Statistical decision theory and all criterion with examples.

1. Statistical Decision Theory
Group 5
Akash Agarwal – 2020005
Anushree Saxena – 2020014
Shubham Bhadauria – 2020020
Girish Khatri – 2020026
Meghana Mahabal Hegde – 2020038
Rachna Gupta – 2020046
Sarthak Baluni – 2020054
Vignesh Kumar R – 2020062

2. Introduction to Statistical Decision Theory
 Statistical Decision Theory – Process concerned with decision making with the presence of
statistical knowledge which provides further information in the process of decision making
 What is a Decision?
It is a conclusion of a process designed to weigh the relative utilities or merits of a set of available
alternatives so that the most preferred course of action can be selected for implementation
 Why must decisions be made?
When there is limited availability of resources and certain objectives must be met, decision theory
helps in making certain decisions that can help in meeting the set objectives in the most optimal way

3. Classification of Decisions
 General Decisions
1. Strategic Decisions 2. Administrative Decisions
 Depending on Nature of Problem
1. Programmed Decisions 2. Non-Programmed Decisions
 Depending on Area of Interest
1. Political Decisions 2. Economic Decisions 3. Scientific Decisions
 Can also be Classified as
1. Static Decisions 2. Dynamic Decisions

4. Phases and Steps of Decisions Making
 Phases of Decision Making
1. How to formulate goals and objectives, enumerate the constraints, identify the various alternatives involved,
and the relevant payoffs
2. How to choose the optimal strategy given the objectives, strategies, and payoffs
 Steps Involved in Decision Making
1. Clearly define the problem
2. List all the possible alternatives (strategies) that can be considered in the decision
3. Identify all outcomes or the states of nature for each alternative (These are not in the control of the decision-
maker)
4. Identify the payoff and construct a payoff table for each alternative and outcome combination
5. Use a decision modelling technique to choose an alternative

5. Types of Decision-Making Environment
 Decision Making Under Certainty (DMUC)
 Decision Making Under Risk (DMUR)
 Decision Making Under Uncertainty (DMUU)

6. Laplace Criterion
 This criterion is based on the principle of insufficient reason and was developed by Thomas
Bayles and supported by Simon de Laplace
 As there is no objective evidence, we can assign equal probabilities to each of the state of
nature
 This subjective assumption of equal probabilities is known as Laplace criterion, or criterion of
insufficient reason in management literature
 If there are n states of nature, each can be assigned a probability of occurrence = 1/n. Using
these probabilities, we compute the expected payoff for each course of action and the action
with maximum expected value is regarded as optimal

7. Illustration – Laplace Criterion
A farmer wants to decide which of the three crops he should plant on his 100 Acre farm. The
profit from each is dependent on the rainfall during the growing seasons. The farmer has
categorized the amount of rainfall as high, medium, low. His estimated profit for each is show in
the table:
Rainfall Crop A Crop B Crop C
High 8000 3500 5000
Medium 4500 4500 5000
Low 2000 5000 4000
If the farmer wishes to plant only one crop, decide which will be his choice using Laplace criterion

8. Solution
As Crop A has the highest EMV (Profits), it is the optimal strategy
Probabilities
State of nature
( Rainfall ) Expected Monetary Value
(EMV)
Profits
High Medium Low
1/3 1/3 1/3
Strategies Utility or Payoffs
Crop A 8000 3500 5000 8000x1/3 + 3500x1/3 + 5000x
1/3= 4833 Maximum
Crop B 3500 4500 5000 3500x1/3 + 4500x1/3 + 5000x1/3=
4333
Crop C 2000 5000 4000 2000x1/3 + 5000x1/3 + 4000x
1/3= 4666

9. Maximin Criterion
 One of the criteria for decision making when probability information regarding the likelihood
of the states of nature is unavailable
 In maximin approach, one looks at the worst that could happen under each action and then
choose the action with the largest payoff.
 It is assumed that the worst will happen, and the action will be taken by choosing the best
among the worst cases, so this approach means ‘best of the worst’
 It is also known as the Pessimist or Conservative approach

10. Illustration – Maximin Approach
Identify where the money should be invested using the Maximin/conservative approach.
Economy
Alternatives Growing Stable Declining
Bonds 40 45 5
Stock 70 30 -13
Mutual Funds 53 45 -5

11. Solution
 Step 1: Identify the lowest value corresponding to each decision alternative and name the
column as Worst
Economy
Alternatives Growing Stable Declining Worst
Bonds 40 45 5 5
Stock 70 30 -13 -13
Mutual Funds 53 45 -5 -5

