This document provides an overview of decision theory and decision making under uncertainty. It discusses structuring decision problems using decision trees and different types of decision making environments including uncertainty, risk, and certainty. It then covers various decision making criteria for uncertainty including optimistic, conservative, minimax regret, equally likely, and criterion of realism approaches. Expected values of perfect and sample information are also introduced. Examples are provided to illustrate key concepts such as calculating expected values and values of information.

1. 1Slide
Decision TheoryDecision Theory
Professor Ahmadi

2. 2Slide
Learning ObjectivesLearning Objectives
Structuring the decision problem and decision treesStructuring the decision problem and decision trees
Types of decision making environments:Types of decision making environments:
• Decision making under uncertainty whenDecision making under uncertainty when
probabilities are not knownprobabilities are not known
• Decision making under risk when probabilitiesDecision making under risk when probabilities
are knownare known
Expected Value of Perfect InformationExpected Value of Perfect Information
Decision Analysis with Sample InformationDecision Analysis with Sample Information
Developing a Decision StrategyDeveloping a Decision Strategy
Expected Value of Sample InformationExpected Value of Sample Information

3. 3Slide
Types Of Decision Making EnvironmentsTypes Of Decision Making Environments
Type 1: Decision Making under CertaintyType 1: Decision Making under Certainty..
Decision maker know for sure (that is, withDecision maker know for sure (that is, with
certainty) outcome or consequence of every decisioncertainty) outcome or consequence of every decision
alternative.alternative.
TType 2: Decision Making under Uncertainty.ype 2: Decision Making under Uncertainty.
Decision maker has no information at all aboutDecision maker has no information at all about
various outcomes or states of nature.various outcomes or states of nature.
Type 3: Decision Making under RiskType 3: Decision Making under Risk..
Decision maker has some knowledge regardingDecision maker has some knowledge regarding
probability of occurrence of each outcome or state ofprobability of occurrence of each outcome or state of
nature.nature.

4. 4Slide
Decision TreesDecision Trees
AA decision treedecision tree is a chronological representation ofis a chronological representation of
the decision problem.the decision problem.
Each decision tree has two types of nodes;Each decision tree has two types of nodes; roundround
nodesnodes correspond to the states of nature whilecorrespond to the states of nature while squaresquare
nodesnodes correspond to the decision alternatives.correspond to the decision alternatives.
TheThe branchesbranches leaving each round node represent theleaving each round node represent the
different states of nature while the branches leavingdifferent states of nature while the branches leaving
each square node represent the different decisioneach square node represent the different decision
alternatives.alternatives.
At the end of each limb of a tree are the payoffsAt the end of each limb of a tree are the payoffs
attained from the series of branches making up thatattained from the series of branches making up that
limb.limb.

5. 5Slide
Decision Making Under UncertaintyDecision Making Under Uncertainty
If the decision maker does not know with certaintyIf the decision maker does not know with certainty
which state of nature will occur, then he/she is saidwhich state of nature will occur, then he/she is said
to beto be making decision under uncertaintymaking decision under uncertainty..
The five commonly used criteria for decision makingThe five commonly used criteria for decision making
under uncertainty are:under uncertainty are:
1.1. thethe optimisticoptimistic approach (Maximax)approach (Maximax)
2.2. thethe conservativeconservative approach (Maximin)approach (Maximin)
3.3. thethe minimax regretminimax regret approach (approach (Minimax regret)Minimax regret)
4.4. Equally likely (Equally likely (LaplaceLaplace criterion)criterion)
5.5. Criterion of realism withCriterion of realism with αα ((Hurwicz criterionHurwicz criterion))

6. 6Slide
Optimistic ApproachOptimistic Approach
TheThe optimistic approachoptimistic approach would be used by anwould be used by an
optimistic decision maker.optimistic decision maker.
TheThe decision with the largest possible payoffdecision with the largest possible payoff isis
chosen.chosen.
If the payoff table was in terms of costs, theIf the payoff table was in terms of costs, the decisiondecision
with the lowest costwith the lowest cost would be chosen.would be chosen.

