0.4 0.04 0.071 Large Fall 0.1 0 0 0.000 X = 0.16 0.56 The probability that the stock market shows “Large Rise” given that the forecast is “positive” • The tabular approach to calculating posterior probabilities for “positive” economical forecast TOM BROWN – Solution TOM BROWN – Solution Bayes’ Theorem Bayes’ Theorem
- 45. 45 States of Prior Prob. Joint Posterior Nature Prob. (State|Positive) Prob. Prob. Large Rise 0.2 0.8 0.16 0.286 Small Rise 0.3 0.7 0.21 0.375 No Change 0.3 0.5 0.15 0.268 Small Fall 0.1 0.4 0.04 0.071 Large Fall 0.1 0 0 0.000 X = TOM BROWN – Solution TOM BROWN – Solution Bayes’ Theorem Bayes’ Theorem Observe the revision in the prior probabilities Probability(Forecast = positive) = .56 • The tabular approach to calculating posterior probabilities for “positive” economical forecast
- 46. 46 States of Prior Prob. Joint Posterior Nature Prob. (State|negative) Probab. Probab. Large Rise 0.2 0.2 0.04 0.091 Small Rise 0.3 0.3 0.09 0.205 No Change 0.3 0.5 0.15 0.341 Small Fall 0.1 0.6 0.06 0.136 Large Fall 0.1 1 0.1 0.227 TOM BROWN – Solution TOM BROWN – Solution Bayes’ Theorem Bayes’ Theorem Probability(Forecast = negative) = .44 • The tabular approach to calculating posterior probabilities for “negative” economical forecast
- 47. 47 Posterior (revised) Probabilities Posterior (revised) Probabilities spreadsheet template spreadsheet template Bayesian Analysis Indicator 1 Indicator 2 States Prior Conditional Joint Posterior States Prior Conditional Joint Posterior of Nature Probabilities Probabilities Probabilities Probabilites of Nature Probabilities Probabilities Probabilities Probabilites Large Rise 0.2 0.8 0.16 0.286 Large Rise 0.2 0.2 0.04 0.091 Small Rise 0.3 0.7 0.21 0.375 Small Rise 0.3 0.3 0.09 0.205 No Change 0.3 0.5 0.15 0.268 No Change 0.3 0.5 0.15 0.341 Small Fall 0.1 0.4 0.04 0.071 Small Fall 0.1 0.6 0.06 0.136 Large Fall 0.1 0 0 0.000 Large Fall 0.1 1 0.1 0.227 s6 0 0 0.000 s6 0 0 0.000 s7 0 0 0.000 s7 0 0 0.000 s8 0 0 0.000 s8 0 0 0.000 P(Indicator 1) 0.56 P(Indicator 2) 0.44
- 48. 48 • This is the expected gain from making decisions based on Sample Information. • Revise the expected return for each decision using the posterior probabilities as follows: Expected Value of Sample Expected Value of Sample Information Information EVSI EVSI
- 49. 49 The revised probabilities payoff table Decision Large rise Small rise No change Small fall Large 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 P(State|Positive) 0.286 0.375 0.268 0.071 0 P(State|negative) 0.091 0.205 0.341 0.136 0.227 EV(Invest in……. |“Positive” forecast) = =.286( )+.375( )+.268( )+.071( )+0( ) = EV(Invest in ……. | “Negative” forecast) = =.091( )+.205( )+.341( )+.136( )+.227( ) = -100 100 200 300 $84 0 GOLD -100 100 200 300 0 GOLD $120 TOM BROWN – Conditional Expected Values TOM BROWN – Conditional Expected Values
- 50. 50 • The revised expected values for each decision: Positive forecast Negative forecast EV(Gold|Positive) = 84 EV(Gold|Negative) = 120 EV(Bond|Positive) = 180 EV(Bond|Negative) = 65 EV(Stock|Positive) = 250 EV(Stock|Negative) = -37 EV(C/D|Positive) = 60 EV(C/D|Negative) = 60 If the forecast is “Positive” Invest in Stock. If the forecast is “Negative” Invest in Gold. TOM BROWN – Conditional Expected Values TOM BROWN – Conditional Expected Values