12. Solution
 Step 2: From the column worst, choose the maximum possible value
So, the money must be invested in Bonds
Economy
Alternatives Growing Stable Declining Worst
Bonds 40 45 5 5
Stock 70 30 -13 -13
Mutual Funds 53 45 -5 -5

13. Hurwicz Criterion
 The Hurwicz criterion attempts to find a middle ground between the extremes posed by the
optimist and pessimist criteria. It incorporates a measure of pessimism and optimism by assigning
a certain percentage weight to optimism and the balance to pessimism
 A weighted average can be computed for every action alternative with an alpha‐weight α, called
the coefficient of realism
 The term Realism here means that the unbridled optimism of maximax is replaced by an
attenuated optimism as denoted by the α. Note that 0 ≤ α ≤ 1
 Thus, a better name for the coefficient of realism is coefficient of optimism. An α=1 implies
absolute optimism (maximax), while an α = 0 implies absolute pessimism (maximin)
 Selecting a value for α simultaneously produces a coefficient of pessimism 1 ‐ α, which reflects the
decision maker's aversion to risk

14. Hurwicz Criterion
A Hurwicz weighted average a can now be computed for every action alternative ai in a as
below:
 If payoff v(ai ,θj) represent profits or income (positive flow payoffs) then amax alternative is
selected
Where amax = αmax{v(ai ,θj)} + (1‐α)min{ v(ai ,θj)}
 If payoff v(ai ,θj) represent loss or cost (negative flow payoffs), then amin alternative is
selected
Where amin = αmin{v(ai ,θj)} + (1‐α)max{ v(ai ,θj)}

15. Illustration 1 – Hurwicz Criterion
A company should decide on the number of supplies to be purchased to meet customer needs during the holidays. The exact
number of customers is unknown but is expected to belong to one of the following categories: 200, 250, 300 or 400 customers.
Four supply levels are proposed, with level i being ideal if the number of customers meets category i. Deviations from the ideal
category, lead to additional costs, either because they keep extra supplies that are not needed, or because the demand is not
met.
The following Table shows the payoff values (costs) for the supplies, where states θ1, θ2, θ3, θ4 corresponds to 200, 250, 300 and
400 costumers while a1, a2, a3, a4 corresponds to supply levels of 200, 250, 300 and 400 pieces. Assume that the company’s
managers have agreed to assess their level of optimism at 60% and select the best alternative.
States
Alternatives
θ1 θ2 θ3 θ4
a1 5 10 18 25
a2 8 7 8 23
a3 21 18 12 21
a4 30 22 19 15

16. Solution 1
Procedure:
1) From the question it is inferred that the coefficient
of optimism α is 0.6
2) For every action alternative compute, the
minimum and maximum payoff value
3) For every action alternative compute, the Hurwicz
weighted average using the weights α and (1-α)
4) The payoff represents cost, which is a negative
flow pay off
5) Thus, the formula used is, amin = αmin{v(ai ,θj)} +
(1‐α)max{ v(ai ,θj)}
6) Choose the alternative with the Minimum
Hurwicz value
Solution:
The Hurwicz Value is calculated as follows:
 For a1
amin = 0.6 x min(ai , θj) + (1-0.6) x max(ai , θj)
= 0.6 x 5 + 0.4 x 25 = 13
 Similarly, for a2, amin= 13.4
 Similarly, for a3, amin= 15.6
 Similarly, for a4, amin= 21
Therefore, the best alternative to follow is a1.

17. Illustration 2 – Hurwicz Criterion
The following matrix gives the payoff in rupees of different strategies (alternatives) A, B, and C against conditions (events) W, X,
Y and Z. Identify the decision taken under the Hurwicz criterion. The decision maker’s degree of optimism (α) being 0.7
EventsAlternatives
W (θ1) X (θ2) Y (θ3) Z (θ4)
A (a1) 4000 -100 6000 18000
B (a2) 20000 5000 400 0
C (a3) 20000 15000 -2000 1000

