Decisions under uncertainty and risk Decision Making Decisions under uncertainty and risk

Example Problem Actions? Demands/Events? A bookstore sells a particular book of tax laws for Rs.100. 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 Since some of the tax laws change every year, the copies unsold at the end of the year become outdated and can be disposed off for Rs.30 each.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?

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

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

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

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

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

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

Summary of Decision Rules Under Conditions of Uncertainty Maximax rule Maximin rule Minimax regret rule Equal probability rule Hurwicz Principle Max: 360 380 400 420 440 460 Min : 360 310 260 210 160 110 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=(360+360+360+360+360+360)/6=360 A2=(310+380+380+380+380+380)/6=368.3 A6110+180+250+320+390+460)/6=285

Decision making with risk

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

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

Multi stage decision making: decision tree

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

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

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

-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

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

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

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

-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

Utility theory

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

An Example: You bet your what? You just won $1,000,000 BUT You are offered a gamble: Bet your $1,000,000.00 on a fair coin flip. Heads: $3,000,000 Tails: $0.00 What should you do?

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;

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