The document discusses statistical decision theory, including individual decision-making, game theory, and social-choice theory, emphasizing the lack of contemporary textbook options. It outlines concepts such as payoff tables, decision trees, Bayesian theorem, and the expected value of sample information, providing a framework for making decisions based on probabilities and utilities. An example illustrates how a bicycle shop can use decision theory to determine the best purchasing strategy based on varying demand scenarios.

1. Statistical Decision TheoryStatistical Decision Theory
©
Prepared by :Prepared by :
Hemn husein yasinHemn husein yasin
University of Salahaddin
College of Administrations and Economics
Department of Statistics
facebook :hemn blbas
Youtube:https://www.youtube.com/channel/UC5YZb6jWP9
dsgpSjhKwsJqg

2.  Introduction:
Decision theory is commonly understood to comprise three
largely separable topics: individual decision-making
(where the theory of maximizing expected utility is the
dominant paradigm), game theory (with its characteristic
concern with scenarios such as the prisoner's dilemma
and solution concepts such as equilibrium strategies),
and social-choice theory (featuring results such as
Arrow's impossibility theorem). For courses covering
these topics, the textbook options have been lacking.
Seminal works -- such as von Neumann and
Morgenstern's Theory of Games and Economic
Behaviour (1944 and 1947), Savage's The Foundations of
Statistics (1954), and Luce and Raffia's Games and
Decisions (1957) -- are dated and idiosyncratic in

3. Framework for a Decision Problem
i. Decision maker has available K possible courses of actionaction: a1,
a2, . . ., aK. Actions are sometime called alternatives.
ii. There are H possible uncertain states of naturestates of nature: s1, s2, . . .,
sH. States of nature are the possible outcomes over which
the decision maker has no control. Sometimes states of
nature are called events.
iii. For each possible action-state of nature combination, there
is an associated outcome representing either profit or loss,
called the monetary payoffpayoff, MI, that corresponds to action ai
and state of nature sj. The table of all such outcomes for a
decision problem is called a payoff tablepayoff table.

4. ai
si
s1 s2 . . . sH
a1
a2
.
.
.
aK
M11
M21
.
.
.
MK1
M12
M22
.
.
.
MK2
. . .
. . .
.
.
.
. . .
M1H
M2H
.
.
.
MKH
ACTIONS STATES OF NATURE

5. Suppose that a decision maker has to choose from K admissible actions a1,
a2, . . ., aK , given H possible states of nature s1, s2, . . ., sH. Let Mij denote the
payoff corresponding to the ith
action and jth
state of nature. For each action,
seek the smallest possible payoff. For action a1, for example, this is the
smallest of M11, M12, . . .M1H . Let us denote the minimum M1
*
where
),,,( 11211
*
1 HMMMMinM =
More generally, the smallest possible payoff for action ai is
given by
The maxim in criterion then selects the action ai for which
the corresponding Mi
*
is largest (that is, the action for which
the minimum payoff is highest).
),,,( 11211
*
Hi MMMM =

6. State of nature/
decision
It rains It does not rain
Umbrella Dry clothes
Don't wets
Dry clothes
Don’t wets
No umbrella Wet clothes
Its wets
Wet clothes
Its wets

7. tree diagramtree diagram:: is the a graphical device that forces
the decision-maker to examine all possible outcomes,
including unfavorable ones.
All decision trees contain:All decision trees contain:
Decision (or action) nodes.
Event (or state-of-nature) nodes.
Terminal nodes.

8. Decision Trees
Process A
Process B
Process C
States of natureActions

10.  Bayes’ Theorem :Bayes’ Theorem : Let s1, s2, . . ., sH be H mutually exclusive
and collectively exhaustive events, corresponding to the H
states of nature of a decision problem. Let A be some other
event. Denote the conditional probability that si will occur,
given that A occurs, by P(si|A), and the probability of A,
given si, by P(A|si )states that the conditional probability of
si, given A, can be expressed as
)(
)()|(
)|(
AP
sPsAP
AsP ii
i =
)()|()()|()()|(
)()|(
)|(
2211 HH
ii
i
sPsAPsPsAPsPsAP
sPsAP
AsP
+++
=

In the terminology of this section, P(si) is the prior probability of si
and is modified to the posterior probability, P(si|A), given the
sample information that event A has occurred.

11. Expected Value of Sample Information,
EVSISupposeSuppose:: that a decision maker has to choose from
among K possible actions, in the face of H states
of nature, s1, s2, . . ., sH. The decision-maker
may obtain sample information. Let there be M
possible sample results, A1, A2, . . . , AM. The
expected value of sample information is obtained
as follows:
(i) Determine which action will be chosen if only the
prior probabilities were used.
(ii) Determine the probabilities of obtaining each
sample result:
)()|()()|()()|()( 2211 HHiiii sPsAPsPsAPsPsAPAP +++= 

12. Expected Value of Sample Information,
EVSI
(continued)
(iii) For each possible sample result, Ai, find the
difference, Vi, between the expected monetary
value for the optimal action and that for the
action chosen if only the prior probabilities are
used. This is the value of the sample
information, given that Ai was observed.
(iv) The expected value of sample information, EVPIexpected value of sample information, EVPI,
is then:
HH WsPWsPWsPEVPI )()()( 2211 +++= 

