1. Chapter 3: DECISION ANALYSIS Part 2 1

2. 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 2 (Probability)(Payoff) Over States of Nature Expected Payoff = 

3. Decision Making Under Risk (cont.)  Select the decision with the best expected payoff 3

4. TOM BROWN - continued (0.2)(250) + (0.3)(200) + (0.3)(150) + (0.1)(-100) + (0.1)(-150) = 130 The Optimal decision 4

5.  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.) 5

6. 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 (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

7. 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 1 st state of nature)*(Probability of 1 st state of nature) + ….. +(best outcome of last state of nature)*(Probability of last state of nature) 7

8. TOM BROWN - continued 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, 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 = $141 8

9. 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). 9

10. Expected Value of Perfect Information (cont.) If Tom knows in advance the market would undergo His optimal decisionWith gain of payoff A large risestock500-250= $250 A small risestock250-200= $ 50 No changegold200-150= $ 50 A small fallgold300-(-100)=$400 A large fallC/D60-(-150)= 210 10 EVPI= 0.2(250) + 0.3(50) +0.3(50)+ 0.1(400)+ 0.1(210)= 141

11. 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. 11

12. TOM BROWN - continued  Tom can purchase econometric forecast results for $50.  The forecast predicts “negative” or “positive” econometric growth.  Statistics regarding the forecast. When the stock market showed a large rise the forecast was “positive growth” 80% of the time. 12

13. TOM BROWN - continued  P(forecast predicts “positive” | small rise in market) = 0.7  P(forecast predicts “ negative” | small rise in market) = 0.3 13 Should Tom purchase the Forecast ?

14. 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. 14

15. SOLUTION  To find Expected payoff with forecast Tom should determine what to do when:  The forecast is “positive growth”  The forecast is “negative growth” 15

16. 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”) 16

17. 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”) 17 Bayes’ Theorem provides a procedure to calculate these probabilities

18. 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) 18

19. 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. 19 Prior probabilities New information Posterior probabilities

20. Bayes’ Theorem(cont.) 20 P(B | A i )P(A i ) [ P(B | A 1 )P(A 1 )+ P(B | A 2 )P(A 2 )+…+ P(B | A n )P(A n ) ] P(A i | B) = Posterior probabilities Probabilities determined after the additional info becomes available Prior probabilities Probabilities estimated Determined based on Current info, before New info becomes available

21.  The tabular approach to calculating posterior probabilities for positive economical forecast 21 X Ai: large rise B: forecast positive P(Bi |Ai )P(Ai) P(forecast= Positive| large rise)P( large rise)

22. 22 X = 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”.

23.  The tabular approach to calculating posterior probabilities for “negative” ecnomical forecast  Probability (forecast= negative) = 0.44 23 X =

24. 24 WINQSB printout for the calculation of the Posterior probabilities

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36.  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 forecast as follows. EV(Invest in……. | “Positive” forecast) = =.286( )+.375( )+.268( )+.071( )+0( ) = EV(Invest in ……. | “Negative” forecast) = =.091( )+.205( )+.341( )+.136( )+.227( ) = Gold -1001002003000 $84 -100 100 200 300 0 0 0 0 $120 Bond 250 200 150 -100 -150 $180 $ 65 36 Expected Value of Sample Information EVSI

37.  The rest of the revised EV s are calculated in a similar manner. 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 = 130 Expected Value of Sample Information - Excel So, Should Tom purchase the Forecast ? 37

38.  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 38

39. 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. 39

40. 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) 40

41. Game Theory (cont.)  Sequence of Moves  Sequential - Each player gets a play in a given sequence.  Simultaneous - All players play simultaneously. 41