18. Solution 2
Procedure:
1) From the question it is inferred that the coefficient
of optimism α is 0.7
2) For every action alternative compute, the
minimum and maximum payoff value
3) For every action alternative compute, the Hurwicz
weighted average using the weights α and (1-α)
4) The payoff data is not given, so assume it is a
positive flow pay off
5) Thus, the formula used is, amax = αmax{v(ai ,θj)} +
(1‐α)min{ v(ai ,θj)}
6) Choose the alternative with the Maximum
Hurwicz value
Solution:
The Hurwicz Value is calculated as follows:
 For a1:
amax = 0.7 x max(ai , θj) + (1-0.7) x min(ai , θj)
= 0.7 x 18000 + 0.3 x -100 = 12570
 Similarly, for a2, amax = 14000
 Similarly, for a3, amax = 13400
Under Hurwicz rule, alternative B is the optimal
strategy as it gives highest payoff

19. Savage Criterion
 The decision-maker might experience regret after the decision has been made and the states of nature i.e., events occurred. Thus, the
decision-maker should attempt to minimize regret before selecting a particular alternative (strategy). The Opportunity Loss = Regret
 Under Savage Criterion, decision making is based on opportunistic loss. Regret/Opportunity Loss is the difference between the payoff
associated with a particular decision alternative and the payoff associated with the decision that would yield the most desirable payoff
for a given state of nature
 Thus, regret represents how much potential payoff one would forgo by selecting a particular decision alternative given that a specific
state of nature will occur. Therefore, regret is often referred to as opportunity loss
 Under the regret approach to decision making, one would choose the decision alternative that minimizes the maximum state of regret
that could occur over all possible states of nature
 The basic steps involved in this criterion are:
1. Determine the amount of regret corresponding to each event for every alternative. The regret for the event corresponding to ith
alternative is given by: ith regret = (maximum payoff - ith payoff) for the jth event
2. Determine the maximum regret amount for each alternative
3. Choose the alternative which corresponds to the minimum regrets

20. Illustration - Savage Criterion
A steel manufacturing company is concerned with the possibility of a strike. It will cost an extra
20,000 to acquire an adequate stockpile. If there is a strike and the company has not stockpiled,
management estimates an additional expense of 60,000 on account of lost sales. Should the
company stockpile or not if it is to use Savage Criterion?
Conditional Cost Table
Alternatives
States of Nature
Strike No-Strike
Stockpile -20000 -20000
Do Not Stockpile -60000 0

21. Solution
 First, we will construct the conditional regret table
 Find the Maximum payoff for Strike (S1) and subtract it from each payoff in S1
 Similarly, for No-Strike (S2) column we find the maximum payoff and subtract it from each payoff in S2
 We get the conditional regret table
 Maximum regret for alternative Stockpile, is 20,000 and for Do not Stockpile it is 40,000
 Therefore, company should choose alternative Stockpile, with minimax regret of 20,000
Conditional Regret Table
Alternatives
States of Nature
Maximum of Row
Strike (S1) No-Strike (S2)
Stockpile (A1) 0 20,000 20,000
Do Not Stockpile
(A2)
40,000 0 40,000

22. Expected Monetary Value
 Expected monetary value is a statistical technique used to quantify risks and calculate the
contingency reserve
 This technique is used in medium to high-cost projects, where the stakes are too high to
risk the project failing
 EMV = ∑(Probability of outcome) x (Payoff of outcome)
 Benefit: It helps you with a make or buy decision during the plan procurement process.
 Drawback: This technique involves expert opinions to finalize the probability and impact
of the risk. Therefore, personal bias may affect the result.

23. Illustration - Expected Monetary Value
A marketing manager of an insurance company has kept complete records of the sales effort of the sales
personnel. These records contain data regarding the number of insurance policies sold and net revenues
received by the company as a function of four different sales strategies. The manager has constructed the
conditional payoff matrix given below, based on his records. (The state of nature refers to the number of
policies sold). The number within the table represents sales. Suppose you are a new salesperson and that
you have access to the original records as well as the payoff matrix. Which strategy would you follow?
State of Nature N1 N2 N3
Probability 0.2 0.5 0.3
Strategies Sales
S1 40 60 100
S2 60 50 90
S3 20 100 80
S4 90 30 70

24. Solution
State of Nature N1 N2 N3
Probability 0.2 0.5 0.3
Strategies Sales Expected Monetary Value
S1 40 60 100 40*0.2+ 60*0.5 + 100*0.3 = 68
S2 60 50 90 60*0.2 + 50*0.5 + 90*0.3 = 64
S3 20 100 80 20*0.2 + 100*0.5 + 80*0.3 = 78
S4 90 30 70 90*0.2 + 30*0.5 + 70*0.3 = 54
As the decision is to be made under risk, multiplying the probability and utility and summing
them up give the expected utility for the strategy
As the third strategy gives highest expected monetary value than other three, strategy three
is optimal

