3 Elements of Decision Theory and Decision Analysis
Elements of Decision Taking and Decision Analysis
l Introduction :
The basic task of manager is to take decision in business. The manager has certain information on the basis of which decisions are to be taken. A decision is defined as selection from two or more courses of action. The manager is uncertain about the actual consequences that will occur for each course of action being considered. The study of decision theory helps the decision maker (manager) to choose an optimal course of action among several alternatives where the outcome or consequences associated with an action is uncertain. Decision theory is concerned with the techniques for making decisions especially under condition of uncertainty and risk.
l Elements of Decision Taking
Before we discuss techniques (criteria) of decision making, it is necessary to define the following basic elements used in its study.
(1) Acts : A decision always involves a selection from two or more alternatives. These several alternatives are called acts. For example, an investor is faced with the problem of choosing one alternative out of three alternatives i bonds, stocks and mutual funds for investment. It means there are three acts out of which one act is to be chosen. Acts are generally denoted by A1, A2, A3 ….. An.
(2) States of Nature (or Events) : In every act, there are events or states of nature which are uncertain and beyond the control of the decision maker. In the above example, where the investor is faced with the problem of choosing one of the three alternatives for investment, the states of nature (or called the potential demand) for the product may be good, moderate or poor. The potential demand are called events or states of nature which are uncertain. The state of nature or events are denoted by S1, S2, S3 ….. S4.
(3) Pay-off Table (or Pay-offs Matrix) : Pay off is the gain or losses received by the decision maker from a given combination of course of action and event (or state of nature). Pay off is also known as outcome, consequence, gains or losses. A pay off table (or pay off matrix) consists of alternative course of action like A1, A2, A3 ….. An and various states of nature (or events) like S1, S2, S3 ……. Sn involved in every act.
The specimen of a pay off table or pay off matrix is shown below :
| States of Nature | Acts | ||
| A1 | A2 | A3 | |
| S1 | p11 | p12 | p13 |
| S2 | p21 | p22 | p23 |
| S3 | p31 | p32 | p33 |
In the above table, column represents acts, row represents events (or states of nature) and p11, p12, ….. p33 are the pay off for a given combination of act and event.
(4) Opportunity Loss Table (or Regret Matrix) : Opportunity loss is the loss incurred due to failure of not choosing the best possible course of action. It is the difference between the highest possible pay off for a state of nature and the actual pay off obtained for a particular action taken. In terms of formula.
Opportunity Loss or Regret for any act = Highest possible pay off for a state of nature – pay off of that course of action.
An opportunity loss table (or regret matrix) gives the opportunity loss values or regret for each combination of a course of action and state of nature.
A specimen of the opportunity loss table (or regret table) is shown below :
Opportunity loss (or Regret) Table
| States of Nature | Acts | ||
| A1 | A2 | A3 | |
| S1 | M1 – p11 | M1 – p12 | M1 – p12 |
| S2 | M2 – p21 | M2 – p21 | M2 – p21 |
| S3 | M3 – p31 | M3– p31 | M3 – p33 |
where Mi (i = 1, 2, 3) = Maximum profit for each state of nature pij = actual pay off of each act.
Decision Making Criteria
Decision Making are being classified in the following three categories:
(1) Decision Making under Certainity : In this environment only one state of nature exists. For each alternative there is one and only one value of the pay-off. Suppose an investor wish to invest Rs, 1,00,000 for 5 years period. The alternatives is to open a saving bank account in a bank giving 7.5% interest and the other alternative is to invest in PPF pay, at 8.5% interest. Since both investments are secured, then PPF will be the better option. It is easy to take decision under conditions of certainity.
(2) Decision Making under Uncertainity : Here more than one state of nature exist but the decision maker is not sure what event or state of natue will take place and he also lack sufficient knowledge to assign probability to various states of nature
(3) Decision Making under Risk : Here also, more than one state of nature or event exist but the decision maker has sufficient knowledge to assign probabilities to various states of nature.
