Decision Analysis

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Decision Analysis
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I begin here with an example. In the table below
you see that a firm has three alternatives that
it can choose from, but it does not know what
future conditions will be. States of
Nature Decision Alternatives s1 s2
d1 8 7 d2 14 5 d3 20 -9 In this
example, the states of nature represent the
future conditions and the firm has no control of
what will happen here. Say the s1 a strong
demand for condominiums, and s2 a weak demand
for condos.
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Now, in the example the firm has a tract of land
and it has to decide how many condominiums it
should build on the land. Say d1 30 units, d2
60 units and d3 90 units. Notation di means
the ith decision alternative, sj means the jth
state of nature, and vij means the value that
would occur if the firm choose di and state of
nature sj occurred. In the current example the
vij refers to profit. In fact, the information
presented is in payoff table format and here the
values would represent profit. The values could
represent other ideas, depending on the problem.
We could have costs, revenue, or other items in
the table.
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The information in a problem such as the one
considered can also be presented in decision tree
format. At the far left we show the decision
alternatives as the main branches of the tree.
Off of each alternative we have each state of
nature and at the end we put the value. For our
example we would have the following
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Strong weak strong weak
7
30 units 60 units 90 units
14
5
Strong weak
20
-9
The decision maker must choose one from these
three alternatives
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Now, there are times when we might have some
information about how likely the states of nature
might be and sometimes we dont have that
information. For now, say we do not have this
information. How do we decide what to do? One
approach used to decide which alternative to
choose is called the Optimistic Approach. The
method would have you find the best value in
the table and then choose the alternative with
that value. The idea of best depends on the
context of the problem. If the problem is about
profit best would mean highest and if the problem
is about cost, for example, then best means
lowest. In our example, since we have profit best
means highest and thus we would choose d3, build
90 units.
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I have reproduced our example here because I want
to show you the optimistic approach in a little
more sophisticated way. States of
Nature Decision Alternatives s1 s2
d1 8 7 d2 14 5 d3 20 -9 Let us now
list for each alternative the maximum profit that
could occur d1 8 d2 14 d3 20. Now, the option
to choose is the one with the maximum profit.
Here we would choose d3. When profit is involved
the optimistic approach is often called the
maximax approach we choose the maximum of the
maximums. If we have costs (or other ideas where
low is best) then we look for the minimin.
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Conservative Approach States of
Nature Decision Alternatives s1 s2
d1 8 7 d2 14 5 d3 20 -9 Let us now
list for each alternative the minimum profit that
could occur d1 7 d2 5 d3 -9. Now, the option
to choose is the one with the maximum profit.
Here we would choose d1. When profit is involved
the conservative approach is often called the
maximin approach we choose the maximum of the
minimums. If we have costs (or other ideas where
low is best) then we look for the minimax. This
method is conservative in that we start by
looking at the worst that can happen in each case
and then we choose the best of the worst case
scenarios.
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Minimax Regret Approach Regret implies thinking
about what might have been. In this scenario we
look at each state of nature and ask what would
be the best alternative. You can see if s1
occurs the best would be d3 with a value of 20
and if s2 occurs the best would be d1 with a
value of 7. So, basically, we look at the
highest value in each column. Next we will
modify all the numbers in the table by taking the
column maximum number and subtract off the
original number in the cell and then take the
absolute value of the result. The original table
States of Nature Decision Alternatives
s1 s2 d1 8 7 d2 14 5 d3 20 -9
becomes States of Nature Decision
Alternatives s1 s2 d1 12 0 d2 6 2
d3 0 16 The 16 at the bottom right is
absolute(7 - -9) 16
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Next, I have a list of the maximum regret for
each alternative d1 12 d2 6 d3 16 Choose the
minimum of the maximum regret values chooses d2
the minimax. So, for this one problem, each
method yielded a different optimal solution.
It seems to me the temperament of the decision
maker will influence which method would be
used. Next lets turn to a situation where we
know the probability of occurrence of each state
of nature.