1. Decisions under uncertainty and risk
Decision Making
Decisions under uncertainty and risk

2. Example Problem Actions? Demands/Events?
A bookstore sells a particular book of tax laws for Rs It purchases the book for Rs.80 per copy. Since some of the tax laws change every year, the copies unsold at the end of the year become outdated and can be disposed off for Rs.30 each. According to past experience, the annual demand for this book is between 18 and 23 copies. Assuming that the order for this book can be placed only once during the year how many copies of the book should be purchased for the next year?
Actions?
Demands/Events?

3. Construction of Pay off matrix
Two possibilities
D  Q
D<Q
Profit = 20Q , When DQ
Profit = 70D-50Q (or 20D – 50(Q-D))when D<Q
P=100D+30(Q-D)-80Q

4. Payoff matrix for D  Q Event/Act A1=18 A2=19 20 21 22 23 E1:18 360

5. Payoff Matrix for D<Q
20D – 50(Q-D)
20*18 – 50(19-18) = 310
Event/Act
A1=18
A2=19 20 21 22 23
E1:18
360
310
E2:19

6. Complete Payoff matrix
Event/Act
A1=18
A2=19 20 21 22 23
E1:18
360
310
260
210
160
110
E2:19
380
330
280
230
180
400
350
300
250
420
370
320
440
390
460

7. Regret Matrix
Event/Act
A1=18
A2=19 20 21 22 23
E1:18 50
100
150
200
250
E2:19 40 60 80
The amount of payoff foregone by not following the optimal outcome

8. Summary of Decision Rules Under Conditions of Uncertainty
Maximax rule
Maximin rule
Minimax regret rule
Equal probability rule (Laplace rule)
Hurwicz Principle
Identify best outcome for each possible decision & choose decision with maximum payoff.
Identify worst outcome for each decision & choose decision with maximum worst payoff.
Determine worst potential regret associated with each decision, where potential regret with any decision & state of nature is the improvement in payoff the manager could have received had the decision been the best one when the state of nature actually occurred. Manager chooses decision with minimum worst potential regret.
Assume each state of nature is equally likely to occur & compute average payoff for each. Choose decision with highest average payoff.
The decision may fall somewhere between extreme pessimism and extreme optimism

9. Summary of Decision Rules Under Conditions of Uncertainty
Maximax rule
Maximin rule
Minimax regret rule
Equal probability rule
Hurwicz Principle
Max:
Min : criterian value = (Max. value)+(1-)Min value
A1:360, A2:380, A3:400, A4:420, A5:440, A6:460
A1:360, A2:310, A3:260, A4:210, A5:160, A6:110
A1:100, A2:80, A3:100, A4:150, A5:200, A6:250
A1=( )/6=360
A2=( )/6=368.3
A6 )/6=285

10. Decision making with risk

11. Payoff matrix Event/Act Prob A1=18 A2=19 20 21 22 23 E1:18 0.05 360
310
260
210
160
110
E2:19
0.1
380
330
280
230
180
0.3
400
350
300
250
0.4
420
370
320
440
390
460
Expected Payoff
376.5
386
374.5
335
288.5

12. Expected Regret value Event/Act Prob A1=18 A2=19 20 21 22 23 E1:18
0.05 50
100
150
200
250
E2:19
0.1
0.3 40
0.4 60 80 51
34.5 25
36.5 76
122.5

13. Multi stage decision making: decision tree

14. Example problem
An oil company has recently acquired rights in a certain area to conduct surveys and test drillings to lead to lifting oil if it is found in commercially exploitable quantities.
The area is considered to have good potential for finding oil in commercial quantities. At the outset, the company has the choice to conduct further geological test or to carry out a drilling programme immediately. On the known conditions, the company estimates that there is a 70:30 chance of further tests showing a success
Whether the tests show the possibility of utlimate success or not or even if no tests are undertaken at all, the company could still pursue its drilling programme or alternatively consider selling its rights to drill in the area. Thereafter, however if it carries out the drilling programme the likelihood of final success or failure is considered dependent on the foregoing stages

15. Thus,
If “successful” tests have been carried out, the expectation of success in drilling is given as 80:20
If the tests indicate “failure”, then the expectation of success in drilling is given as 20:80
If no tests have been carried out at all, the expectation of success in drilling is given as 55:45
Costs and revenues have been estimated for al possible outcomes and the net present value of each is given in the next slide

16. Costs and Revenues
Draw the decision tree diagram to represent the above information
Evaluate the tree and obtain the best course of action
Outcome
Net present value
Success
With Prior tests
100
Without prior tests
120
Failure
-50
-40
Sale of exploitation rights:
Prior tests show ‘success’ 65
Prior tests show ‘failure’ 15 45

17. -40 65
Failure (0.45)
-50
120
Sell
Failure (0.2)
Success (0.55)
Drill
100
Drill 2
Positive (0.7)
Success (0.8)
Test 3
Success (0.2)
100
Negative (0.3)
Drill 1
Sell
Failure (0.8)
-50
Sell 15 45

18. Evaluation of Decision Node 1
Alternative
Outcome
Prob
NPV
Expected value 1
Drill
Success
0.2
100 20
Failure
0.8
(50)
-40
(20) 2
Sell
1.0 15

19. Evaluation of Node 2 Alternative Outcome Prob NPV Expected value 1
Drill
Success
0.8
100 80
Failure
0.2
(50)
-10 70 2
Sell
1.0 65

20. Evaluation of Node 3 Alternative Outcome Prob NPV Expected value 1
Drill
Success
0.55
120 66
Failure
0.45
(40)
(18) 48 2
Test
Positive
0.7 70 49
Negative
0.3 15
4.5
53.5
Sell
1.0 43

21. -40 65
Failure (0.45)
-50
Sell 48
120
(0.2)
Success (0.55)
Drill 70
100
Drill 70
(0.8)
Positive (0.7)
Test
53.5
53.5
(0.2)
100
-20
Negative (0.3)
Drill 15
Sell
(0.8)
-50
Sell 15 45

22. Utility theory

23. An Example: You bet your what?
You just won $1,000,000

24. An Example: You bet your what?
You just won $1,000,000
BUT
You are offered a gamble:
Bet your $1,000, on a fair coin flip.
Heads: $3,000,000
Tails: $0.00
What should you do?

25. Problem Analysis Expected monetary gain = 0.5* $0 + 0.5* $ 3,000,000 =
$1,500,000
$1,500,000 > $ 1,000,000 !
Will you take the bet now?
How much do you need as a pay off?
Utility theory posits lotteries that result in indifference, and in taking the bet.
Let Sk be your current wealth. Let U(Sk) = 5;
U(Sk +3,000,000) = 10; U(Sk+1,000,000) = 8;

26. Utility Theory
Theory of utility postulates that a rational decision maker will always decide to maximize utility or expected utility
The expected utility of a risky alternative is defined as the aggregate of the products of the utility values of all its possible outcomes and their respective probabilities
For an alternative A, if there are two possible outcomes x1 and x2 with respective probabilities of  and (1- ), and that the respective utilities are Ux1 and Ux2, we have Expected utility of A.
EUA = Ux1 + (1- )Ux2