7. 7Slide
Conservative ApproachConservative Approach
TheThe conservative approachconservative approach would be used by awould be used by a
conservative decision maker.conservative decision maker.
For each decision the minimum payoff is listed andFor each decision the minimum payoff is listed and
then the decision corresponding to the maximum ofthen the decision corresponding to the maximum of
these minimum payoffs is selected. (Hence, thethese minimum payoffs is selected. (Hence, the
minimum possible payoff is maximizedminimum possible payoff is maximized.).)
If the payoff was in terms of costs, the maximumIf the payoff was in terms of costs, the maximum
costs would be determined for each decision andcosts would be determined for each decision and
then the decision corresponding to the minimum ofthen the decision corresponding to the minimum of
these maximum costs is selected. (Hence, thethese maximum costs is selected. (Hence, the
maximum possible cost is minimizedmaximum possible cost is minimized.).)

8. 8Slide
Minimax Regret ApproachMinimax Regret Approach
The minimax regret approach requires theThe minimax regret approach requires the
construction of aconstruction of a regret tableregret table or anor an opportunity lossopportunity loss
tabletable..
This is done by calculating for each state of nature theThis is done by calculating for each state of nature the
difference between each payoff and the largest payoffdifference between each payoff and the largest payoff
for that state of nature.for that state of nature.
Then, using this regret table, the maximum regret forThen, using this regret table, the maximum regret for
each possible decision is listed.each possible decision is listed.
The decision chosen is the one corresponding to theThe decision chosen is the one corresponding to the
minimum of the maximum regretsminimum of the maximum regrets..

9. 9Slide
Example: Marketing StrategyExample: Marketing Strategy
Consider the following problem with two decisionConsider the following problem with two decision
alternatives (dalternatives (d11 & d& d22) and two states of nature S) and two states of nature S11 (Market(Market
Receptive) and SReceptive) and S22 (Market Unfavorable) with the(Market Unfavorable) with the
following payoff table representing profits ( $1000):following payoff table representing profits ( $1000):
States of NatureStates of Nature
ss11 ss33
dd11 2020 66
DecisionsDecisions
dd22 2525 33

10. 10Slide
Example:Example: Optimistic ApproachOptimistic Approach
An optimistic decision maker would use the optimisticAn optimistic decision maker would use the optimistic
approach. All we really need to do is to choose theapproach. All we really need to do is to choose the
decision that has the largest single value in the payoffdecision that has the largest single value in the payoff
table. This largest value is 25, and hence the optimaltable. This largest value is 25, and hence the optimal
decision isdecision is dd22..
MaximumMaximum
DecisionDecision PayoffPayoff
dd11 2020
choosechoose dd22 dd22 25 maximum25 maximum

11. 11Slide
Example:Example: Conservative ApproachConservative Approach
A conservative decision maker would use theA conservative decision maker would use the
conservative approach. List the minimum payoff forconservative approach. List the minimum payoff for
each decision. Choose the decision with the maximumeach decision. Choose the decision with the maximum
of these minimum payoffs.of these minimum payoffs.
MinimumMinimum
DecisionDecision PayoffPayoff
choosechoose dd11 dd11 6 maximum6 maximum
dd22 33

12. 12Slide
Example:Example: Minimax Regret ApproachMinimax Regret Approach
For the minimax regret approach, first compute aFor the minimax regret approach, first compute a
regret table by subtracting each payoff in a columnregret table by subtracting each payoff in a column
from the largest payoff in that column. The resultingfrom the largest payoff in that column. The resulting
regret table is:regret table is:
ss11 ss22 MaximumMaximum
dd11 5 05 0 55
dd22 0 30 3 33 minimumminimum
Then, select the decision with minimum regret.Then, select the decision with minimum regret.

13. 13Slide
Example: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 forAverage for dd11 = (20 + 6)/2 = 13= (20 + 6)/2 = 13
Average forAverage for dd22 = (25 + 3)/2 = 14 Thus,= (25 + 3)/2 = 14 Thus, dd22 is selectedis selected