- 51. 51 • Since the forecast is unknown before it is purchased, Tom can only calculate the expected return from purchasing it. • Expected return when buying the forecast = ERSI = P(Forecast is positive)(EV(Stock|Forecast is positive)) + P(Forecast is negative”)(EV(Gold|Forecast is negative)) = (.56)(250) + (.44)(120) = $192.5 TOM BROWN – Conditional Expected Values TOM BROWN – Conditional Expected Values
- 52. 52 • The expected gain from buying the forecast is: EVSI = ERSI – EREV = 192.5 – 130 = $62.5 • Tom should purchase the forecast. His expected gain is greater than the forecast cost. • Efficiency = EVSI / EVPI = 63 / 141 = 0.45 Expected Value of Sampling Expected Value of Sampling Information (EVSI) Information (EVSI)
- 53. 53 TOM BROWN – Solution TOM BROWN – Solution EVSI spreadsheet template EVSI spreadsheet template Payoff Table Large Rise Small Rise No Change Small Fall Large Fall s6 s7 s8 EV(prior) EV(ind. 1) EV(ind. 2) Gold -100 100 200 300 0 100 83.93 120.45 Bond 250 200 150 -100 -150 130 179.46 67.05 Stock 500 250 100 -200 -600 125 249.11 -32.95 C/D Account 60 60 60 60 60 60 60.00 60.00 d5 d6 d7 d8 Prior Prob. 0.2 0.3 0.3 0.1 0.1 Ind. 1 Prob. 0.286 0.375 0.268 0.071 0.000 #### ### ## 0.56 Ind 2. Prob. 0.091 0.205 0.341 0.136 0.227 #### ### ## 0.44 Ind. 3 Prob. Ind 4 Prob. RESULTS Prior Ind. 1 Ind. 2 Ind. 3 Ind. 4 optimal payoff 130.00 249.11 120.45 0.00 0.00 optimal decision Bond Stock Gold EVSI = 62.5 EVPI = 141 Efficiency= 0.44
- 54. 54 6.6 6.6 Decision Trees Decision Trees • The Payoff Table approach is useful for a non- sequential or single stage. • Many real-world decision problems consists of a sequence of dependent decisions. • Decision Trees are useful in analyzing multi- stage decision processes.
- 55. 55 • A Decision Tree is a chronological representation of the decision process. • The tree is composed of nodes and branches. Characteristics of a decision tree Characteristics of a decision tree A branch emanating from a state of nature (chance) node corresponds to a particular state of nature, and includes the probability of this state of nature. Decision node Chance node Decision 1 Cost 1 Decision 2 Cost 2 P(S2) P(S1 ) P(S3 ) P(S2) P(S1 ) P(S3 ) A branch emanating from a decision node corresponds to a decision alternative. It includes a cost or benefit value.
- 56. 56 BILL GALLEN DEVELOPMENT COMPANY BILL GALLEN DEVELOPMENT COMPANY – BGD plans to do a commercial development on a property. – Relevant data • Asking price for the property is 300,000 dollars. • Construction cost is 500,000 dollars. • Selling price is approximated at 950,000 dollars. • Variance application costs 30,000 dollars in fees and expenses – There is only 40% chance that the variance will be approved. – If BGD purchases the property and the variance is denied, the property can be sold for a net return of 260,000 dollars. – A three month option on the property costs 20,000 dollars, which will allow BGD to apply for the variance.
- 57. 57 – A consultant can be hired for 5000 dollars. – The consultant will provide an opinion about the approval of the application • P (Consultant predicts approval | approval granted) = 0.70 • P (Consultant predicts denial | approval denied) = 0.80 • BGD wishes to determine the optimal strategy – Hire/ not hire the consultant now, – Other decisions that follow sequentially. BILL GALLEN DEVELOPMENT COMPANY BILL GALLEN DEVELOPMENT COMPANY
- 58. 58 BILL GALLEN - Solution BILL GALLEN - Solution • 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.
