MCom I Semester Statistical analysis Decision Theory Study Material Notes ( Part 3 )
Table of Contents
MCom I Semester Statistical analysis Decision Theory Study Material Notes ( Part 3 ) : Decision Making under Uncertainty When Event Probabilities are Not Knows Opportunity Loss Table Miscellaneous Illustration Computation of Expected Cost Statistical decision Theory at a Glance Examination Questions Long Answer Questions Short Answer Questions Objective Questions :
MCom I Semester Statistical Analysis Test Significance Large Samples Study Material Notes
(II) Decision-making under Uncertainty When
Event-Probabilities are Not Known If the event probabilities are not known then we cannot use the EMV or EOL criterion. In such cases, the choice of a course of action is very largely depend on the personality of the decision-maker and the policy of an organization. When event-probabilities are not known then following criterion are mainly used for decision-making :
(1) Maximin Criterion or Criterion of Pessimism;
(2) Maximax Criterion or Criterion of Optimism;
(3) Minimax Regret Criterion;
(4) Hurwitz Criterion or Criterion of Balance;
(5) Laplace Criterion or Criterion of Rationality.
Decision Theory Study Material
(1) The Criterion of Pessimism or Maximin : This is called ‘Waldian Criterion’ because it was suggested by Arabham Wald. This is also known as the criterion of pessimism. Under this rule the decision-maker is completely pessimistic. He assumes that the situation will always be disadvantageous. Using this approach, the minimum pay-offs resulting from adoption of various strategies are considered and among these values, the maximum one is selected. It involves, therefore, choosing the best (the maximum) profit from the set of worst (the minimum) profits.
When dealing with the costs, the maximum cost associated with each alternative is considered and the alternative which minimizes this maximum cost is chosen. In this context, therefore, the principle used is minimax—the best the minimum cost) of the worst (the maximum cost).
The basic rational behind this criterion is that a pessimism is not irrational under the state of uncertainty. The idea is to avoid the worst. Probably in this criteria the motive of self-preservation is considered more important than that of a windfall gain. Let us have an example :
It is clear from the above table that minimum pay-off associated with various acts is 18 (A1), 30 (A) and 22 (A3). Since the maximum of these is 30 (A). Therefore, according to maximin criterion the choice will be A, act which has the maximum pay-off amongst all the minimum of three acts.
(2) The Criterion of Optimism or Maxima : This criteria is just reverse to the above criterion and was suggested by Leonid Hurwitz. Under this rule the decision-maker is quite optimistic. He assumes that the situation will always be to his advantage. He, therefore, selects the strategy which yields him the best possible pay-off or the best of bests. Since this decision criterion locates the alternative with the highest possible gain, it has also been called an optimistic decision criterion.
In case of profit matrix, this criterion corresponds to identifying the best possible outcome (maximum pay-off) associated with each course of action and then choose the maximum of the maximum values in order to select the optimal course of action. For example, if we consider the pay-off table given in maximin criterion, the maximum pay-off associated with the different acts is as follows:
A1:50, A2:40, Az = 42
Since acts A has the maximum pay-off amongst all the maximum pay-offs of three acts, so the decision-maker will choose A, according to the maximax criterion.
(3) Minimax Regret Criterion : This criterion is attributed to Leonard Savage and is used to identify the regret (or opportunity loss) associated with each state of nature if a particular course of action is undertaken. Since this criterion calls for selecting the course of action that minimizes the maximum regret, it is known as the minimax regret criterion. The term ‘minimax’ is an abbreviation of the phrase ‘minimum of the maximum’. Under this criterion, the decision-maker first observes the maximum opportunity loss over all the various states of nature. He then selects that action for which the maximum opportunity loss is minimum. This criterion gives the greatest protection against the largest loss.
Computation Procedure : The various steps for calculating minimax regret criterion are as follows:
(i) The regret matrix is derived from the pay-off matrix, as explained earlier, i.e., determine the opportunity loss for each strategy by subtracting from maximum pay-off of a state of nature, the pay-offs of all the strategies.
(ii) Determine the maximum regret value (maximum opportunity loss) corresponding to each of the strategies.
(iii) Select the strategy which minimizes the maximum regret.
Illustration 15.
From the following pay-off table, decide as to which strategy should be chosen on the basis of (i) Maximin Criterion, (ii) Maximax Criterion and (iii) Minimax Regret Criterion.
(4) Hurwitz Criterion : Leonid Hurwitz has developed a criteria to overcome the disadvantages of pessimism of maximin and optimism of maximax criteria. According to this criterion, a rational decision-maker should be neither completely optimistic nor pesimistic. Thus, this criterion is a compromise between an optimistic and pessimistic decision criterion.