14. 14Slide
Example:Example: Criterion of Realism (Hurwicz)Criterion of Realism (Hurwicz)
Often calledOften called weighted averageweighted average, the, the criterion of realismcriterion of realism
(or(or HurwiczHurwicz) decision criterion is a compromise between) decision criterion is a compromise between
optimistic and aoptimistic and a pessimisticpessimistic decision.decision.
• First, selectFirst, select coefficient of realismcoefficient of realism,, αα, with a value, with a value
between 0 and 1. Whenbetween 0 and 1. When αα is close to 1, decision makeris close to 1, decision maker
is optimistic about future, and whenis optimistic about future, and when αα is close to 0,is close to 0,
decision maker is pessimistic about future.decision maker is pessimistic about future.
• Payoff =Payoff = αα x (maximum payoff) + (1-x (maximum payoff) + (1-αα) x (minimum payoff)) x (minimum payoff)
In our example letIn our example let αα = 0.8= 0.8
Payoff for dPayoff for d11 = 0.8*20+0.2*6=17.2= 0.8*20+0.2*6=17.2
Payoff for dPayoff for d22 = 0.8*25+0.2*3=20.6 Thus, select d2= 0.8*25+0.2*3=20.6 Thus, select d2

15. 15Slide
Decision Making with ProbabilitiesDecision Making with Probabilities
Expected Value ApproachExpected Value Approach
• If probabilistic information regarding the states ofIf probabilistic information regarding the states of
nature is available, one may use thenature is available, one may use the expectedexpected
Monetary value (EMV) approach (also known asMonetary value (EMV) approach (also known as
Expected Value or EV)Expected Value or EV)..
• Here the expected return for each decision isHere the expected return for each decision is
calculated by summing the products of the payoffcalculated by summing the products of the payoff
under each state of nature and the probability ofunder each state of nature and the probability of
the respective state of nature occurring.the respective state of nature occurring.
• The decision yielding theThe decision yielding the best expected returnbest expected return isis
chosen.chosen.

16. 16Slide
Expected Value of a Decision AlternativeExpected Value of a Decision Alternative
TheThe expected value of a decision alternativeexpected value of a decision alternative is the sumis the sum
of weighted payoffs for the decision alternative.of weighted payoffs for the decision alternative.
The expected value (EV) of decision alternativeThe expected value (EV) of decision alternative ddii isis
defined as:defined as:
where:where: NN = the number of states of nature= the number of states of nature
PP((ssjj) = the probability of state of nature) = the probability of state of nature ssjj
VVijij = the payoff corresponding to decision= the payoff corresponding to decision
alternativealternative ddii and state of natureand state of nature ssjj
EV( ) ( )d P s Vi j ij
j
N
=
=
∑1
EV( ) ( )d P s Vi j ij
j
N
=
=
∑1

17. 17Slide
Example: Marketing StrategyExample: Marketing Strategy
Expected Value ApproachExpected Value Approach
Refer to the previous problem. Assume theRefer to the previous problem. Assume the
probability of the market being receptive is known toprobability of the market being receptive is known to
be 0.75. Use the expected monetary value criterion tobe 0.75. Use the expected monetary value criterion to
determine the optimal decision.determine the optimal decision.

18. 18Slide
Expected Value of Perfect InformationExpected Value of Perfect Information
Frequently information is available that can improveFrequently information is available that can improve
the probability estimates for the states of nature.the probability estimates for the states of nature.
TheThe expected value of perfect informationexpected value of perfect information (EVPI) is(EVPI) is
the increase in the expected profit that would result ifthe increase in the expected profit that would result if
one knew with certainty which state of nature wouldone knew with certainty which state of nature would
occur.occur.
The EVPI provides anThe EVPI provides an upper bound on the expectedupper bound on the expected
value of any sample or survey informationvalue of any sample or survey information..

19. 19Slide
Expected Value of Perfect InformationExpected Value of Perfect Information
EVPI CalculationEVPI Calculation
• Step 1:Step 1:
Determine the optimal return corresponding to eachDetermine the optimal return corresponding to each
state of nature.state of nature.
• Step 2:Step 2:
Compute the expected value of these optimalCompute the expected value of these optimal
returns.returns.
• Step 3:Step 3:
Subtract the EV of the optimal decision from theSubtract the EV of the optimal decision from the
amount determined in step (2).amount determined in step (2).

