14 DECISION THEORY CHAPTER

Learning Objectives Decision Making Environment Decision Making Under Uncertainty Decision Making Under Risk Decision Matrix Expected Value of Perfect information Utility as a decision criterion Decision Trees Expected Value of Sample information

Decisions A decision is a choice amongst alternatives Steps in Decision Making List out possible alternatives / options available. These are called strategies. List out future events which can occur and over which the decision maker has no control. These are called States of Nature. Construct a payoff table. Evaluate options for the best payoff using various criteria.

Decision Making Environment Decision Making Under Conditions of Certainty. In such situations only one ‘state of nature’ exists. There is complete certainty about the future. Decision Making Under Conditions of Uncertainty. In such an environment, more than one state of nature exists but the decision maker has little knowledge about them and is unable to assign any probability for their occurrence. Decision Making Under risk. In such a situation, more than one state of nature exists, but the decision maker has sufficient knowledge and information to be able to assign probabilities to the occurrence of each state.

Decision Making Under Uncertainty - Example Consider the case of a fashion designer, who is planning her fall collection. She has three options open to her with regard to the length of skirts: Maxi skirts. Midi skirts Mini skirts. Future events related to demand as perceived by the fashion designer are High demand. The fashion is accepted by the masses. Medium demand. Only a segment of the customers accept the fashion. Low demand. The length of the skirt is not acceptable to most people.

The payoff table is given below: Alternative Strategies States of Nature High Demand Moderate Demand Low Demand Maxi Skirts Rs 5,00,000 Rs 3,20,000 - Rs 1,00,000 Midi Skirts Rs 7,50,000 Rs 3,00,000 - Rs 2,00,000 Mini Skirts Rs 4,00,000 Rs 1,50,000 - Rs 1,25,000 Decision is taken according to certain rules. The decision maker adopts a rule based on his personality and risk taking profile.

Alternative Strategies States of Nature High Demand Moderate Demand Low Demand Maxi Skirts Rs 5,00,000 Rs 3,20,000 - Rs 1,00,000 Midi Skirts Rs 7,50,000 Rs 3,00,000 - Rs 2,00,000 Mini Skirts Rs 4,00,000 Rs 1,50,000 - Rs 1,25,000 The Maximax Criterion /Rule of Optimism / best of the best Choose the maximum payoff for each strategy and choose the maximum of the maximums. Launch Midi skirts (Maximum pay off Rs 7,50,000)

Alternative Strategies States of Nature High Demand Moderate Demand Low Demand Maxi Skirts Rs 5,00,000 Rs 3,20,000 - Rs 1,00,000 Midi Skirts Rs 7,50,000 Rs 3,00,000 - Rs 2,00,000 Mini Skirts Rs 4,00,000 Rs 1,50,000 - Rs 1,25,000 The Maximin Criterion /Rule of Pessimism / best of the worst Choose the minimum payoff for each strategy and choose the maximum of the minimums. Launch Maxi skirts (Worst pay off Rs – 1,00,000)

Alternative Strategies States of Nature High Demand Moderate Demand Low Demand Maxi Skirts Rs 5,00,000 Rs 2,50,000 Rs 3,20,000 Rs 0 Rs 0 - Rs 1,00,000 Midi Skirts Rs 0 Rs 7,50,000 Rs 3,00,000 Rs 20,000 - Rs 2,00,000 Rs 1,00,000 Mini Skirts Rs 4,00,000 Rs 3,50,000 Rs 1,50,000 Rs 1,70,000 - Rs 1,25,000 Rs 25,000 The Minimax Regret Criterion /Savage’s Rule Take the highest value for each state of nature and subtract other values of payoffs for that state of nature from it. Make a new table Take the maximum regret for each alternative and select the minimum of the maximums. Launch Midi skirts.

Alternative Strategies States of Nature High Demand Moderate Demand Low Demand Maxi Skirts Rs 5,00,000 Rs 3,20,000 - Rs 1,00,000 Midi Skirts Rs 7,50,000 Rs 3,00,000 - Rs 2,00,000 Mini Skirts Rs 4,00,000 Rs 1,50,000 - Rs 1,25,000 The Realism Criterion / Hurwicz’s Rule. Payoff for Mini skirts Attempts a balance between optimism and pessimism. Decision maker chooses an index of optimism between 0 and 1. For each alternative If is 0.7 then payoff for maxi skirts = Payoff for Midi skirts Launch Midi Skirts as it has maximum payoff

Alternative Strategies States of Nature High Demand Moderate Demand Low Demand Maxi Skirts Rs 5,00,000 Rs 3,20,000 - Rs 1,00,000 Midi Skirts Rs 7,50,000 Rs 3,00,000 - Rs 2,00,000 Mini Skirts Rs 4,00,000 Rs 1,50,000 - Rs 1,25,000 Criterion of Insufficient Reason / Laplace’s Rule Assumes that there is equal likelihood of occurrence of all states of nature. The mean value for all alternatives is calculated and the alternative yielding the maximum value is selected. Maxi Skirts = 2,40,000; Midi Skirts = 2,83,333; Mini skirts = 1,41,667 Launch Midi skirts.

