1. Decision Theory
Dr. T. T. Kachwala

2. Decision Theory - Introduction
In an environment of uncertainty, a decision making process leads a manager to one or more optimum solutions from amongst the alternates available within the constraints of the available resources and such that the manager optimizes the value of the objective function and simultaneously minimizes the risk involved.
Operations Research or Management Science or Decision Science is a science of application of mathematical models and statistical theories for the betterment or improvement of management decision making process.

3. Decision Theory – Basic Elements
Decision Theory comprises of two basic elements:
States of Nature (Future Events)
Strategies (Course of Action)
State of Nature: refers to that element of the decision making process which is not in the control of decision maker; example demand of a product. We use the term state of nature because the decision maker is an “innocent bystander” in the determination of which state of nature occurs
Strategy: refers to that element of the decision making process which is in the control of the decision maker, example Product mix strategy, Investment strategy, Marketing mix strategy.

4. Decision Theory – Basic Elements
Payoff: is a monetary value associated with an outcome corresponding to a combination of a possible state of nature that occurs and the strategy the decision maker selects. Since this value is condition to (depends on) the state of nature that occurs, it is referred as conditional payoff.
Payoff Matrix: is a tabular compilation of the payoffs for all the possible combinations of state of nature that occurs and the strategies the decision maker selects.

5. Conditional Payoff Matrix
Strategy (Course of Action)
Pij is the conditional payoff corresponding to the ith state of Nature that occurs and jth strategy the decision maker selects.
If this value is positive, it means the decision maker is the gainer and the outcome is favorable to the decision maker. However, if this value is negative, it means the decision maker is a loser and the outcome is unfavorable to the decision maker. S1 S2 Sj Sn N1
P11
P12
P1n N2
P21
P22
P2n Ni Nm
Pm1
Pm2
Pmn
State of Nature (Future Events)
Pij

6. Selection of an Optimum Strategy
Selection of an Optimum strategy depends on the environments or situations of Decision making.
Decision – making environment signifies a condition or a situation of decision-making based on the knowledge or information the decision maker has on the possible States of Nature that occurs.
There are three important environments defined in decision-making:
Decision making under Certainty (Perfect Information)
Decision making under Risk (Less than Perfect Information)
Decision making under Uncertainty (No Information)

7. Decision making under Certainty
Decision making under Certainty: signifies a situation of decision-making where a decision maker has perfect information or complete knowledge on the possible States of Nature that occurs.
The decision-making in this situation is to select a strategy that offers maximum conditional payoff.
In practical situations of Decision making it is difficult (almost impossible) to obtain perfect information on the possible states of nature that occurs

8. Decision making under Risk
Decision making under risk: signifies a situation of decision making where a decision maker has less than perfect information on the possible states of nature that occurs. In this situation, the decision maker is able to associate probability values for the States of Nature that occurs based on the past data using relative frequency theory.
The decision making in this situation is to select a strategy that maximizes weighted payoff (EMV or Expected Monetary Value) or minimizes weighted opportunity loss (EOL or Expected Opportunity Loss)

9. EMV Criterion
EMV is an acronym for Expected Monetary Value. It is mathematically defined as follows:
EMV = Pij pi
Where Pij is the conditional payoff corresponding to State of Nature Ni & Strategy Sj & pi is the probability of obtaining the State of Nature Ni
The following is the procedure for EMV Criterion:
Compile the Conditional Payoff Matrix
Calculate EMV corresponding to each strategy
Select the optimum strategy corresponding to the maximum value of EMV

10. EMV Criterion
The decision maker selects the optimum strategy that maximizes the expected payoff, which gives importance to both conditional payoff and the values of probability. The values of probability are the weights. The higher the value of probability, the more the weightage for that state of nature. EMV therefore signifies weighted payoff.
Distinction between a good decision and good outcome is important. Decision analysis is a logical framework for obtaining good decisions, but does not guarantee good outcomes. Even after a good decision has been made, state of nature will play a role in determining whether the decision results in a good or bad outcome.
There are many examples in real life situations where the decisions have been good but the outcome was not good & similarly there are examples where decision were not good but the outcome was good

11. EMV Criterion
Expected Payoff for Perfect Information = EPPI = Pijmax pi
where Pijmax is the maximum value of payoff corresponding to State of Nature Ni & pi is the probability of obtaining the State of Nature Ni
EPPI signifies the theoretical maximum average payoff the decision maker can obtain assuming that he has perfect information on the possible states of nature that occurs
Expected Value of Perfect Information = EVPI = EPPI – EMVmax
EVPI signifies the cost of uncertainty. It signifies the maximum average cost a decision maker may not mind incurring to obtain perfect information on the possible state of nature that occurs
Alternately, EVPI signifies the loss to the decision maker for not having perfect information on the possible states of nature that occurs

12. EOL Criterion Expected Opportunity Loss = EOL = lij pi
Where lij is the conditional opportunity loss corresponding to State of Nature Ni & Strategy Sj and is defined as lij = Pijmax – Pij
The procedure for EOL calculation is as follows:
Compile Conditional Opportunity Loss Matrix (Regret Matrix) from Conditional Payoff Matrix
Calculate EOL for each strategy
Select an optimum strategy corresponding to minimum value of EOL

13. EOL Criterion Interpretation of COL Matrix:
COL signifies the loss in payoff for selecting a particular strategy in preference to the best available strategy for a particular state of nature. The higher the value of COL, the more the regret of the decision maker for selecting that strategy over the best available strategy
 COL Matrix signifies a compilation of all the possible values of COL for all the possible States of Nature that occurs and Strategy the decision maker selects.
COL matrix is also referred as regret matrix as it indicates the regret of the decision maker for not selecting the best strategy for a particular state of nature that occurs

14. EOL Criterion
EOL signifies the average opportunity loss. It is also the weighted opportunity loss. Importance is given to both COL and the value of probability which signify the weights
 EOL = EVPI; EOL / EVPI signifies the expected opportunity loss for selecting a particular strategy in preference to the best available strategy for that State of Nature
The optimum strategy for both EMV & EOL criterion are the same. In practical application of Management, it is sufficient to apply any one of the two criterion. Of the two criterion, EMV criterion is more popular

15. Decision making under Uncertainty
Decision making under Uncertainty: signifies a situation of decision-making where a decision maker has no knowledge on the possible state of nature that occurs, for example launching a new product in the market.
Since there is no past data available, probability values cannot be obtained. EMV criterion cannot be applied for selecting the optimum strategies

16. Decision Making Under Uncertainty
The decision-making in this situation is to select a strategy based on conditional payoff together with one or more of the following criterion or principle.
Criterion of Optimism (Maximax)
Criterion of Pessimism (Maximin)
Hurwicz Criterion (Max H)
Laplace Criterion (Max Average)
Criterion of Regret (Minimax)

17. Decision Making Under Uncertainty