25. Expected opportunity loss (EOL)
 A statistical calculation used primarily in the business field to help determine optimal courses
of action. Doing business is full of decision making. Any decision consists of a choice between
two or more events. For each event, there are two or more possible courses of action that
you might take. Calculating the EOL is an organized way of using a mathematical model to
compare these choices and outcomes, to make the most profitable decision
 EOL (Expected Opportunity Loss) can be computed by multiplying the probability of each of
state of nature with the appropriate loss value and adding the resulting products

26. Illustration - Expected Opportunity Loss
State of nature Probability A1 A2 A3 A4
S1=1 0.4 1 0 -1 -2
S2=2 0.3 1 2 1 0
S3=3 0.2 1 2 3 2
S4=4 0.1 1 2 3 4
Step 1 - Find the highest
value in each row (s1, s2
etc.)
Step 2 - Subtract each
column in that row with
the identified numberRegret table
State of nature Probability A1 A2 A3 A4
S1=1 0.4 0 1 2 3
S2=2 0.3 1 0 1 2
S3=3 0.2 2 1 0 1
S4=4 0.1 3 2 1 0
Step 3 - Multiply the
probability with the
respective outcomes row
wise
Step 4 - Add the all the
figures obtained for each
act, the act with the
lowest E.O.L is the act
with least expected loss
State of nature Probability A1 A2 A3 A4
S1=1 0.4 0 0.4 0.8 1.2
S2=2 0.3 0.3 0 0.3 0.6
S3=3 0.2 0.4 0.2 0 0.2
S4=4 0.1 0.3 0.2 0.1 0
Expected
Opportunity Loss
1 0.8 1.2 2

27. Expected Value of Perfect Information (EVPI)
 The amount by which perfect information would increase our expected payoff. In decision theory,
the expected value of perfect information (EVPI) is the price that one would be willing to pay in order
to gain access to perfect information. It provides an upper bound on what to pay for additional
information
 In general, the Expected Value of Perfect Information (EVPI) is computed as follows:
EVPI = |EVwPI – EMV|
Where,
 EVPI = Expected value of perfect information
 EVwPI = Expected value with perfect information about the states of nature
 EMV = Expected Monetary Value

28. Expected Value of Perfect Information (EVPI)
• Expected Value With Perfect Information (EVwPI): The expected payoff of having perfect
information before making a decision.
EVwPI = ∑ (Best payoff of outcome) x (Probability of outcome)
• Expected Monetary Value (EMV) uses the probabilities to calculate the average payoff for each
alternative.
EMV (For alternative i) = ∑ (Probability of outcome) x (Payoff of outcome)

29. Illustration – Expected Value of Perfect Information
Suppose we have an option of 3 plants (Large, Small and No plant) and their respective Outcomes (Demand)
is divided into 3 categories High, Moderate and Low with their probability of outcomes known, we can
calculate EMV, EVwPI and then arrive at EVPI
Alternatives
Outcomes (Demand)
High Moderate Low EMV
Large plant 2,00,000 1,00,000 -1,20,000 86,000
Small plant 90,000 50,000 -20,000 48,000
No plant 0 0 0 0
Probability of
outcome
0.3 0.5 0.2

30. Solution
Solution: Step 1: Calculate EMV
EMV (For alternative i) = ∑ (Probability of outcome) x (Payoff of outcome)
EMV for Large plant= (2,00,000 * 0.3) + (1,00,000 * 0.5) + (-1,20,000 * 0.2) = 86,000
EMV for Small plant= (90,000 * 0.3) + (50,000 * 0.5) + (-20,000 * 0.2) = 48,000
EMV for No plant= 0
Choose the large plant, as it has highest EMV

31. Solution
Step 2: Calculate EVwPI
Payoffs in green would be chosen based on perfect information (knowing demand level)
EVwPI = ∑ (Best payoff of outcome) x (Probability of outcome)
EVwPI for large plant= (2,00,000 * 0.3) + (1,00,000 * 0.5) + (0*0.2) = 1,10,000
Step 3: Calculate EVPI
EVPI = |EVwPI – EMV| = 1,10,000 – 86,000 =24,000
In other words, the “perfect information” increases the expected value by 24,000
Alternatives
Outcomes (Demand)
High Moderate Low
Large plant 2,00,000 1,00,000 -1,20,000
Small plant 90,000 50,000 -20,000
No plant 0 0 0
Probability of
outcome
0.3 0.5 0.2

32. Thank You