Formal decision analysis is often used when a decision must be made under conditions of
significant uncertainty. SmartDrill can assist management with any of a variety of decision analysis
approaches, depending on the situation.
Here we discuss various possible approaches to decision analysis, as well as their strengths and limitations,
using a hypothetical example of a company that is trying to decide where to purchase land on which to build a new
restaurant. The decision is based on where the company believes a new sports stadium will be built, because
the presence of the stadium would greatly increase the restaurant's profitability.
Based on existing research, the company knows that the city will build the new sports stadium in either the
eastern or western suburb of the city, but the city has not made a final decision. The company needs to
buy the land before the stadium siting decision is made public, in order to save money on the purchase price of the
land, which will probably increase substantially when the location for the new stadium is made
public.
Here is the statement of the problem:
A wellsituated parcel of land in the eastern suburb currently costs $30 million; a parcel in the western
suburb currently costs $21 million. The present value of future cash flows of the East location is $52
million if the stadium is built there; the value of the West location is $40 million if the stadium is built
there. The value of the East parcel if the stadium is not built there is only $10 million; and the
corresponding value for the West parcel is $7 million.
Using a Decision Matrix
One common method of portraying decision analysis involves decision matrices. Here is the basic
payoff matrix for this problem:
The table above shows us that if the company buys the East parcel and the stadium is built there, the expected
payoff is $22 million. If they buy the West parcel and the stadium is built in the East, then they will lose
$20 million. If they buy the West parcel and the stadium is built there, then the payoff will be
$19 million. And if they buy the West parcel and the stadium is built in the East, then they will lose $14
million.
The company could also hedge its bets and buy both parcels of land. If they do this and the stadium is
built in the East, then the payoff is $8 million; but if the stadium is built in the West, then they lose $1
million. And of course, if the company decides not to buy either parcel, then they will neither gain nor lose
anything.
So the question is: What should the company do? To answer this question, there are several
decision rules that they could consider. Let's examine these various rules.
Maximax
If the company is optimistic and believes that events will tend to favor their decision, then they could
use the maximax decision rule, as shown below:
Here the assumption is that the company should make the decision that has the maximum
payoff. Thus they would buy the East parcel and hope to make $22 million. But of course there
is no guarantee that the company actually will have made the right choice, and they could lose $20 million
if they're wrong. And what if the $20 million loss in the above table were actually a much larger
loss, e.g., $100 million? Unfortunately, the maximax rule doesn't take into account the magnitude of the
downside of a wrong decision; so it can lead to some
really poor decisions.
Maximin
If the company is conservative and pessimistic, and believes that events will tend to go
against their decision, then they could use the maximin decision rule, as shown
below:
Here the company is aiming for the least loss. In this case, that would mean that the
company wouldn't buy either parcel of land. This decision rule can also lead to some poor
decisions, because it doesn't consider the magnitude of the potential upside. For example, what if the $22
million cell in the above table were actually $100 million?
Minimax Regret
Another approach to the problem involves using the minimax regret decision rule, that focuses
on what is sometimes referred to as opportunity cost or opportunity
loss. Here the strategy is to make the decision that, if wrong, will result in the least amount of
regret or opportunity loss, as shown below:
So if the company buys the East parcel, but the stadium is built in the West, then the maximum regret or
opportunity loss would be $39 million. If they buy the West parcel, but the stadium is built in the
East, then the maximum opportunity loss would be $36 million. If they buy both East and West
parcels, then the maximum regret would be $20 million. And if they decide to purchase neither
parcel, then the opportunity loss would be $22 million. So if they pick the minimum of these four
maximum regret values ($20 million), they would decide to buy both East and West parcels.
This decision rule is arguably the most defensible of the various decision rules we have considered so
far. But it can also lead to some poor decisions if just one more option is added to the
problem. For example, suppose a third attractive parcel of land suddenly becomes available in either the East
or West suburb. Depending on the values assigned to this third parcel, it is possible, according to
the conditions of the minimax regret rule, that even if the newly available third parcel is
not chosen, simply applying the rule to this new situation could actually change the original
decision. So changes in the situation can lead to some inconsistent results, which can be quite
troubling and problematic for the decisionmaker.