Decision Making under Uncertainity
A decision process is said to be under the condition of uncertainity when the probabilities associated with the occurence of different events or states of nature are not known to the decision-maker. For decision making under uncertainity, the following different decision criteira are usually adopted :
(1) Maximax Criterion
(2) Maximin Criterion
(3) Minimax Regret Criterion
(4) Laplace Criteria
(5) Hurwictz Criterian
(1) Maximax Criterion : This criterion is based on extreme optimism. Under this crieterion, the decision maker find the maximum pay-offs for each act and then select the act that represents maximum of the maxima. In decision problem dealing with costs, the minimum for each act is considered and then the act that minimise the above minimum costs is selected. Thus the criterion used is minimin.
(2)Maximin Criterion : This criterion is based on extreme pessimism. It is also called Wald criterion named after Abraham Wald. Under this criteria the decision maker find the minimum pay off for each act and then select the act for which the minimum pay off is highest. In decision problem dealing with costs, maximum cost associated with each act is considered and that the act that minimises the maximum costs is chosen. Thus,the criteria used is the minimax crieterion.
(3) Minimum Regret Crietrion : This criterion was given by Leonard Savage and it is therefore called Savage criteria. This crieteria uses opportunity losses or regret matrix. Under this criterion, the pay off matrix is converted into opportunity losses (or regret) matrix. Thereafter, the decision maker find the maximum regret for each act and then selects the act for which maximum regret is minimum.
(4) Hurwicz Criterion : This criterion is the mixture of maximax and maximin criteria and that is why it is called criterion of realism. This criterion was given by Leonard Hurwicz and therefore called Hurwicz Criterion. In this criteria, it is assumed that decision maker possesses a specified degree of optimism represented by coefficient of optimism (α) 0 < α < 1 (α < 1) and coefficient of pessimism is obtained by subtracting α from 1 i.e. (1 – α) is the coefficient of pessismism. Under this crieterion, the decision maker find the maximum and minimum pay off for each act and estimate the coefficient of optimism denoted by α and coefficient of pessimism denoted by 1 – α. The expected value for each act is obtained by using the following formula.
Expected Value (Ai) = α × maximum pay off + (1 – α) minimum pay off
The decision maker select the act which has maximum expected value. The act selected is called optimal act.
(5) Laplace Criterion : This crieterion is based on the concept of equal likehood which means that the probabilities of different states of nature for a given act are all equal. This criteria was given by Semonde Laplace and therfore called Laplace Criterion. Under this criterian the decision maker determines the average pay off by using the formula :
Expected Value (Ai) = 1 (x + x + …… xn) where i = 1, 2, 3 …..
n12 where EV (Ai) = Expected Value for act Ai.n the number of states of nature or events.x1, x2 ….. xn are pay offs of each act corresponding to different states of nature.The decision maker select that act which has largest expected value. The act selected is called optimal act.Note : If the number of states of nature are 4, the probability of each state of nature is 1/4. Thus, the probability depends upon the number of states of nature.
The following example illustrate the above decision crieteria.
Example 1 : Suppose that a decision maker faced with three decision alternatives and four states of nature constructs the following pay off table :
Pay off Table (Rs. ‘000)
| States of Nature | Acts | |||
| A1 | A2 | A3 |
A4 | |
| S1 | 5 | 10 | 18 | 25 |
| S2 | 8 | 7 | 8 | 23 |
| S3 | 21 | 18 | 12 | 21 |
| S4 | 30 | 22 | 19 | 20 |
Determine the alternative (or act) to be chosen under :
(i) Maximax Criterion (ii) Maximin Criterion (iii) Minimax Regret Criterion (iv) Hurwicz Criteron (Assuming α = 0.8) and (v) Laplace Criterion
Solution : (i) Maximax Criterion : In this criterion the decision maker selects the alternative (act) which maximses the maximnum profits
Acts Maximum Pay offs
A1 30 – Maximax
1 A2 22
A3 19
A4 25
Thus, A1 is the optimal act
(ii) Maximin Criterion : In this criterion the decision maker selects the alternative (or act) which maximises the minimum pay off.