The Hurwitz criterion of decision-making stipulates that a decision-maker’s view may fall somewhere between the extreme pessimism of the maximin principle and the extreme optimism ofthe maximax principle. This assumes that a decision-maker possesses a specified degree of optimism indicated by the coefficient of optimism (a). The value of a always lies between zero and one i.e., 0 sasl. The value of a nearer to one (1) means the decision-maker is optimistic, and nearer to zero reflects a pessimistic decision-maker and a = 0.5 reflects a neutralist.
For taking a decision using this principle, first the decision-maker’s degree of optimism is indicated on the scale. Assuming that the decision-maker is able to reflect a degree of optimism by assigning a particular value of a, we multiply the maximum profit for each strategy A by a, and the minimum profit for it by 1 – a. The sum of these products, called the Hurwitz Criterion, is obtained for each strategy and we select the alternative which maximizes this value. Obviously, when a = 0, only the minimum of the profits for each strategy would be considered and the decision would in effect be according to the maximin criterion. Similarly, when a = 1, the decision would be identical to that arrived through the maximax principle. In short, According to Hurwitz criterion, select that strategy which maximizes :
H= a (Maximum Pay-off in Column)
+ (1 – a)(Minimum Pay-off in Column)
In the case of costs, the principle works like this. The minimum of the costs for each course of action is multiplied by a (the indicator of the degree of optimism of the decision-maker), and the maximum of the costs for each alternative is multiplied by 1 – a. Then, the sum of the products for each action strategy is obtained. The alternative for which the sum is the least is selected.
The major difficulty in applying this rule is the measurement of the value of a. Although more information than in minimax is being used while applying Hurwitz rule, yet only the two extreme pay-offs are considered and the remaining information is ignored.
(5) Criterion of Rationality or Laplace Criterion : This criterion was developed by Thomas Bayes and supported by Simor de Laplace. This criterion is based on the principle of equal likelihood or insufficient reason. According to this criterion, since probabilities of future states of nature are not known, therefore, there is no reason to consider any one outcome more likely than another i.e., all outcomes should be considered equally likely. In other words, the Laplace criterion is based on the simple philosophy that if we are uncertain about the various events then we may treat them as equally probable. Thus, with n outcomes, each outcome has a probability of 1/n. Using these probabilities, choose a course of action with the highest expected profit or the lowest expected loss. Since all outcomes are equally weighted, therefore, calculate the average outcome for every course of action i.e., add for each course of action the pay-offs for all outcomes (states of nature) and then divide by the number of course actions. If the pay-offs are in terms of costs, we choose the strategy with the lowest average cost.
Decision Theory Study Material
Conclusion : From the above description it should become clear that there is no single best rule for decision-making under the situation of uncertainty. There are several models (decision rules) for the purpose. The choice for the selection of a model should be left to the decision-maker who should ultimately decide as per his own skill and experience by considering the environment, firm’s policy and other relevant factors. Illustration 16.
From the pay-off table given in Illustration 15, decide as to which strategy should be chosen on the basis of : (i) Hurwitz criterion assuming coefficient of optimism (a) to be 0.80 and (ii) Laplace criterion.
Solution.
Coefficient of Optimism (a) = 0.80 (as given in question) ..
Coefficient of Pessimism = 1 – a=1 -0.80 = 0.20
According to Hurwitz criterion, select that strategy which maximizes:
H= a (Maximum Pay-off in Column)
+ (1 – a)(Minimum Pay-off in Column)
Pay-off Table
(Profit in Rs.)
Statistical Decision Theory At a Glance
(I) When Event-Probability is Known (1) EMV Criterion: This criterion requires the decision-maker to compute the expected pay-off for each decision alternative and then select the course of action that yields the highest EMV.
(2) EOL Criterion : This criterion requires the decision-maker to compute the expected opportunity loss for each decision alternative and then choose the strategy with minimum expected opportunity loss. Opportunity loss is the difference between the highest pay-off for a state of nature and the actual pay-off obtained for the particular action taken.
(3) EVPI (The Expected Value of Perfect Information) :
EVPI = EPPI – EMV (max.)
where EPPI = Highest pay-off for each event x Probability for corresponding event EMV (max.) = Expected monetary value of the optimal action
(II) When Event-Probability is Not Known
(1) Maximin Criterion : This is also known as the criterion of pessimism. Under this criterion the minimum pay-offs resulting from adoption of various strategies are considered and among these values the maximum one is selected. When dealing with the costs, the maximum cost associated with each alternative is considered and the alternative which minimizes this maximum cost is chosen.