Decision Rules All decisions are correct as per the rule applied. Generally a decision maker applies a decision rule consistently. A study of his/her past behaviour can help to predict the decision that he / she will take in a particular situation. This is helpful if he/she is a competitor.

Decision Making Under Risk Assumes that the decision maker can list out the various states of nature and can also assign probabilities to the event of their occurrence. Most commonly used criterion for decision making under risk is the criterion of expected value or the Baye’s criterion. Expected payoff for each alternative is the sum of the weighted payoffs for that alternative where the weights are the probability values assigned to the occurrence of different states of nature by the decision maker.

Decision Making Under Risk Example Assume that the fashion designer has assigned probabilities to the occurrence of various states of nature. Alternative Strategies States of Nature High Demand Moderate Demand Low Demand Expected Value   Probability 0.6 0.3 0.1 Maxi Skirts Rs 5,00,000 Rs 3,20,000 - Rs 1,00,000 Rs 3,86,000 Midi Skirts Rs 7,50,000 Rs 3,00,000 - Rs 2,00,000 Rs 5,20,000 Mini Skirts Rs 4,00,000 Rs 1,50,000 - Rs 1,25,000 Rs 2,72,500

Decision Making Under Risk Different probabilities of occurrence may be assigned to different states of nature for different alternatives. Alternative Strategies States of Nature High Demand Moderate Demand Low Demand Expected Value   Probability 0.6 0.3 0.1 Maxi Skirts Rs 5,00,000 Rs 3,20,000 - Rs 1,00,000 Rs 3,86,000 0.5 0.2 Midi Skirts Rs 7,50,000 Rs 3,00,000 - Rs 2,00,000 Rs 3,75,000  0.4 Mini Skirts Rs 4,00,000 Rs 1,50,000 - Rs 1,25,000 Rs 1,67,500 

Example Raju is a milk vendor who supplies milk in a small colony. He buys milk from a local dairy farm at Rs 10 per litre and sells it for Rs 12 per litre. Any unsold milk at the end of the day has to be thrown away as Raju does not have any refrigerating facility and the milk curdles. Raju’s demand fluctuates between 31 and 36 litres a day. Past data on demand reveals Demand in litres 31 32 33 34 35 36 No of Days 15 20 30 10 5 Probability of Demand 0.15 0.20 0.30 0.10 0.05

Construct a payoff table (conditional profit table). Stock Policy Demand Expected Value 31 32 33 34 35 36 Probability 0.15 0.20 0.30 0.10 0.05 62 62.00 52 64 62.20 42 54 66 60.00 44 56 68 54.20 22 46 58 70 46.00 12 24 48 60 72 36.60 If Raju stocks 32 litres and demand is 31 litres, he will be able to sell 31 litres and make a profit of Rs 62. But he will have to throw away 1 litre and incur a loss of Rs 10. Net profit Rs 52. Use expected value criterion. Stock 32 litres.

Expected Value of Perfect Information If Raju knows the exact demand for any day he will buy only that much milk. Stock Policy Demand Expected Value 31 32 33 34 35 36 Probability 0.15 0.20 0.30 0.10 0.05 62 9.30 64 12.80 66 19.80 68 13.60 70 7.00 72 3.60 He can now earn an average profit of Rs 66.10. Without this information his profit was Rs 62.20. The value of perfect information is Rs 3.90.

Minimising Expected Loss Obsolescence Loss – Loss due to over stocking by way of having to throw away surplus items, or sell them at a discount. Opportunity Loss – Loss due to understocking and thus losing an opportunity to make more profit.