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Expected Value
The previous examples of decision matrices were nonprobabilistic, because they assumed that the
stadium was equally likely to be built in either the East or the West suburb. But in many cases where
historical data exist, or research can be conducted, or the services of an experienced expert consultant can be
contracted, we can try to assign probabilities to the possible stadium siting outcomes, thus creating a
probabilistic analysis. In our example above, we implicitly assigned probabilities of
0.50 to each possible reality (the stadium's being built in either the East or West). But we can
use explicit probabilities in what is called expected value or expected
monetary value (EMV) analysis.
First, here is the original payoff matrix, to which we have added probabilities for where the stadium is likely
to be built:
Here we have simply multiplied the original payoff values by the probability of the respective states of nature,
and then summed across each row. For example, for the first row we multiply 22 x 0.50 to get 11; and we
multiply 20 x 0.50 to get 10. When we sum these two products, we get the 1.0 shown in the EMV column.
Note that the EMV solution to the decision problem gives us the same result as the previous minimax
regret solution: the company chooses to buy both parcels of land to hedge their bets. But we can also
transform the EMV approach into an expected regret or expected opportunity loss
(EOL) approach, just as we did previously for the nonprobabilisic minimax regret matrix.
Here is the result:
Here we see that the EMV and EOL methods lead us to the same decision. And, in fact, these two
approaches will always arrive at the same decision. They are two equivalent ways of choosing
the same decision alternative.
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Sensitivity Analysis
Now let us examine one of the particular benefits of the probabilistic EMV decision analysis. Sometimes,
based on historical data, additional research and/or expert advice, we can adjust the probabilities away from
0.50 to reflect this new information. And sometimes relatively small adjustments can make a big
difference in our final decision.
Below is a sensitivity analysis table that shows us how the decision would change based on a
change in the probabilities of the two possible realities or states of nature:
In the table above, the first row of numbers are simply the EMV values we saw before that
caused us to buy both parcels of land. The leftmost column of numbers are a range of probabilities
for the stadium's being built at the East location. Notice how the EMVs for the various
possible decisions change as we change these probabilities. Our original problem assumed a 50/50
chance of the stadium's being built at the East location, and the EMVs for this scenario are shown in the row
highlighted in red.
But if the probabilities begin to deviate enough from the original 50/50 scenario, then we see shifts in the
decision choice. The rows of the table highlighted in green show the range of probabilities that would lead
us to choose to buy only the West parcel; and the rows highlighted in blue show us the range of probabilities that
would lead us to choose to buy only the East parcel. The line graph below gives us another way to visualize
the data in the table:
So now let's see what happens if the probability of the stadium's being built at the East location changes
from 0.50 to 0.45. Although this is a small change in probability, it ends up changing the
decision. Now, instead of buying both parcels of land, the company chooses to buy just the West parcel, thus
saving a substantial amount of investment capital:
The Expected Value of Perfect Information
So the additional data, research and/or expert advice has allowed us to adjust the probabilties,
which has in turn allowed us to change our decision with confidence. But how much should we be willing to pay
for the new data, research and/or expert advice, which is sometimes referred to as the expected value
of perfect information?
In the table above we have subtracted the maximum EMV from the "EMV with perfect information" to arrive at the
value of perfect information, which in this case is about $16.2 million. [Note: Given the same input
probability data, the expected value of perfect information (EVPI) will always be equal to the minimum
expected opportunity loss (EOL) that we saw previously.]
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Decision Trees
Another general method of portraying decision analyses is via the use of decision trees instead of decision
matrices. The various nonprobabilistic decision rules that we discussed previously can be applied to decision
trees as well as to payoff matrices. However, decision trees are most commonly used to portray the results of
either the probabilistic EMV decision rule or rules based on utility theory (which is discussed
below).
Decision trees have some advantages over decision matrices. First, the graphical nature of decision trees
makes it very easy to identify the decision alternative having the highest EMV; and decision trees can clearly
portray much more complicated decision situations than can be easily understood from a matrix. In addition,
it is easy to "roll back" complicated decision trees and then show analyses based not only on different
probabilities but also on different theoretical analytic approaches (such as EMV rules vs. utility theory).
And lastly, a big advantage of decision trees is that they can easily portray multistage decision
processes, where a subsequent decision may depend on earlier ones in a chain of sequential
decisions. Such situations arise very often, making decision trees a more attractive method of
displaying results instead of via matrices.
For the restaurant siting problem, the decision tree below shows us the various combinations of decision
alternatives (where to buy land) and possible states of nature (where the stadium could be built), and
shows us the outcomes clearly:
The above decision tree shows us what decision is recommended when the states of nature have equal
probabilities of occurring (the green box). The yellow boxes represent the possible decision
alternatives; and the gray boxes represent the two possible states of nature that can be combined with the four
decision alternatives to give us eight possible combinations.
