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DECISION THEORY
DEFINITION 1.1 :
DECISION THEORY (DT) is a set of concepts, principles, tools and
techniques that aid the decision maker in dealing with compl~x decision
problems under uncertainty.
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COMPONENTS OF A DT PROBLEM:
1. THE DECISION MAKER
2. ALTERNATIVE COURSES OF ACTION
This is the controllable aspect of the problem.
3. STATES OF NATURE OR EVENTS
These are the scenarios or states of the environmen.tnot under the
control of the decision maker. The events defmed should be
mutually exclusive and collectively exhaustive.
4. CONSEQUENCES
The consequences that must be assessed by the decision maker are
measures of the net bertefit, payoff, cost or revenue received by the
decision maker. There is a consequence (or vector of
consequences) associated with each action-event pair. The
consequences resutilmarized a decisionmatrix.
a in .
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DECISION THEORY EDGAR L. DE CASTRO PAGE 1
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CLASSIFICATIONS OF DT PROBLEMS:
1. Single Stage Decision Problems
A decision is m~de only once. ~"
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2. Multiple Stage/Sequential Decision Problems
Decisions are made one after another.
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3. Discrete DT Problems
The alternative courses of actions and states of nature are finite.
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4. Continuous DT Problems
The alternative courses of actions and states of nature are infinite.
DT Problems can also be classified as those with or without
experimentation. Experimentation is perfoaned to obtain additional
information that will aid the decision maker.
I. DISCRETE DECISION THEORY PROBLEMS
DECISION TREES
A discrete DT problem can be represented pictorially using a tree.
diagram or decision tree. It chronologically depicts the sequence of
actions and events as they unfold.
A square node ( D) precedes the set of possible actions that can
be taken by the decision maker. A round node (0 ) precedes the set of
events or states of nature that could be encountered after a decision is
made. The nodes are connectedby branches. (""- )
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DECISION THEORY EDGAR L. DE CASTRO PAGE 2
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EXAMPLE: i.
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DECISIONS UNDER RISK AND UNCERTAINTY
Consider a DT problem with m alternative courses of actions and a
maximum of n events or states of nature for each alternativecourse of
action.
Defme: Ai = alternative course of action i; i = 1, 2, . . .,m
q)j=state of nature j; j = 1, 2, . . .,n
The decision matrix of payoffs is given by :
q)l q)2 ... q)11
Al v(A1,q)1) veAl ,q)2) ... v(Abn)
A2 v(A2'1) v(A2'2) ... v(A2 '11)
. . . . .
. . . . .
. . . . .
Am v(Am '1) V(Am'2) ... v(An"l1 )
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DECISION THEORY EDGAR L. DE CASTRO PAGE 3
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A. LAPLACE CRITERION
This criterion is based on what is mown as the principle of
insufficientreason. Her~the probabilitiesassociatedwith the'occurrence
of the event is uriknown.We do not have,Su(.ticie~~,leason
to
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rjJj , "
conclude that the probabilities are different. Hence we 'assume that all "
events are equally likely, i.e. ....
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P( t/J = rjJ . ) = - ~.:-.
} n
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Then, the optimal decision rule is to select action '51t corresponding to "
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B. MINIMAX (MAXIMIN) CRITERION
This is the most conservative criterion since it is based on making
the best out of the worst possible conditions. For each possible decision
alternative, we select the worst condition and then select the alternative
corresponding to the best of the worst conditions.
The MINIMAX strategy is given by:
min
A; f max{V(Ai,rjJ
L. r/J.
J
j }
]
The MAXIMIN strategy is given by:
max
A; [min {v( Ai , rjJ ]
j r/J
j }
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DECISION THEORY EDGAR L. DE CASTRO PAGE 4
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5. C. SAVAGE MINIMAX REGRET CRITERION
The MINIMAX rule is an extremely conservative type of decision
rule. The savage MINIMAx regret criterion assUmes that a new .loss ,.
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r
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matrix is constructed id which v(Ai, (Jj) is replaced by r(A~,..(J)' which
j 1.
is defined by:
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max { v( Ak , (Jj )} - v( Ai , (Jj ), if v is profit
Ak
, r
t.
v(Ai,(J j) - min{v(Ak,(J )}, if v i~loss
j II
Ak .,/
. .
Once the loss matrix is constructed using the above fonnula, we can
now apply the MINIMAX criterion defined in b.
D. HURWICZ CRITERION
This criterion represents a range of attitudes from the most
optimistic to the most pessimistic.
Under the most optimistic conditions, one would choose the action
yielding:
max max {v( Ai , (Jj }
Ai { I } t/>
Under the most pessimistic conditions, the chosen action cOlTesponds
to: .
max min {v( Ai , (Jj }
A. A..
t { '1') }
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DECISIONTHEORY EDGARL. DE CASTRO PAGE5
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6. The Hurwicz criterionstrikes a balancebetween extremepessimism and
extreme optimism by weighing the above conditions by respective I
weights a and (1- a), where 0 < a <1. That is the action selected is t
,
that which yields:' .