Acts Minimum Pay off
A1 5
A2 7
A3 8
A4 20 – Maximin
Thus, A4 is the optimal act.
(iii) Minimax Regret Criterion : In this criterion, the decision maker selects that alternative which minimises the maximum of the oppotunity loss. To apply this criteria we construct opportunity loss table.
Opportunity Loss Table
| States of Nature | Acts | |||
| A1 | A2 | A3 | A4 | |
| S1 | 25 – 5 = 20 | 25 – 10 =15 | 25 – 18 = 7 | 25 – 25 = 0 |
| S2 | 23 – 8 = 15 | 23 – 7 = 16 | 23 – 8 = 15 | 23 – 23 = 0 |
| S3 | 21 – 21 = 0 | 21 – 18 = 3 | 21 – 12 = 9 | 21 – 21 = 0 |
| S4 | 30 – 30 = 0 | 30 – 22 = 8 | 30 – 19 = 11 | 30 – 20 = 10 |
ActsMaximum Opportunity Loss
A120
A216
A315
A410* Minimax Regret
Thus, A4 is the optimal act. It minimises the maximum opportunity losses.
(iv) Hurwicz Criterion : (Given α = 0.8 and 1 – α = 1 – 0.8 = 0.2)
ActsMaximum Pay offMinimum Pay offExpected Value
A1 30 5 30 x 0.8 x 0.2 = 25
A2 22 7 22 x 0.8 + 7 x 0.2 = 19
A3 19 8 19 x 0.8 + 8 x 0.2 = 16.8
A4 25 20 25 x 0.8 + 20 x 0.2 = 24
Thus, according to Hurwicz criterion A1 act is selected because the expected value of A1 act is maximum.
(v) Laplace Criterion : In this criterion we assign equal probability to each state of nature
| States of Nature | Acts | ||||
| Probability | A1 | A2 | A3 | A4 | |
| S1 | 1/4 | 5 | 10 | 18 | 25 |
| S2 | 1/4 | 8 | 7 | 8 | 23 |
| S3 | 1/4 | 21 | 18 | 12 | 21 |
| S4 | 1/4 | 30 | 22 | 19 | 20 |
Expected Value (A1) = 1/4 [5 + 8 + 21 + 30) = 16
Expected Value (A2) = 1/4 [10 + 7 + 18 + 22) = 14.25
Expected Value (A3) = 1/4 [18 + 8 + 12 + 19] = 14.25 Expected Value (A4) = 1/4 [25 + 23 + 21 + 20] = 22.25
Thus, the decision maker should select act A4 for which it maximises the expected pay off.
Decision Making under Risk
A decision process is said to be under the condition of risk when the probablities associated with the occurence of different events or states of nature are known to the decision maker. For decision making under risk the following criteria are usually adopted :
(1) Expected Monetary Value (EMV) Criterion
(2) Expected Opportunity Loss (EOL) Criterion
(3) Expected Value of Perfect Information (EVPI)
(1) Expected Monetary Value (EMV) Criterion : Under this criterion, the decision maker maximise the expected monetary value of each act. This criterion requires the calculation of expected monetary value of each act which is obtained by multiplying the pay offs for the act by the assigned probabilities of various states of nature. The decision maker selects the act that yields the highest EMV. Symbolically,
EMV (Ai) = p1x11 + p2x21 + p3X31 + ….. px Xn1
where X11, X21 and X31 …. Xn1 denote pay off of each act for S1, S2, S3 ….. Sn states of nature and p1, p2, p3 ….. pn denote the probability of occurence of S1, S2, S3 …. Sn states of nature.
(2) Expected Opportunity Loss (EOL) Criterion : This criterion is an alternative to EMV criteria. Under this criterion the decision minimises the expected opportunity losses of each act. The EOL of an act is obtained by multiplying the opportunity loss for that act by the probabilities assigned to various states of nature. The decision maker selects the act with minimise EOL. Symbolically,
EOL (Ai ) = p1 L11 + p2L21 + p3 L31 + …. pn Ln1
where L11, L21 = L31 …… Ln1 denote the opportunity loss of each act for S1, S2 …..
