Decision Theory
Professor Ahmadi
Slide 1
Learning Objectives
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Structuring the decision problem and decision trees
Types of decision making environments:
• Decision making under uncertainty when
probabilities are not known
• Decision making under risk when probabilities
are known
Expected Value of Perfect Information
Decision Analysis with Sample Information
Developing a Decision Strategy
Expected Value of Sample Information
Slide 2
Types Of Decision Making Environments
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Type 1: Decision Making under Certainty.
Decision maker know for sure (that is, with
certainty) outcome or consequence of every decision
alternative.
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Type 2: Decision Making under Uncertainty.
Decision maker has no information at all about
various outcomes or states of nature.
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Type 3: Decision Making under Risk.
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Decision maker has some knowledge regarding
probability of occurrence of each outcome or state of
nature.
Slide 3
Decision Trees
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A decision tree is a chronological representation of
the decision problem.
Each decision tree has two types of nodes; round
nodes correspond to the states of nature while square
nodes correspond to the decision alternatives.
The branches leaving each round node represent the
different states of nature while the branches leaving
each square node represent the different decision
alternatives.
At the end of each limb of a tree are the payoffs
attained from the series of branches making up that
limb.
Slide 4
Decision Making Under Uncertainty
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If the decision maker does not know with certainty
which state of nature will occur, then he/she is said
to be making decision under uncertainty.
The five commonly used criteria for decision making
under uncertainty are:
1. the optimistic approach (Maximax)
2. the conservative approach (Maximin)
3.
the minimax regret approach (Minimax regret)
4. Equally likely (Laplace criterion)
5. Criterion of realism with  (Hurwicz criterion)
Slide 5
Optimistic Approach
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The optimistic approach would be used by an
optimistic decision maker.
The decision with the largest possible payoff is
chosen.
If the payoff table was in terms of costs, the decision
with the lowest cost would be chosen.
Slide 6
Conservative Approach
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The conservative approach would be used by a
conservative decision maker.
For each decision the minimum payoff is listed and
then the decision corresponding to the maximum of
these minimum payoffs is selected. (Hence, the
minimum possible payoff is maximized.)
If the payoff was in terms of costs, the maximum
costs would be determined for each decision and
then the decision corresponding to the minimum of
these maximum costs is selected. (Hence, the
maximum possible cost is minimized.)
Slide 7
Minimax Regret Approach
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The minimax regret approach requires the
construction of a regret table or an opportunity loss
table.
This is done by calculating for each state of nature
the difference between each payoff and the largest
payoff for that state of nature.
Then, using this regret table, the maximum regret for
each possible decision is listed.
The decision chosen is the one corresponding to the
minimum of the maximum regrets.
Slide 8
Example: Marketing Strategy
Consider the following problem with two decision
alternatives (d1 & d2) and two states of nature S1
(Market Receptive) and S2 (Market Unfavorable) with
the following payoff table representing profits ( $1000):
States of Nature
s1
s3
Decisions
d1
20
6
d2
25
3
Slide 9
Example: Optimistic Approach
An optimistic decision maker would use the optimistic
approach. All we really need to do is to choose the
decision that has the largest single value in the payoff
table. This largest value is 25, and hence the optimal
decision is d2.
Maximum
Decision
Payoff
d1
20
choose d2
d2
25
maximum
Slide 10
Example: Conservative Approach
A conservative decision maker would use the
conservative approach. List the minimum payoff for
each decision. Choose the decision with the maximum
of these minimum payoffs.
Minimum
Decision
Payoff
choose d1
d1
d2
6
3
maximum
Slide 11
Example: Minimax Regret Approach
For the minimax regret approach, first compute a
regret table by subtracting each payoff in a column
from the largest payoff in that column. The resulting
regret table is:
s1
d1
d2
5
0
s2
0
3
Maximum
5
3
minimum
Then, select the decision with minimum regret.
Slide 12
Example: Equally Likely (Laplace) Criterion
Equally likely, also called Laplace, criterion finds
decision alternative with highest average payoff.
• First calculate average payoff for every alternative.
• Then pick alternative with maximum average payoff.
Average for d1 = (20 + 6)/2 = 13
Average for d2 = (25 + 3)/2 = 14 Thus, d2 is selected
Slide 13
Example: Criterion of Realism (Hurwicz)
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Often called weighted average, the criterion of realism
(or Hurwicz) decision criterion is a compromise between
optimistic and a pessimistic decision.