(2) Maximax Criterion : This criterion corresponds to identify the best possible outcome (maximum pay-off) associated with each course of action and then choose the maximum of the maximum values in order to select the optimal course of action.
(3) Minimax Regret Criterion : Under this criterion, the decision-maker first observes the maximum opportunity loss over all the various states of nature. He then selects that action for which the maximum opportunity loss is minimum.
(4) Hurwitz Criterion : According to Hurwitz criterion, select that strategy which maximizes :
H= a (Maximum pay-off in column) + (1 – a) (Minimum pay-off in column)
(5) Criterion of Rationality or Laplace Criterion: The decision maker first calculate the average outcome for every course of action (which is the sum of all outcomes divided by the number of outcomes) and then select that with the maximum number. If the pay-offs are in terms of costs, we choose the strategy with the lowest average cost.
EXAMINATION QUESTIONS
Long Answer Theoretical Questions
1 Explain the ‘Statistical Decision Theory’. Discuss its scope, utility and limitations.
2. What do you understand by decision theory? How does decision theory differ from other theories ?
3. Explain the various decision criteria used in decision theory.
4. State the different kinds of situations under which business decisions are taken. Indicate the difference between decision-making under risk and uncertainty in statistical decision theory.
5. Discuss the various criteria for decision-making under uncertainty when event-probabilities are known.
6. Discuss the various criteria for decisions under uncertainty when probabilities of states of nature are not known. Illustrate each method by suitable example.
7. Decision criteria under situation of uncertainty are governed by the attitude of the decision maker. Explain.
8. What is a pay-off table ? Explain pay-off table with an example and convert that into an opportunity loss table.
9. How would you select a decision criterion suitable for a decision problem ? Illustrate your answer with suitable examples.
10 (i) What is pay-off matrix ? What is its utility in decision-making process ?
(ii) Describe briefly the ‘maximin’, ‘maximax’ and ‘minimax regret’ criteria of decision making.
Decision Theory Study Material
Short Answer Theoretical Questions
1 Explain clearly the difference between the following:
(i) Pay-off and Opportunity Loss;
(ii) Expected Monetary Value and Expected Opportunity Loss:
(iii) Prior Analysis and Pre-posterior Analysis:
(iv) Expected Profits with Perfect Information and Expected Value of Perfect Information.
(v) Maximin and Maximax decision-rules.
2. Write short notes on the following:
(i) States of Nature;
(ii) Decision Tree Diagram;
(iii) Inadmissible Acts;
(iv) Expected Value of Perfect Information:
(V) Coefficient of Optimum;
(vi) Laplace Decision Criterion.
3. Explain :
(a) EMV;
(b) Expected profit on Perfect knowledge;
(c) Decision Tree;
(d) Regret Matrix.
4. What is Opportunity Loss Table? Explain with an example.
5. What do you mean by Minimax Regret?
Objective Questions State whether the following statements are ‘True’ or ‘False’:
1 Hypothesis testing is sometimes called classical decision theory. (True)
2. Statistical decision theory and Bayesian theory are different. (False)
3. The prior distribution is said to be subjectively prior distribution. (False)
4. There are various criteria for selecting an act that is optimal in some sense from among the set of acts. (True)
5. The coefficient of optimism criterion and Herwitz coefficient of optimism mean the same thing. (False)
6. Min-Max criterion mean minimum of the maximum decision criterion. (True)
7. Decision theory is an art of decision-making in the face of uncertainty. (True)
8. A pay-off table cannot be converted to an opportunity loss table. (False)
9. The EOL for each act is calculated in the same way as the EMV. (False)
10. A decision tree is simply a graphic device for displaying information about the acts, the states and the consequences. (False)
11. EVPI and EVSI refer to the same thing. (True)
12. The term ‘risk’ and Uncertainty refer to two different types of information available to the decision-maker. (True)
13. States of nature affecting a particular decision are not dichotomous. (False)
Fill in the Blanks :
1 Pay-off tables can also be constructed in terms of ………. or ………
2. The ………. criterion is an expected value criterion.
3. EVPI refers to ………
4. A table in which all the consequences are shown in terms of opportunity loss is called an ……… table.
5. Statistical decision theory introduces a set of procedures by which statistical data can be used for making ………. decision.
Ans. (1) Losses, Costs (2) Bayes (3) Expected value of perfect information (4) Opportunity loss (5) Optimal.
