Decision Theory NEW - notes PDF

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DECISION THEORY The problem of statistical decision theory is that given a situation where there are several available alternative courses of action, each of which may lead to a set of mutually exclusive outcomes associated with certain probabilities, which course of action should a decision maker take? Structure of Decision Making Problem: 1. The Decision maker: The decision maker refers to the individual or a group of individuals responsible for making a choice of an appropriate course of action from the available courses of action. 2. Courses of action: Courses of action are also called actions, alternatives, acts or strategies. 3. States of nature: States of nature are sometimes called outcomes or events. The decision maker must develop an exhaustive list of all possible future events. However decision maker has no control over the occurrence of specific event. 4. Pay off: Each combination of a course of action and an event is associated with a payoff, which measures the net benefits to the decision maker that occurs from a given combination of decision alternatives and events. 5. Payoff table: Courses action

of A1

A2

…..

An

Events E1 P11 P12 P1n … Em Pm1 …. Pmn 6. Regret ( Opportunity loss): The opportunity loss has been defined to be the difference between the highest possible profit for an event (state of nature) and the actual profit obtained for the particular action taken, i.e, an opportunity loss is the loss incurred because of failure to take the best possible action. Types of decision making environment: 1. Decision making Under the conditions of Certainty: Here only one state of nature exist 2. Decision making under the conditions of Uncertainty 3. Decision making under the conditions of Risk

Decision making under the conditions of uncertainty: 1. 2. 3. 4. 5.

Maximax(Optimistic) Maximin(Pessimistic) Minimax Regret Criterion Laplace Criterion Hurwicz Criterion (Criterion of Realism)

Example: The following matrix gives the payoff of different strategies S1, S2 and S3 against conditions N1,N2, N3,N4 N1 N2 N3 S1 4000 -100 6000 S2 20000 5000 400 S3 20000 15000 -2000 Indicate the decision taken under the following approaches. i. ii. iii. iv. v.

N4 18000 0 1000

Pessimistic Optimistic Equal probability Regret Hurwicz criterion, his degree of optimism being .7

SOLUTION: 1. Pessimistic: Let us consider the following table

S1 S2 S3

Minimum Payoff -100 0 -2000

Maximin 0

Decision S2

Maximax 20000

Decision S2 or S3

2. Optimistic: Let us consider the following table

S1

Maximum Payoff 18000

S2 20000 S3 20000 4. Regret The following table represents the regrets for every event and for each alternative calculated by the expression Ith regret=(max payoff-ith pay off) for the jth event Let us consider the following table

S1 S2 S3

N1 16000 0 0

Regret N2 N3 15100 0 10000 5600 0 8000

MAX Regret N4 0 18000 17000

Decision (Minmax Regret) S1

16000 18000 17000

3. Equal Probability:

Expected Payoff

¼(4000100+6000+18000)=6975 ¼(20000+5000+400+0)=635 0 8500

S1 S2 S3

Max. Expected payoff 8500

Decision

S3

5. Hurwicz criterion, his degree of optimism being .7 For the given payoff matrix the minimum and maximum payoffs for each alternative are given below. Alternative

Max Payoff

Min Payoff

S1

18000

-100

S2 S3

20000 20000

0 -2000

Exp. Payoff=α.max payoff+(1-α) min payoff . 7*18000-.3*100= 12570 14000 13400

Max. expected payoff

Decision

14000

S2

Solution: For the given payoff matrix, the values corresponding to the pessimistic, optimistic equal probability criteria are given below in the following payoff table

S1

S2 S3 Decision

Pessimistic(min payoff) -100

Optimistic(Max. Payoff) 18000

0 -2000 Maximin “0” S2

20000 20000 Maximax 20000 S2, S3

Equal prob. ¼(4000100+6000+18000) = 6350 8500 Max Payoff 8500 S3

IV. The following table represents the regrets for every event and for each alternative calculated by the expression Ith regret = (max payoff-ith payoff) for the jth event

S1 S2 S3

N1 Regret 16000 0 0

N2 Regret 15100 10000 0

N3 Regret 0 5600 8000

N4 Regret 0 18000 17000

Max Regret 16000 18000 17000

The decision alternative S1 would be chosen since it corresponds to the minimal of the maximum possible regrets. V. Hurwicz criterion, his degree of optimism being .7 For the given payoff matrix the minimum and the maximum payoff for each alternative are given below Alternative

Max payoff

Min payoff

S1

18000

-100

α .max+( 1−α ¿ Payoff= .min .7*18000-.3*100=12570

S2 S3

20000 20000

0 -2000

14000 13400

Thus under Hurwicz rule S2 should be chosen as it is associated with the highest payoff 14000.

2. Decision Making Under the Conditions of Risk: The decision making under risk is a probabilistic decision situation. Several possible states of nature may occur, each with a given probability. One of the popular method of making decision under risk is, selective the alternative with the highest expected monetary value. We will also look at the concepts of perfect information and opportunity loss. i.