- 59. 59 BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree Let us consider the decision to not hire a consultant Do not hire consultant Hire consultant Cost = -5000 Cost = 0 Do nothing 0 Buy land -300,000 Purchase option -20,000 Apply for variance Apply for variance -30,000 -30,000 0 3
- 60. 60 Approved Denied 0.4 0.6 12 Approved Denied 0.4 0.6 -300,000 -500,000 950,000 Buy land Build Sell -50,000 100,000 -70,000 260,000 Sell Build Sell 950,000 -500,000 120,000 Buy land and apply for variance -300000 – 30000 + 260000 = -300000 – 30000 – 500000 + 950000 = Purchase option and apply for variance BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
- 61. 61 60 Do not hire consultant Hire consultant Cost = -5000 Cost = 0 Do nothing 0 Buy land -300,000 Purchase option -20,000 Apply for variance Apply for variance -30,000 -30,000 0 61 12 -300,000 -500,000 950,000 Buy land Build Sell -50,000 100,000 -70,000 260,000 Sell Build Sell 950,000 -500,000 120,000 Buy land and apply for variance -300000 – 30000 + 260000 = -300000 – 30000 – 500000 + 950000 = Purchase option and apply for variance This is where we are at this stage Let us consider the decision to hire a consultant BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree
- 62. 62 Do not hire consultant 0 Hire consultant -5000 Predict Approval P r e d i c t D e n i a l 0 . 4 0 . 6 -5000 Apply for variance Apply for variance Apply for variance Apply for variance -5000 -30,000 -30,000 -30,000 -30,000 BILL GALLEN – BILL GALLEN – The Decision Tree The Decision Tree Let us consider the decision to hire a consultant Done Do Nothing Buy land -300,000 Purchase option -20,000 Do Nothing Buy land -300,000 Purchase option -20,000
- 63. 63 BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree Approved Denied Consultant predicts an approval ? ? Build Sell 950,000 -500,000 260,000 Sell -75,000 115,000
- 64. 64 BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree Approved Denied ? ? Build Sell 950,000 -500,000 260,000 Sell -75,000 115,000 The consultant serves as a source for additional information about denial or approval of the variance.
- 65. 65 ? ? BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree Approved Denied Build Sell 950,000 -500,000 260,000 Sell -75,000 115,000 Therefore, at this point we need to calculate the posterior probabilities for the approval and denial of the variance application
- 66. 66 BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree 22 Approved Denied Build Sell 950,000 -500,000 260,000 Sell -75,000 27 25 115,000 23 24 26 The rest of the Decision Tree is built in a similar manner. Posterior Probability of (approval | consultant predicts approval) = 0.70 Posterior Probability of (denial | consultant predicts approval) = 0.30 ? ? .7 .3
- 67. 67 • 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 represents the optimal decision. The Decision Tree The Decision Tree Determining the Optimal Strategy Determining the Optimal Strategy
- 68. 68 22 Approved Denied 27 25 23 24 26 -75,000 115,000 115,000 -75,000 115,000 -75,000 115,000 -75,000 115,000 -75,000 22 115,000 -75,000 (115,000)(0.7)=80500 (-75,000)(0.3)= -22500 -22500 80500 80500 -22500 80500 -22500 58,000 ? ? 0.30 0.70 Build Sell 950,000 -500,000 260,000 Sell -75,000 115,000 With 58,000 as the chance node value, we continue backward to evaluate the previous nodes. BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree Determining the Optimal Strategy Determining the Optimal Strategy
- 69. 69 Predicts approval Hire Do nothing BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree Determining the Optimal Strategy Determining the Optimal Strategy .4 .6 $10,000 $58,000 $-5,000 $20,000 $20,000 Buy land; Apply for variance Predicts denial D e n i e d Build, Sell Sell land Do not hire $-75,000 $115,000 .7 .3 A p p r o v e d
- 70. 70 BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree Excel add-in: Tree Plan Excel add-in: Tree Plan
- 71. 71 BILL GALLEN - The Decision Tree BILL GALLEN - The Decision Tree Excel add-in: Tree Plan Excel add-in: Tree Plan
- 72. 72 6.7 6.7 Decision Making and Utility Decision Making and Utility • Introduction – The expected value criterion may not be appropriate if the decision is a one-time opportunity with substantial risks. – Decision makers do not always choose decisions based on the expected value criterion. • A lottery ticket has a negative net expected return. • Insurance policies cost more than the present value of the expected loss the insurance company pays to cover insured losses.