Loss table for Raju Stock Policy Demand Expected Value 31 32 33 34 35 36 Probability 0.15 0.20 0.30 0.10 0.05 2 4 6 8 10 4.1 3.9 20 6.1 30 11.9 40 20.1 50 29.5 If Raju stocks 32 litres and demand is 31 litres, he has to throw away 1 litre of milk and loses Rs 10. If demand is 33 litres, he loses Rs 2 of profit that he would have made if he had stocked 33 litres. Take least expected value of loss. He should stock 32 litres

Marginal Analysis Let p be the probability of selling an additional unit Then 1 – p is the probability that the unit will not be sold. Let MP be the profit of selling one additional unit (marginal profit) and ML be the loss if the unit is not sold (marginal loss). Expected profit is p(MP) and expected loss is (1 – p)(ML). It would be worthwhile stocking an additional unit as long as there is no loss; or

Applying this to Raju’s problem As long as the probability of selling an additional unit is 0.833 or more, an additional unit of milk can be stocked. Demand Probability that demand will be Probability that demand will be equal to or greater than 31 0.15 1.0 32 0.20 0.85 33 0.30 0.65 34 0.35 35 0.10 36 0.05 Raju should stock 32 litres as the probability of selling more is less than 0.833

Utility as a Decision Criterion Money is not entirely an objective measure of value. Money does not mean the same thing to everybody. One may argue that Rs 50 should have the same value for every one, but a pay raise of Rs 50 per month means little to someone earning Rs 40000 per month but means a great deal to another person who may be earning a meagre Rs 1000 per month. Each of us assign a particular value to money. The utility that we attach to it, that determines its value for us. The value varies according to the risk taking profile of the decision maker.

Risk Taking Profile To the risk avoider losses are much more important in terms of utility than gains. For instance if his losses increased from – Rs 8000 to – Rs 10000, the utility would reduce from -6 to -10 as can be seen from the above diagram. If his gains increased from Rs 8000 to Rs 10000 his utility would only increase from 9.5 to 10. For the risk taker, gains mean much more than losses. If his losses increased from – Rs 8000 to – Rs 10000 the utility would change from -9.5 to -10, whereas a gain from Rs 800 to Rs 10000 would increase his utility from 6 to 10. RISK AVOIDER RISK TAKER

Decision Trees The decision tree is a means of representing the sequential, multi-stage logic of a decision problem. It uses two symbols – a box to represent a decision node and a circle to represent a chance node. The outcomes emanating from chance nodes are the various states of nature.

Decision Tree - Example A company is introducing a new product. It may set up a commercial plant now or set up a pilot plant at present and set up a commercial plant later depending on the performance of the product. Present cost of setting up the pilot plant will be Rs 2 lakhs and the cost of setting up a commercial plant will be Rs 21 lakhs. If the commercial plant is set up three months later it will cost Rs 25 lakhs. The probability of the product giving a high yield during the pilot stage is 0.9 and that of giving a low yield is 0.1. If the product is introduced commercially without going through a pilot plant stage, it is likely to give a high yield of profits with a probability of 0.7. If the pilot plant shows a high yield, then the probability that the commercial plant will also give a high yield is 0.85; but if the pilot plant gives a low yield the probability that the commercial plant will give a high yield is only 0.1. The estimated profits from high yield at the commercial stage are Rs 122.5 lakhs and if the yield is low the company will suffer a loss of Rs 15.25 lakhs.

Cost of pilot plant Rs 2 Lakhs Cost of plant Rs 21 Lakhs Build Commercial Plant High 0.7 81.17 122.5 2 - 21 Low 0.3 - 15.25 High 0.85 60.17 122.5 Build Com Plant 101.84 6 76.84 - 25 76.84 Low 0.15 1 High 0.9 4 -15.25 67.16 Build Pilot Plant Stop 69.16 3 Stop - 2 67.16 Low 0.1 5 Build Com Plant High 0.1 Value at Node 7 Value at Node 2 Value at Node 6 Value at Node 3 Value at Node 1 Higher of the two branch values 67.16 Value at Node 4 Higher of the two branch values 76.84 122.5 Decisions Build a pilot plant If yield is high, build main plant else stop. - 1.47 Value at Node 5 Higher of the two branch values 7 - 25 Low 0.9 -15.25 -26.47 Cost of pilot plant Rs 2 Lakhs Net Value Rs 67.16 lakhs Cost of plant Rs 21 Lakhs Net Value Rs 60.17 lakhs Cost of plant Rs 25 Lakhs Net Value Rs 76.84 lakhs Cost of plant Rs 25 Lakhs Net Value Rs -26.47 lakhs

Expected Value of Sample Information The building of the pilot plant has provided us with additional information If commercial production of the product is started without going through the pilot plant stage, the expected value would be Rs 60.17 lakhs. The information provided by the pilot plant has increased this value to Rs 67.16 lakhs. The cost of the pilot plant is Rs 2 lakhs. Net increase in value is 69.16 – 60.17 = 8.99 lakhs. If the cost of the pilot plant is more than Rs 8.99 lakhs then it is not worth while to construct a pilot plant. The expected cost of sample information is Rs 8.99 lakhs.