1. Example We want to determine the best real estate investment project given the following table of payoffs for three possible interest rate scenarios. Interest Rates ProjectsDeclineStableIncrease Office park0.51.74.5 Office Bldg1.51.92.4 Warehouse1.71.41.0 Shop. Ctr.0.72.43.6 Condos3.21.50.6

2. Maximax Criteria The most optimistic Decision Criteria. Step 1: take the Maximum of each row Step 2: take the Maximum of the column of maxima Interest Rates ProjectsDeclineStableIncrease Office park0.51.74.5 Office Bldg1.51.92.4 Warehouse1.71.41.0 Shop. Ctr.0.72.43.6 Condos3.21.50.6

3. Maximax Criteria The most optimistic Decision Criteria. Step 1: take the Maximum of each row Step 2: take the Maximum of the column of maxima Interest Rates ProjectsDeclineStableIncreaseMax Office park0.51.74.54.5 Office Bldg1.51.92.42.4 Warehouse1.71.41.01.7 Shop. Ctr.0.72.43.63.6 Condos3.21.50.63.2

4. Maximax Criteria The most optimistic Decision Criteria. Step 1: take the Maximum of each row Step 2: take the Maximum of the column of maxima Interest Rates ProjectsDeclineStableIncreaseMax Office park0.51.74.54.5 Office Bldg1.51.92.42.4 Warehouse1.71.41.01.7 Shop. Ctr.0.72.43.63.6 Condos3.21.50.63.2 Maximax Criteria says build the Office Park

5. Maximin Criteria The most pessimistic (but also most safe) Decision Criteria. Step 1: take the Minimum of each row Step 2: take the Maximum of the column of minima Interest Rates ProjectsDeclineStableIncrease Office park0.51.74.5 Office Bldg1.51.92.4 Warehouse1.71.41.0 Shop. Ctr.0.72.43.6 Condos3.21.50.6

6. Maximin Criteria The most pessimistic (but also most safe) Decision Criteria. Step 1: take the Minimum of each row Step 2: take the Maximum of the column of minima Interest Rates ProjectsDeclineStableIncreaseMin Office park0.51.74.50.5 Office Bldg1.51.92.41.5 Warehouse1.71.41.01.0 Shop. Ctr.0.72.43.60.7 Condos3.21.50.60.6

7. Maximin Criteria The most pessimistic Decision Criteria. Step 1: take the Minimum of each row Step 2: take the Maximum of the column of minima Interest Rates ProjectsDeclineStableIncreaseMin Office park0.51.74.50.5 Office Bldg1.51.92.41.5 Warehouse1.71.41.01.0 Shop. Ctr.0.72.43.60.7 Condos3.21.50.60.6 Maximin Criteria says build the Office Bldg

8. Equal likelihood Criteria With no forecast or probabilities telling us what to expect, we assume that every possibility has equal likelihood. Step 1: For each entry, multiply by 1 / (number of outcomes) Step 2: Sum the weighted values in each row Step 3: Take the Maximum of the expected values for each row Interest Rates ProjectsDeclineStableIncrease Office park0.51.74.5 Office Bldg1.51.92.4 Warehouse1.71.41.0 Shop. Ctr.0.72.43.6 Condos3.21.50.6

9. Equal likelihood Criteria With no forecast or probabilities telling us what to expect, we assume that every possibility has equal likelihood. Step 1: For each entry, multiply by 1 / (number of outcomes) Step 2: Sum the weighted values in each row Step 2: Take the Maximum of the expected values for each row Interest Rates ProjectsDeclineStableIncrease Exp. Office park0.5 x 1/3 +1.7 x 1/3 + 4.5 x 1/3 = 2.23 Office Bldg1.51.92.4 Warehouse1.71.41.0 Shop. Ctr.0.72.43.6 Condos3.21.50.6

10. Equal likelihood Criteria With no forecast or probabilities telling us what to expect, we assume that every possibility has equal likelihood. Step 1: For each entry, multiply by 1 / (number of outcomes) Step 2: Sum the weighted values in each row Step 2: Take the Maximum of the expected values for each row Interest Rates ProjectsDeclineStableIncrease Exp. Office park0.51.7 4.5 2.23 Office Bldg1.51.92.4 1.93 Warehouse1.71.41.0 1.37 Shop. Ctr.0.72.43.6 2.23 Condos3.21.50.6 1.77

11. Equal likelihood Criteria With no forecast or probabilities telling us what to expect, we assume that every possibility has equal likelihood. Step 1: For each entry, multiply by 1 / (number of outcomes) Step 2: Sum the weighted values in each row Step 2: Take the Maximum of the expected values for each row Interest Rates ProjectsDeclineStableIncrease Exp. Office park0.51.7 4.5 2.23 Office Bldg1.51.92.4 1.93 Warehouse1.71.41.0 1.37 Shop. Ctr.0.72.43.6 2.23 Condos3.21.50.6 1.77 Under Equal likelihood Criteria we will select either the Office Park or the Shopping center