.,
ma..'Xa m~ v( Ai , r/J ) + (1 - a )min v( Ai , r/J )
j j
Ai { t/J j t/J
j }
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[Note the above formulas represent the case where payoffs are expressed
as profits]
If a = 1, the decision rule is referred to as the MAXIMAX RULE, and if
a = 0, the decision rule becomes the MAXIMIN RULE. For the case
where the payoff represent costs, the decision rule is given by:
min a min v( Ai , f/Jj ) + (1- a) max v( Ai , f/J )
j
~ { ~ ~ }
E. BAYES' RULE
Here, weasswne that the probabilities associated with each state
of natureareknown.Let . .
P{ljJ = ljJj} = Pj
The action which minimizes (maximizes) the cost (profit) is selected.
This is given by:
The backward induction approach is used. With the aid of a decision
tree, expected values are computed each time a round node is
encountered and the above decision rule is utilized each time a square
node is encountered ,i.e., a decision is made each time a square node is
encountered.
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DECISIONTHEORY EDGARL. DE CASTRO PAGE6
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F. EXPECTED VALUE-VARIANCE CRITERION
This is an extension of the expected value criterion. Here we
simultaneously maximi~e profit and minimize the variance of the profit. . .
If Z represents profit as a random variable with variance q:, then the
criterion is given by:
maximize E(z) - Kvar(z)
where K is any specified constant. If Z represents cost: .' .
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minimize E(z) + Kvar(z)
G. DECISION MAKING WITH EXPERIMENTATION
In some situations, it may be viable to secure additional
information tp revise the original estimates of the probability of
occurrence of the state of nature.
DEFINITION 1.2 :
Preposterior Analysis considers the question of deciding whether or not
it would be worthwhile to get additional information or to perfonn
further experimentation.
DEFINITION 1.3 :
Posterior Analysis deals with the optimal choice and evaluation of an
action subsequent to all experimentation and testing using the
experimental results.
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DECISION THEORY EDGAR L. DE CASTRO PAGE 7
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8. ---.......-----
DEFINITION 1.4 :
Prior probabilities are the initial probabilities assumed without the
benefit of experiment~tion. Posterior probabilities refer to the revised
probability values obtamed after experimentation. . -
Let: Pj =prior probability estimate of event (Jj
P{Zkl(Jj} = conditional probability of experimental (jutcome Zk
P{(Jj IZk} = posterior probability of event (Jj
The experimental results are assumed to be given by Zk, k = 1, 2, ... 1.
The conditional probability can be considered to be a measure of the
reliability of the ~xperiment. The idea is to calculate the posterior
probabilities by combining the prior probabilities and the conditional
probabilities of experimental outcome Zk. The posterior probabilities are
given by:
m
L P{Zk l(Ji}P{(Ji}
i=1
Once the posterior probabilities are calculated, the original problem can
be viewed as a multiple stage/sequential DT problem. The first stage
involves the decision of whether to perform additional experimentation
or not. Once this is decided, the outcomes of the experiment are
considered together with the original set of decision alternatives and
events.
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DECISIONTHEORY EDGARL. DE CASTRO PAGE8
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9. DEFINITION 1.5 :
A perfect infonnation source would provide, with 100% reliability,
which of the states of nature wouldoccur. .
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Define: EPPI = expected profit from a perfect information source
EVPI = expected value of the perfect infonnation source
EP = Bayes' expected profit without experimentation
Then:
.,
EVPI = EPPI - EP
where:
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E VPI = L Pj * max { v( Ai , fjJ ) }
j
. j=1 Ai
EVPI is easily seen as a measure of the maximum amount a decision
maker should be willing to pay for additional infonnation.
Define: EVSI = expected value of sample information
ENOS = expected net gain from sampling
CAI = cost of getting additional information
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DECISIONTHEORY EDGARL. DE CASTRO PAGE9
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Then: I
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ENGS = EVSI - CAI
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The information source would be viable if ENGS > O.
II. CONTINUOUS DECISION THEORY.
As previously mentioned, continuous decision theory problems
refer to those where the number of alternatives and/or states of nature
can be considered infmite. The optimization model in this case is given
by: .
max f(A) =J: v(A,t/J)htjJ(t/J)dt/J
where:
htjJ(t/J) = prior distribution function of the states of nature
In the above model, it is assumed that no additional infonnation is
available and the expectation is evaluated with respect to the prior
distribution of the states of nature. If additional infonnation is available,
we update the prior distribution of the states of nature by detennining its
posterior distribution, which is nothing but the conditional distribution
of the states of nature given the experimental outcome. Hence, the
optimization converts to:
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DECISIONTHEORY EDGARL. DE CASTRO PAGE 10
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. max f(A) = f: v(A,f/J)ht/>IZ=z (f/J)df/J
where:
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I hfj)IZ=z (rjJ) = conditional distribution of the state of nature' given the
experimental outcome
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hZIfj)(z) = conditional distribution of the experimental outcome given the
state of nature
hz (z) = marginal distribution function of the experimental outcomes
where:
LEIBNIZ' RULE
LEIBNIZ' Rule is applied to find the derivative'of a function
which contains integrals. Consider a function in one variable A:
d b ig
b db da
- fa g(A,rjJ)drjJ=fa ~rjJ+g(A,b)--g(A,a)-
ciA 8A ciA ciA
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DECISION THEORY EDGAR L. DE CASTRO PAGE 11
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