Sn … states of nature and p1, p2 ….. pn denote the probability of occurence of S1, S2 …….
Sn states of nature.
Note : Decision using EMV and EOL criteria are the same.
(3) Expected Value of Perfect Information (EVPI) : Under this criterion it is assumed that the decision maker has authentic and perfect information about the future,With perfect information the decision maker would know in advance the demand for each day and will store the exact number as per demand.The expected value of pefect information (EVPI) is the difference between expected profit under perfect information (Certainity) and expected pay-off with uncertainity (or EMV of Best Act). Symbolically,EVPI = EPPI – EMV of Best Act Where EPPI = (Best pay-off the Ist state of nature x probability of Ist state of nature) + Best pay off for 2nd state of nature x probability of 2nd state of nature + ….. Best pay off for last state of nature x probability of last state of nature.
The following example illustrates the EMV, EOL and EVPI criteria.
Example : Pay-offs of three acts A, B and C and states of nature S1, S2, S3 are given below :
Pay off (in Rs.)
| State of Nature | Acts | ||
| A | B | C | |
| S1 | -20 | -50 | 200 |
| S2 | 200 | -100 | -50 |
| S3 | 400 | 600 | 300 |
The probabilities of the states of nature are 0.3, 0.4 and 0.3. Calculate the EMV and EOL for the data given and select the best act. Also find the expected value of perfect information (EVPI).
Solution (i) Calculation of EMV
Pay Off Table
| States of Nature | Probability | Acts | ||
| A | B | C | ||
| S1 | 0.3 | -20 | -50 | 200 |
| S2 | 0.4 | 200 | -100 | -50 |
| S3 | 0.3 | 400 | 600 | 300 |
The expected monetary value (EMV) for the acts A, B and C are calculated below :
EMV (A) = –20 × 0.3 + 200 × 0.4 + 400 × 0.3 = Rs. 194
EMV (B) = –50 × 0.3 + –100 × 0.4 + 600 × 0.3 = Rs. 125
EMV (C) = 200 × 0.3 + –50 × 0-.4 + 300 × 0.3 = Rs. 130
Since EMV of Act A is highest, Act A can be chosen as the best act.
(ii) Calculation of EOL
For calculating EOL, we first convert the pay-off matrix into opportunity losses (or regret) matrix
Opportunity Loss (or Regret) Table
The above data is re-written in the form of the following table :
The expected opportunity losses (EOL) for the acts A, B and C are calculated as :
EOL (A) = 220 x 0.3 + 0 x 0.4 + 0.3 x 200 = Rs. 126
EOL (B) = 250 x 0.3 + 300 x 0.4 + 0 x 0.3 = Rs. 195
EOL (C) = 0 x 0.3 + 250 x 0.4 + 300 x 0.3 = Rs. 190
Since EOl of act A is minimum, Act A can be chosen as best act.
Note : Decision using EMV and EOL criteria are the same
(iii) Calculation of EVPI
To calculate EVPI, first we calculate EPPI
Expected Profit with Perfect Information
EPPI = 200 x 0.3 + 200 x 0.4 + 600 x 0.3 = Rs. 320
EVPI = Expected Profit with Perfect Information – Expected Monetary Value of Best Act under Risk
- = EPPI – EMV of Best Act
- = 320 – 194 = Rs. 126
Example 3 : A newspaper boy has estimated the following probability of selling a magazine:
Cost of copy is Rs. 3 and sale price is Rs. 5. He cannot return unsold copies.
How many copies should he order to maximise probability ? Also calculate EVPI.
Solution : It is clear from the given problem that the newspaper to boy would not purchase less than 10 copies and more than 14 copies. We can calculate the profit values for each purchase action – event (demand) combination using the formula.