12. Using Probabilities If we have probabilities for each state of nature we can use them to calculate an expected monetary value (EMV) also called just expected value (EV) This is just like taking a weighted sum to find your grade in a class (for example) This is also just like the Equal likelihood criterion, except the events may not have equal weight

13. Expected Value Assume for our previous example that there is a 50% chance of a decline, a 35% of stable rates, and a 15% chance of an increase. We compute the following expected values for each decision. Interest Rates ProjectsDeclineStableIncreaseEV Office park0.5 x.5 +1.7 x.35 +4.5 x.15 =1.52 Office Bldg1.51.92.41.775 Warehouse1.71.41.01.49 Shop. Ctr.0.72.43.61.73 Condos3.21.50.62.215 Using these probabilities we would select Condos

14. What would we do if we knew which scenario would come true? Pick the decision that has the highest value for that column! If we then weight these values by the corresponding probabilities and sum we get EVUC. Interest Rates ProjectsDeclineStableIncrease Office park0.51.74.5 Office Bldg1.51.92.4 Warehouse1.71.41.0 Shop. Ctr.0.72.43.6 Condos3.21.50.6 Expected Value Under Certainty

15. Pick the decision that has the highest value for that column! If we then weight these values by the corresponding probabilities and sum we get EVUC. Interest Rates ProjectsDeclineStableIncrease Office park0.51.74.5 Office Bldg1.51.92.4 Warehouse1.71.41.0 Shop. Ctr.0.72.43.6 Condos3.21.50.6 Expected Value Under Certainty EVUC = 3.2 x.5 + 2.4 x.35 + 4.5 x.15 = 3.115

16. Expected Value Under Certainty is not much good on its own, but can be used to compute the Expected Value of Perfect Information = EVPI We simply subtract what we get without perfect information (EMV) from EVUC In this case we have: EVPI = 3.115 – 2.215 = 0.9 Expected Value of Perfect Information

17. Using Excel OM Tools – Decision Tables

18. Using a Decision Tree Should the team go for two points (for the win) or one point (for the tie) with no time remaining? Two point success rate: 30% One point success rate: 98% Chances in overtime: 20% PayoffsWin: Sugar Bowl = 9.2M Lose: Gator Bowl = 1.5M

19. Making the tree Start at the initial decision and work through each possibility at the next step squares = your decisions, circles = nature decides 1 2 3 Go for 2 Go for 1

20. Making the tree Start at the initial decision and work through each possibility at the next step 1 2 3 Go for 2 Go for 1 + - W: $9.2M L: $1.5M

21. Making the tree Start at the initial decision and work through each possibility at the next step Next add probabilities to each branch from a circle 1 2 3 Go for 2 Go for 1 + - W: $9.2M L: $1.5M 4 Tie W: $9.2M L: $1.5M + - + -

22. Making the tree Next add probabilities to each branch from a circle 1 2 3 Go for 2 Go for 1 + - W: $9.2M L: $1.5M 4 Tie W: $9.2M L: $1.5M + - + -.30.70.98.02.2.8

23. Find Expected Values For each group of terminal nodes (leaves) multiply payoff by probability. Sum to get expected value for preceding node. 1 2 3 Go for 2 Go for 1 + - W: $9.2M L: $1.5M 4 Tie W: $9.2M L: $1.5M + - + -.30.70.98.02.2.8 9.2 x.3 + 1.5 x.7 = 3.81

24. Find Expected Values For each group of terminal nodes (leaves) multiply payoff by probability. Sum to get expected value for preceding node. 1 2 3 Go for 2 Go for 1 + - W: $9.2M L: $1.5M 4 Tie W: $9.2M L: $1.5M + - + -.30.70.98.02.2.8 3.81 3.04 3.009

25. Find Expected Values What must the probability of winning in overtime be to make them indifferent between going for one and going for two? 1 2 3 Go for 2 Go for 1 + - W: $9.2M L: $1.5M 4 Tie W: $9.2M L: $1.5M + - + -.30.70.98.02 p 1-p 3.81 7.7p + 1.5 7.55p + 1.5 7.55p + 1.5= 3.81 p = 0.306

26. Find Expected Values For each group of terminal nodes (leaves) multiply payoff by probability. Sum to get expected value for preceding node. 1 2 3 Go for 2 Go for 1 + - W: 1 L: 0 4 Tie W: 1 L: 0 + - + -.30.70.98.02.2.8 0.3 0.2 0.196