- 73. 73 • It is assumed that a decision maker can rank decisions in a coherent manner. • Utility values, U(V), reflect the decision maker’s perspective and attitude toward risk. • Each payoff is assigned a utility value. Higher payoffs get larger utility value. • The optimal decision is the one that maximizes the expected utility. The Utility Approach The Utility Approach
- 74. 74 • The technique provides an insightful look into the amount of risk the decision maker is willing to take. • The concept is based on the decision maker’s preference to taking a sure payoff versus participating in a lottery. Determining Utility Values Determining Utility Values
- 75. 75 • List every possible payoff in the payoff table in ascending order. • Assign a utility of 0 to the lowest value and a value of 1 to the highest value. • For all other possible payoffs (Rij) ask the decision maker the following question: Determining Utility Values Determining Utility Values Indifference approach for assigning utility values Indifference approach for assigning utility values
- 76. 76 • Suppose you are given the option to select one of the following two alternatives: – Receive $Rij (one of the payoff values) for sure, – Play a game of chance where you receive either • The highest payoff of $Rmax with probability p, or • The lowest payoff of $Rmin with probability 1- p. Determining Utility Values Determining Utility Values Indifference approach for assigning utility values Indifference approach for assigning utility values
- 77. 77 Rmin What value of p would make you indifferent between the two situations?” Determining Utility Values Determining Utility Values Indifference approach for assigning utility values Indifference approach for assigning utility values Rij Rmax p 1-p
- 78. 78 Rmin The answer to this question is the indifference probability for the payoff Rij and is used as the utility values of Rij. Determining Utility Values Determining Utility Values Indifference approach for assigning utility values Indifference approach for assigning utility values Rij Rmax p 1-p
- 79. 79 Determining Utility Values Determining Utility Values Indifference approach for assigning utility values Indifference approach for assigning utility values d1 d2 s1 s1 150 -50 140 100 Alternative 1 A sure event Alternative 2 (Game-of-chance) $100 $150 -50 p 1-p • For p = 1.0, you’ll prefer Alternative 2. • For p = 0.0, you’ll prefer Alternative 1. • Thus, for some p between 0.0 and 1.0 you’ll be indifferent between the alternatives. Example:
- 80. 80 Determining Utility Values Determining Utility Values Indifference approach for assigning utility values Indifference approach for assigning utility values d1 d2 s1 s1 150 -50 140 100 Alternative 1 A sure event Alternative 2 (Game-of-chance) $100 $150 -50 p 1-p • Let’s assume the probability of indifference is p = .7. U(100)=.7U(150)+.3U(-50) = .7(1) + .3(0) = .7
- 81. 81 TOM BROWN TOM BROWN - - Determining Utility Values Determining Utility Values • Data – The highest payoff was $500. Lowest payoff was -$600. – The indifference probabilities provided by Tom are – Tom wishes to determine his optimal investment Decision. Payoff -600 -200 -150 -100 0 60 100 150 200 250 300 500 Prob. 0 0.25 0.3 0.36 0.5 0.6 0.65 0.7 0.75 0.85 0.9 1
- 82. 82 TOM BROWN TOM BROWN – – Optimal decision (utility) Optimal decision (utility) Utility Analysis Certain Payoff Utility -600 0 Large Rise Small Rise No Change Small Fall Large Fall EU -200 0.25 Gold 0.36 0.65 0.75 0.9 0.5 0.632 -150 0.3 Bond 0.85 0.75 0.7 0.36 0.3 0.671 -100 0.36 Stock 1 0.85 0.65 0.25 0 0.675 0 0.5 C/D Account 0.6 0.6 0.6 0.6 0.6 0.6 60 0.6 d5 0 100 0.65 d6 0 150 0.7 d7 0 200 0.75 d8 0 250 0.85 Probability 0.2 0.3 0.3 0.1 0.1 300 0.9 500 1 RESULTS Criteria Decision Value Exp. Utility Stock 0.675
- 83. 83 Three types of Decision Makers Three types of Decision Makers • Risk Averse -Prefers a certain outcome to a chance outcome having the same expected value. • Risk Taking - Prefers a chance outcome to a certain outcome having the same expected value. • Risk Neutral - Is indifferent between a chance outcome and a certain outcome having the same expected value.
