Decision Trees 507profits: the cost of drilling and recovery, the amount of oil discovered, and the
price of oil per barrel. As the tree depicts, the cost of drilling and recovery is
the first uncertainty to be resolved and depends on the depth at which oil is
found (or not found). In the wildcatter’s judgment, oil may be struck at one of
two depths or not at all. Thus, the tree depicts three branches emanating from
the initial chance node. As an example, let’s consider the second branch: oil
found at 5,000 feet. This branch ends in a chance node from which three new
branches emerge. These branches show the possible amounts of oil (barrels
per year) that might be recovered; the third branch, for instance, has a total
recovery of 16,000 barrels. Finally, each recovery branch ends in a chance node
from which three new branches sprout. These indicate the possible different
values of average oil prices over the life of the well. For example, the third
branch lists a $55-per-barrel price. At the end of this branch, the last uncer-
tainty is resolved and the wildcatter’s profit, in this case $180 thousand, is finally
determined. (Simply take the profit figures at face value. We have not supplied
the revenues and costs on which they are based.)
The path from the leftmost chance node to the $180,000 profit outcome
indicates one particular scenario that might occur: finding a 16,000-barrel oil
field at 5,000 feet and selling it at a two-year average price of $55 per barrel.
However, this outcome is but one of many possible outcomes contingent on
the resolution of the multiple risks. In all there are (2)(3)(3) 1 19 possi-
ble profit outcomes, one for each branch tip. The combination of multiple
risks, each with multiple outcomes, means that the corresponding decision tree
will be bushy indeed.
The bushy tree also requires a lengthier process of probability assessment,
because the wildcatter must evaluate probabilities for three distinct risks. The
first three branches of the tree show his chances of striking (or not striking) oil
at different depths. If he finds some oil at a given depth, the next question is
how much. The secondary branches of the tree list the chances of finding dif-
ferent oil quantities. Note that the likelihood of different recovery amounts
depends on the depth at which oil is first found, and the likelihood of very
large deposits is better at 3,000 feet than at 5,000 feet. (Remember that these
recovery probabilities are conditional on some oil being found at all. Shallow
fields are likely to be large fields, but the chance of finding oil at 3,000 feet is
only .13 in the first place.) Finally, once the recovery quantity is ascertained, the
sole remaining uncertainty concerns the market price of oil. The chances listed
on the third-level branches have been obtained from an expert’s prediction of
future prices. Note that the chances of different market prices per barrel are
independent of the quantity of oil recovered (i.e., the chances are the same
regardless of the recovery amount).
What is the wildcatter’s expected value from drilling in the face of these
multiple risks? To answer this question, we calculate the expected value in stages
by “averaging back” the tree, starting at the branch tips and working to the left.
To illustrate, consider the chance node D on the tree: a 5,000-barrel-per-year oilc12DecisionMakingunderUncertainty.qxd 9/29/11 1:34 PM Page 507