15. Obtaining a Utility Function
Utility table
Suppose that a decision maker may receive several alternative
payoffs. The transformation from payoffs to utilities is
obtained as follows:
(i) The units in which utility is measured are arbitrary.
Accordingly, a scale can be fixed in any convenient fashion.
Let L be the lowest and H the highest of all the payoffs. Assign
utility 0 to payoff L and utility 100 to payoff H.
(ii) Let I be any payoff between L and H. Determine the
probability π such that the decision-maker is indifferent
between the following alternatives:
(a) Receive payoff I with certainty
(b) Receive payoff H with probability π and payoff L with
probability (1 - π)
(iii) The utility to the decision-maker of payoff I is then 100π. The
curve relating utility to payoff is called a utility functionutility function..

16. The Expected Utility CriterionSuppose that a decision maker has K possible actions, a1,
a2, . . ., aK and is faced with H states of nature. Let Uij denote
the utility corresponding to the ith
action and jth
state and πj
the probability of occurrence of the jth
state of nature. Then
the expected utility, EU(aexpected utility, EU(aii),), of the action ai is
Given a choice between alternative actions, the expectedexpected
utility criterionutility criterion dictates the choice of the action for which
expected utility is highest. Under generally reasonable
assumptions, it can be shown that the rational decision-maker
should adopt this criterion.
If the decision-maker is indifferent to risk, the expected
utility criterion and expected monetary value criterion are
equivalent.
∑=
=+++=
H
j
ijjiHHiii UUUUEU
1
2211)a( ππππ 

17. Utility Functions; (a) Risk Aversion;
(b) Preference for Risk; (c)
Indifference to Risk
(Figure 19.11)
Payoff
Utility
Payoff
UtilityPayoff
Utility
(a) Risk aversion (b) Preference for risk
(c) Indifference to risk

18. MaxMax Criterion
• Select the decision that results in the maximum of the
maximum rewards
• A very optimistic decision criterion – Decision maker assumes
that the most favorable state of nature for each action will
occur
• Most risk prone agent

19. • Thompson Lumber Co. assumes that the most favorable state
of nature occurs for each decision alternative
• Select the maximum reward for each decision – All three
maximums occur if a favorable economy prevails (a tie in case of
no plant)
• Select the maximum of the maximums – Maximum is $200,000;
corresponding decision is to build the large plant – Potential loss
of $180,000 is completely ignored

20. Example: A bicycle shop
Zed and Adrian and run a small bicycle shop called "Z to A
Bicycles". They must order bicycles for the coming season.
Orders for the bicycles must be placed in quantities of twenty
(20). The cost per bicycle is $70 if they order 20, $67 if they
order 40, $65 if they order 60, and $64 if they order 80. The
bicycles will be sold for $100 each. Any bicycles left over at
the end of the season can be sold (for certain) at $45 each. If
Zed and Adrian run out of bicycles during the season, then
they will suffer a loss of "goodwill" among their customers.
They estimate this goodwill loss to be $5 per customer who
was unable to buy a bicycle. Zed and Adrian estimate that the
demand for bicycles this season will be 10, 30, 50, or 70
bicycles with probabilities of 0.2, 0.4, 0.3, and 0.1
respectively.

21. Actions
There are four actions available to Zed and Adrian. They have to decide
which of the actions is the best one under each criteria.
1.Buy 20 bicycles
2.Buy 40 bicycles
3.Buy 60 bicycles
4.Buy 80 bicycles
Zed and Adrian have control over which action they choose. That is the
whole point of decision theory - deciding which action to take.
States of Nature
There are four possible states of nature. A state of nature is an outcome.
1.The demand is 10 bicycles
3.The demand is 50 bicycles
4.The demand is 70 bicycles
2.The demand is 30 bicycles
Zed and Adrian have no control over which state of nature will occur.
They can only plan and make the best decision based on the appropriate
decision criteria.

22. Payoff Table
After deciding on each action and state of nature, create a payoff table. The numbers in parentheses for each state of nature represent the probability of that state occurring.
Ok, the question on your mind is probably "How the [expletive deleted]
did you come up with those numbers?". Let's take a look at a couple of
examples.
Demand is 50, buy 60:
They bought 60 at $65 each for $3900. That is -$3900 since that is
money they spent. Now, they sell 50 bicycles at $100 each for
$5000. They had 10 bicycles left over at the end of the season, and
they sold those at $45 each of $450. That makes $5000 + 450 - 3900
= $1550.
Demand is 70, buy 40:
They bought 40 at $67 each for $2680. That is a negative $2680 since
that is money they spent. Now, they sell 40 bicycles (that's all they
had) at $100 each for $4000. The other 30 customers that wanted a
bicycle, but couldn't get one, left mad and Zed and Adrian lost $5 in
goodwill for each of them. That's 30 customers at -$5 each or -$150.
That makes $4000 - 2680 - 150 = $1170.

23. Action
State of Nature Buy 20 Buy 40 Buy 60 Buy 80
Demand 10
(0.2)
50 -330 -650 -970
Demand 30
(0.4)
550 770 450 130
Demand 50
(0.3)
450 1270 1550 1230
Demand 70
(0.1)
350 1170 2050 2330