• First, select coefficient of realism, , with a value
between 0 and 1. When  is close to 1, decision maker
is optimistic about future, and when  is close to 0,
decision maker is pessimistic about future.
• Payoff =  x (maximum payoff) + (1-) x (minimum payoff)
In our example let  = 0.8
Payoff for d1 = 0.8*20+0.2*6=17.2
Payoff for d2 = 0.8*25+0.2*3=20.6 Thus, select d2
Slide 14
Decision Making with Probabilities
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Expected Value Approach
• If probabilistic information regarding the states of
nature is available, one may use the expected
Monetary value (EMV) approach (also known as
Expected Value or EV).
• Here the expected return for each decision is
calculated by summing the products of the payoff
under each state of nature and the probability of
the respective state of nature occurring.
• The decision yielding the best expected return is
chosen.
Slide 15
Expected Value of a Decision Alternative
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The expected value of a decision alternative is the sum
of weighted payoffs for the decision alternative.
The expected value (EV) of decision alternative di is
defined as:
N
EV( d i )   P( s j )Vij
j 1
where:
N = the number of states of nature
P(sj) = the probability of state of nature sj
Vij = the payoff corresponding to decision
alternative di and state of nature sj
Slide 16
Example: Marketing Strategy
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Expected Value Approach
Refer to the previous problem. Assume the
probability of the market being receptive is known to
be 0.75. Use the expected monetary value criterion to
determine the optimal decision.
Slide 17
Expected Value of Perfect Information
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Frequently information is available that can improve
the probability estimates for the states of nature.
The expected value of perfect information (EVPI) is
the increase in the expected profit that would result if
one knew with certainty which state of nature would
occur.
The EVPI provides an upper bound on the expected
value of any sample or survey information.
Slide 18
Expected Value of Perfect Information
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EVPI Calculation
• Step 1:
Determine the optimal return corresponding to each
state of nature.
• Step 2:
Compute the expected value of these optimal
returns.
• Step 3:
Subtract the EV of the optimal decision from the
amount determined in step (2).
Slide 19
Example: Marketing Strategy
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Expected Value of Perfect Information
Calculate the expected value for the best action for each
state of nature and subtract the EV of the optimal
decision.
EVPI= .75(25,000) + .25(6,000) - 19,500 = $750
Slide 20
Decision Analysis With Sample Information
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Knowledge of sample or survey information can be
used to revise the probability estimates for the states of
nature.
Prior to obtaining this information, the probability
estimates for the states of nature are called prior
probabilities.
With knowledge of conditional probabilities for the
outcomes or indicators of the sample or survey
information, these prior probabilities can be revised by
employing Bayes' Theorem.
The outcomes of this analysis are called posterior
probabilities.
Slide 21
Posterior Probabilities
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Posterior Probabilities Calculation
• Step 1:
For each state of nature, multiply the prior
probability by its conditional probability for the
indicator -- this gives the joint probabilities for the
states and indicator.
• Step 2:
Sum these joint probabilities over all states -- this
gives the marginal probability for the indicator.
• Step 3:
For each state, divide its joint probability by the
marginal probability for the indicator -- this gives
the posterior probability distribution.
Slide 22
Expected Value of Sample Information
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The expected value of sample information (EVSI) is the
additional expected profit possible through knowledge
of the sample or survey information.
EVSI Calculation
• Step 1: Determine the optimal decision and its
expected return for the possible outcomes of the
sample using the posterior probabilities for the
states of nature.
• Step 2: Compute the expected value of these optimal
returns.
• Step 3: Subtract the EV of the optimal decision
obtained without using the sample information from
the amount determined in step (2).
Slide 23
Efficiency of Sample Information
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Efficiency of sample information is the ratio of EVSI to
EVPI.
As the EVPI provides an upper bound for the EVSI,
efficiency is always a number between 0 and 1.
Slide 24
Refer to the Marketing Strategy Example
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It is known from past experience that of all the cases
when the market was receptive, a research company
predicted it in 90 percent of the cases. (In the other 10
percent, they predicted an unfavorable market). Also,
of all the cases when the market proved to be
unfavorable, the research company predicted it
correctly in 85 percent of the cases. (In the other 15
percent of the cases, they predicted it incorrectly.)
Answer the following questions based on the above
information.
Slide 25
Example: Marketing Strategy
1. Draw a complete probability tree.
2. Find the posterior probabilities of all states of nature.
3. Using the posterior probabilities, which plan would
you recommend?
4. How much should one be willing to pay (maximum)
for the research survey? That is, compute the expected
value of sample information (EVSI).
Slide 26