We are given Cost Price = Rs. 3, Selling Price = Rs. 5
Profit on sale = Rs. 5 – Rs. 3 = Rs. 2
Loss on unsold copies = Rs. 3
The conditional pay off (profit) in Rs. is given by :
Pay Off Table
Expected Monetary Value (EMV) can now be computed
EMV (10 copies) = 0.10 × 20 + 0.15 × 20 + 0.20 × 20 + 0.25 × 20 + 0.30 × 20 = Rs. 20
EMV (11 copies) = 0.10 × 17 + 0.15 × 22 + 0.20 × 22 + 0.25 × 22 + 0.30 × 22 = Rs 21.50
EMV (12 copies) = 0.10 × 14 × 0.15 × 19 + 0.20 × 24 + 0.25 × 24 + 0.30 × 24 = Rs. 22.25
EMV (13 copies) = 0.10 × 11 + 0.15 × 16 + 0.20 × 21 + 0.25 × 26 + 0.30 × 26 = Rs. 22
EMV (14 copies) = 0.10 × 8 + 0.15 × 13 + 0.20 × 18 + 0.25 × 23 + 0.30 × 28 = Rs. 20.50
From the above, we see that the highest value of EMV is Rs. 22.50 which corresponds to the purchase of 12 copies. Hence the newspaper boy must order 12 copies to earn the highest possible average profit of Rs. 22.25.
Calculation of EVPI
To calculate EVPI, we first calculate EPPI
Maximum Profit with Perfect Information
EPPI= 20 x 0.10 + 22 x 0.15 + 24 x 0.20 + 26 x 0.25 + 28 x 0.30
= 2 + 33 + 4.8 + 6.5 + 8.4 = Rs. 25
EVPI= EPPI – EMV of Best Act= 25 – 22.25 = Rs. 2.75
l Decision Tree Technique
A decision tree is a graphic representation of the decison process indicating decision acts, states of nature, probabilities attached to the states of nature and conditional profit and losses. The decision tree consists of nodes and branches. The nodes are of two types, decision nodes and chance node q square indicates a decision node. There are number of branches leading from this square. These branches indicate various course of action available to the decision maker. At the end of each branch there is a m circle which represents chance node. From these chance nodes, chance events emanate in the form of sub-branches. The respective pay-offs and probabilities associated with alternate course of action and the chance events are shown along the sub-branches. At the terminal of sub branches are shown the expected values of the outcomes. Following diagram gives a simplified view of the structure of the decision tree :
Outcomes
Speciment of Decision Tree
A decision tree is highy useful to a decision maker to reach at the optimal act. Using Roll Back Technique, from forward to the backward, we are able to eliminate unprofitable branches and determine optimal decision.
Example 4 : A company is evaluating three alternative investment opportunities whose returns are based on the state of economy. The possible state of the economy and the associated probabilities is as follows :
State of Economy Excellent Good Fair
Probability 0.2 0.3 0.5
The return for each investment opportunity and each state of the economy are as follows:
Pay of Table (in ‘000 Rs.)
Using the decision-tree approach, which alternative investment proposal would you recommend if EMV criteria is to be used.
Since node 1 has highest EMV, so the company will opt for alternative A1 Summary
Decision theory is concerned with choosing an optimal course of action from among several alternatives. Decisions are made under uncertainity and risk. Among the criteria for taking decision under uncertainity are included maximax, maximin, minimax, Hurwicz and Laplace criteria. For taking decisions under risk, the criteria included are EMV and EOL criteria. In addition, the expected value of perfect information (EVPI) is also calculated. A decision problem can also be depicted and solved using decision tree approach ( a graphical represenation). In decision tree approach, decision is taken by computing EMV.
Few important sources to learn more on decision theory.
- Richard I. Levin, David S. Rubin, Statistics for Management, Prentice Hall of India Private Ltd., New Delhi.
- R.S. Bhardwaj, Business Statistics, Excel Books, New Delhi
- U.K. Srivastava, G.V. Shenoy and S.C. Sharma, Quantitative Techniques of Management Decision, New Age International, New Delhi.
- N.D. Vohra, Quantitative Techniques for Management, Tata McGraw Hill
- J.K. Sharma, Business Statistics, Pearson Education
- H.A. Taha, Operations Research, Pearson Publications.
- Hira and Gupta, Operations Research, Sultan Chand, New Delhi.