Expected Monetary Value: The expected value or the mean value is the long run average value that would result if the decision were repeated a large number of times. EMV (alternative i)= ( Pay off of the first state of nature)*( probability of second state of nature)+………………….+ ( Pay off of the last state of nature)*( probability of last state of nature)

ii.

Expected value of perfect information:

EVPI= (Best pay off of the first state of nature)*(probability of second state of nature)+ ………………….+ ( Best pay off of the last state of nature)*( probability of last state of nature)max(EMV) iii.

Expected Opportunity loss:

An alternative approach to maximize EMV is to minimize expected opportunity loss. Opportunity loss, sometimes called regret refers to the difference between the optimal profit or payoff for the given state of nature and actual pay off received. iv.

Use of marginal analysis According to the rule, any additional unit is purchased will be either sold or remain unsold. If P denotes the probability of selling one additional unit then ( 1-P) must be the probability of not selling it. If the additional unit is sold the conditional profit will increase as a result of the profit earned from this unit.. this is termed as incremental (marginal) profit “MP”. If the additional unit is not sold, the conditional profit is reduced and the amount of reduction is called the incremental loss ML.

The expected marginal profit will be P*MP and expected marginal loss will be (1P)*ML. Thus the units should stocked up to the point such that P*MP≥ (1-P)*ML Or, P≥ML/(ML+MP)

EXAMPLE: A newspaper boy has the following probabilities of selling magazine: No. of 10 11 12 13 14 copies sold Prob. .10 .15 .20 .25 .30 Cost of a copy is 30 paisa and sale price is 50 paisa. He cannot return unsold copies. How many copies should he order?? EMV: If CP denotes the conditional profit, S the quantity in stock and D the demand then. CP=⌈

20 S when S ≤ D ⌉ 20 D−( S−D ) 30 when S > D

The resulting payoff is given below Possible Possible Stock probability Demand 10 11 12 13 14 10 .10 200 170 140 110 80 11 .15 200 220 190 160 130 12 .20 200 220 240 210 180 13 .25 200 220 240 260 230 14 .30 200 220 240 260 280 Now the expected value of each decision alternative is obtained by multiplying its conditional profit by the associated probability and adding the resulting value. Possible Demand

probability

10

11

Possible Stock 12

13

14

10

.10

200*.10=20

17

14

11

8

11

.15

200*.15=30

33

28.5

24

19.5

12 13 14

.20 .25 .30

40 50 60

44 55 66

48 60 72

42 65 78

36 57.5 84

200

Total

215

222.5

220

205

DECISION: Stock 12 newspaper

EVPI: EVPI= EPPI- MAX EMV=(200*.10+220*.15+240*.20+260*.25+280*.30)-222.5=27.5 EOL: If CP denotes the conditional profit, S the quantity in stock and D the demand then. CP=⌈

20 S when S ≤ D ⌉ 20 D−( S−D ) 30 when S > D

The resulting payoff is given below Possible Demand 10 11 12 13 14

probability .10 .15 .20 .25 .30

10 200 200 200 200 200

11 170 220 220 220 220

Possible Stock 12 140 190 240 240 240

13 110 160 210 260 260

14 80 130 180 230 280

11 30 0 20 40 60

Possible Stock 12 60 30 0 20 40

13 90 60 30 0 20

14 120 90 60 30 0

13 9 9 6

14 12 13.5 12

Conditional loss table: Possible Demand 10 11 12 13 14

probability .10 .15 .20 .25 .30

10 0 20 40 60 80

EOL values for various stock actions are computed in the following table Possible Demand 10 11 12

probability .10 .15 .20

10 0 20*.15=3 8

11 3 0 4

Possible Stock 12 6 4.5 0

13 14 TOTAL

.25 .30

15 24 50

10 18 35

5 12 27.5

0 6 30

7.5 0 45

DECISION: Stock 12 newspaper

Marginal Analysis: Here MP= (50-30)=20 paisa ML= 30 paisa Let p denotes the probability of selling one additional unit, then (1-p) must be the probability of not selling it. The expected marginal profit will be pMP and expected marginal loss will be (1-p)ML. Thus the units should be stocked up to the point such that PMP≥ (1-P)ML Or,PMP+PML≥ML Or,P(MP+ML)≥ML Or,P≥ ML/(MP+ML) Or, P≥30/(20+30) Or, P≥.60 The value of .60 implies that in order to justify the stocking of an additional unit there must be at least .60 cumulative probability of selling that unit. The cumulative probability of sales are computed in the following table Possible Demand

Probabilit y

10 11 12 13 14 EOL

.10 .15 .20 .25 .30

Cu. Probability 1.00 .90 .75 .55 .30

Since P is less then after stocking 12 unit, so he should stock 12 units of newspaper...