20. 20Slide
Example: Marketing StrategyExample: Marketing Strategy
Expected Value of Perfect InformationExpected Value of Perfect Information
Calculate the expected value for the best action for eachCalculate the expected value for the best action for each
state of nature and subtract the EV of the optimalstate of nature and subtract the EV of the optimal
decision.decision.
EVPI= .75(25,000) + .25(6,000) - 19,500 = $750EVPI= .75(25,000) + .25(6,000) - 19,500 = $750

21. 21Slide
Decision Analysis With Sample InformationDecision Analysis With Sample Information
Knowledge of sample or survey information can beKnowledge of sample or survey information can be
used to revise the probability estimates for the states ofused to revise the probability estimates for the states of
nature.nature.
Prior to obtaining this information, the probabilityPrior to obtaining this information, the probability
estimates for the states of nature are calledestimates for the states of nature are called priorprior
probabilitiesprobabilities..
With knowledge ofWith knowledge of conditional probabilitiesconditional probabilities for thefor the
outcomes or indicators of the sample or surveyoutcomes or indicators of the sample or survey
information, these prior probabilities can be revised byinformation, these prior probabilities can be revised by
employingemploying Bayes' TheoremBayes' Theorem..
The outcomes of this analysis are calledThe outcomes of this analysis are called posteriorposterior
probabilitiesprobabilities..

22. 22Slide
Posterior ProbabilitiesPosterior Probabilities
Posterior Probabilities CalculationPosterior Probabilities Calculation
• Step 1:Step 1:
For each state of nature, multiply the priorFor each state of nature, multiply the prior
probability by its conditional probability for theprobability by its conditional probability for the
indicator -- this gives theindicator -- this gives the joint probabilitiesjoint probabilities for thefor the
states and indicator.states and indicator.
• Step 2:Step 2:
Sum these joint probabilities over all states -- thisSum these joint probabilities over all states -- this
gives thegives the marginal probabilitymarginal probability for the indicator.for the indicator.
• Step 3:Step 3:
For each state, divide its joint probability by theFor each state, divide its joint probability by the
marginal probability for the indicator -- this gives themarginal probability for the indicator -- this gives the
posterior probability distribution.posterior probability distribution.

23. 23Slide
Expected Value of Sample InformationExpected Value of Sample Information
TheThe expected value of sample informationexpected value of sample information (EVSI) is the(EVSI) is the
additional expected profit possible through knowledgeadditional expected profit possible through knowledge
of the sample or survey information.of the sample or survey information.
EVSI CalculationEVSI Calculation
• Step 1:Step 1: Determine the optimal decision and itsDetermine the optimal decision and its
expected return for the possible outcomes of theexpected return for the possible outcomes of the
sample using the posterior probabilities for the statessample using the posterior probabilities for the states
of nature.of nature.
• Step 2:Step 2: Compute the expected value of these optimalCompute the expected value of these optimal
returns.returns.
• Step 3:Step 3: Subtract the EV of the optimal decisionSubtract the EV of the optimal decision
obtained without using the sample information fromobtained without using the sample information from
the amount determined in step (2).the amount determined in step (2).

24. 24Slide
Efficiency of Sample InformationEfficiency of Sample Information
Efficiency of sample informationEfficiency of sample information is the ratio of EVSI tois the ratio of EVSI to
EVPI.EVPI.
As the EVPI provides an upper bound for the EVSI,As the EVPI provides an upper bound for the EVSI,
efficiency is always a number between 0 and 1.efficiency is always a number between 0 and 1.

25. 25Slide
Refer to the Marketing Strategy ExampleRefer to the Marketing Strategy Example
It is known from past experience that of all the casesIt is known from past experience that of all the cases
when the market was receptive, a research companywhen the market was receptive, a research company
predicted it in 90 percent of the cases. (In the other 10predicted it in 90 percent of the cases. (In the other 10
percent, they predicted an unfavorable market). Also,percent, they predicted an unfavorable market). Also,
of all the cases when the market proved to beof all the cases when the market proved to be
unfavorable, the research company predicted itunfavorable, the research company predicted it
correctly in 85 percent of the cases. (In the other 15correctly in 85 percent of the cases. (In the other 15
percent of the cases, they predicted it incorrectly.)percent of the cases, they predicted it incorrectly.)
Answer the following questions based on the aboveAnswer the following questions based on the above
information.information.

26. 26Slide
Example: Marketing StrategyExample: Marketing Strategy
1. Draw a complete probability tree.1. Draw a complete probability tree.
2. Find the posterior probabilities of all states of nature.2. Find the posterior probabilities of all states of nature.
3. Using the posterior probabilities, which plan would3. Using the posterior probabilities, which plan would
you recommend?you recommend?
4.4. How much should one be willing to pay (maximum)How much should one be willing to pay (maximum)
for the research survey? That is, compute the expectedfor the research survey? That is, compute the expected
value of sample information (EVSI).value of sample information (EVSI).