- 84. 84 Payoff Utility The Utility Curve for a Risk Averse Decision Maker 100 0.5 200 0.5 150 The utility of having $150 on hand… U(150) …is larger than the expected utility of a game whose expected value is also $150. EU(Game) U(100) U(200)
- 85. 85 Payoff Utility 100 0.5 200 0.5 150 U(150) EU(Game) U(100) U(200) A risk averse decision maker avoids the thrill of a game-of-chance, whose expected value is EV, if he can have EV on hand for sure. CE Furthermore, a risk averse decision maker is willing to pay a premium… …to buy himself (herself) out of the game-of-chance. The Utility Curve for a Risk Averse Decision Maker
- 86. 86 Risk Neutral Decision Maker Payoff Utility Risk Averse Decision Maker Risk Taking Decision Maker
- 87. 87 6.8 6.8 Game Theory Game Theory • Game theory can be used to determine optimal decisions 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.
- 88. 88 – By number of players • Two players - Chess • Multiplayer – Poker – By 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) – By sequence of moves • Sequential - each player gets a play in a given sequence. • Simultaneous - all players play simultaneously. Classification of Games Classification of Games
- 89. 89 IGA SUPERMARKET 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.
- 90. 90 • 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 Emphasis Meat Produce Grocery Bakery IGA's Meat 2 2 -8 6 Emphasis Produce -2 0 6 -4 Grocery 2 -7 1 -3 IGA SUPERMARKET IGA SUPERMARKET
- 91. 91 IGA needs to determine an advertising emphasis that will maximize its expected change in market share regardless of Sentry’s action.
- 92. 92 IGA SUPERMARKET - Solution IGA SUPERMARKET - Solution • To prevent a sure loss of market share, both IGA and Sentry should select the weekly emphasis randomly. • Thus, the question for both stores is: What proportion of the time each area should be emphasized by each store?
- 93. 93 IGA’s Linear Programming Model IGA’s Linear Programming Model • Decision variables – X1 = the probability IGA’s advertising focus is on meat. – X2 = 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 increase regardless of Sentry’s advertising policy.
- 94. 94 • Constraints – IGA’s market share increase for any given advertising focus selected by Sentry, must be at least V. • The model Max V S.T. Meat 2X1 – 2X2 + 2X3 V Produce 2X1 – 7X3 V Groceries -8X1 – 6X2 + X3 V Bakery 6X1 – 4X2 – 3X3 V Probability X1 + X2 + X3 = 1 IGA’s Perspective IGA’s Perspective IGA’s expected change in market share. Sentry’s advertising emphasis
- 95. 95 Sentry’s Linear Programming Model Sentry’s Linear Programming Model • Decision variables – Y1 = the probability Sentry’s advertising focus is on meat. – Y2 = the probability Sentry’s advertising focus is on produce. – Y 3 = the probability Sentry’s advertising focus is on groceries. – Y4 = the probability Sentry’s advertising focus is on bakery. • Objective Function For Sentry Minimize the changes in market share in favor of IGA
- 96. 96 • Constraints – Sentry’s market share decrease for any given advertising focus selected by IGA, must not exceed V. • The Model Min V S.T. 2Y1 + 2Y2 – 8Y3 + 6Y4 V -2Y1 + 6Y3 – 4Y4 V 2Y1 – 7Y2 + Y3 – 3Y4 V Y1 + Y2 + Y3 + Y4 = 1 Y1, Y2, Y3, Y4 are non-negative; V is unrestricted Sentry’s perspective Sentry’s perspective
- 97. 97 • For IGA – X1 = 0.3889; X2 = 0.5; X3 = 0.1111 • For Sentry – Y1 = .3333; Y2 = 0; Y3 = .3333; Y4 = .3333 • For both players V =0 (a fair game). IGA SUPERMARKET – Optimal Solution IGA SUPERMARKET – Optimal Solution
- 98. 98 Worksheet: [IGA.xls]Sheet1 Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $A$2 X1 0.388888889 0 0 4 6 $B$2 X2 0.5 0 0 4 2 $C$2 X3 0.111111111 0 0 1.5 2 $D$2 V -6.75062E-29 0 1 1E+30 1 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $E$4 -1.11022E-16 -0.333333333 0 0 1E+30 $E$5 6.75062E-29 0 0 0 1E+30 $E$6 3.88578E-16 -0.333333333 0 1E+30 0 $E$7 -2.77556E-16 -0.333333333 0 1E+30 0 $E$8 1 0 1 0.000199941 1E+30 IGA Optimal Solution - worksheet IGA Optimal Solution - worksheet
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