The green box is labeled "Decision" because it's text tells us the final decision ("Buy Both") as well
as the EMV associated with that decision alternative. The yellow boxes are labeled
"Chance" because we do not know what the result of any given decision will be until we learn where the stadium
is actually going to be built. The gray boxes are labeled "Terminal" because they represent the possible
states of reality that end the decision analysis process in the tree diagram.
Notice also that the yellow boxes contain the EMV for each choice; and the 0.50 probability is shown to the left
of each of the Terminal nodes, reflecting the fact that each state of reality is assumed to have an equal
probability. The Terminal boxes also contain the same payoff numbers we saw in the original payoff matrix
near the top of this Web page. And the red arrow points clearly to the decision alternative that the model
has chosen.
Below is the decision tree for the modified problem, where we reduce the probability that the stadium will
be built in the East suburb from 0.50 to 0.45. This causes a change in our decision, from buying both parcels
to buying only the West parcel, which the tree diagram clearly shows:
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Utility Theory
While the EMV approach to decision analysis is often useful, it does not take into account the
possibility that different decisionmakers may have different risk tolerances or risk
profiles. Depending on the situation and/or the decisionmaker, it is useful to be able to
adjust the analyses to suit any of a variety of possible risk profiles. Utility theory allows us to do
that.
Without getting too technical or mathematical, we'll just point out that most business
decisionmakers tend to be either riskneutral or somewhat risk averse; so we'll focus on those types of
decisionmaker. Also, being more riskaverse means that the decisionmaker is willing to settle for a smaller
potential payoff in exchange for reduced risk ("lower risk, lower reward"). Here is a utility function graph
for a riskneutral person:
The riskneutral person follows the EMV decision rule we discussed earlier, resulting in a linear
utility function across the spectrum of possible payoffs ranging from $20 million to +$22 million. The
riskneutral person is said to have constant marginal utility for increased payoffs. In other words, each
additional dollar in potential payoff leads to the same amount of increase in utility.
Here is a utility function graph for a slightly riskaverse person:
Here we can see a slightly concave utility function (a slight exponential curve), indicating a
tendency for the person to have a slightly reduced marginal utility for increased profit beyond a certain
point.
Next we see a utility function graph for a moderately riskaverse person:
Here the person is expressing an even greater diminution in marginal utility with increasing
payoffs.
And finally, here is a graph for a very riskaverse individual:
Here the decisionmaker shows a marked diminution in marginal utility after potential payoffs reach
a certain point. (Note: although we did not show it here, the utility function graph for a risktaker
would be convex instead of concave, indicating a nonlinearly increasing marginal utility
associated with increasing payoffs.)
Now let's examine the utilitytheory versions of our original decision tree diagrams. First,
let's look at how the first decision tree above (that assumed a 50/50 chance of the stadium's being built at the
East location) can be rolled back and portrayed according to utility theory, based on the risk profile of a
riskneutral decisionmaker (i.e., someone who is neither riskaverse nor a risktaker):
The decision tree looks somewhat similar to the first one, except that now the various boxes
contain utilities instead of EMV values. (Note: for the riskneutral decisionmaker, the EMV of
3.5 associated with a 0.50 probability of the stadium's being built in the East suburb is equivalent to the utility
value of 0.56 shown in the above tree. The reader can confirm this by examining the linear utilityfunction
graph for the riskneutral decisionmaker we discussed earlier, as well as the colorcoded table that appears
above at the start of the Sensitivity Analysis section of this page.)
And finally, here is the utilitytheory version of the decision tree in which we assumed only
a 0.45 probability that the stadium would be built in the East suburb. Here again, the model is based on a
riskneutral decisionmaker:
In this case, the 0.575 utility in the second yellow box is equivalent to the 4.2 EMV value shown
in the earlier decision matrix that displayed a 0.45 probability for the East location and also showed an EVPI
(expected value of perfect information) of $16.2 million.
SmartDrill can assist our clients with very difficult decisions using any of the above techniques,
although we have generally found decision trees based on utility functions to be the most useful. And
although we have presented relatively simple singlestage decision trees, we can build very complicated multistage
trees as well. The latter can help to sort out very complex decision processes, and can typically help
clients save a great amount of money.
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