9781118041581

Decision Trees 503

More generally, suppose the decision maker faces a risky prospect that has

n possible monetary outcomes, v 1 , v 2 ,... , vn, predicted to occur with probabili-

ties p 1 , p 2 ,... , pn. Then the expected monetary value of the risky prospect is

E(v) p 1 v 1 p 2 v 2 .. .pnvn.

In the preceding numerical example, we have applied exactly this formula with

respect to the four possible outcomes.

DECISION TREES

The decision treeis a convenient way to represent decisions, chance events,

and possible outcomes in choices under risk and uncertainty. In fact, this sim-

ple diagram can incorporate all the information needed to “solve” the deci-

sion problem once the specific objectives of the decision maker have been

established. The method is extremely versatile. When first encountered,

choices under risk appear messy, ill defined, and puzzling. The actual choices,

the potential risks, and the appropriate objective to pursue may all be far from

clear. The individual should not be blamed for regarding his or her choice as

“a riddle wrapped in a mystery inside an enigma,” to borrow a phrase from

Winston Churchill. However, sketching a crude decision tree almost always will

clarify the options. The very structure of the tree emphasizes the ingredients

(choices, outcomes, and probabilities) necessary for making an informed deci-

sion. The more precise the tree becomes (after drawing and redrawing), the

more precise one’s thinking becomes about the problem. The “finished” tree

can then be evaluated to “solve” the decision problem. Probably more impor-

tant, the decision tree provides a visual explanation for the recommended

choice. One easily can pinpoint the “why” of the decision: which circumstances

or risks weighed in favor of which course of action. And one can undertake

any number of sensitivity analyses, altering the facts of the decision to deter-

mine the impact on the recommended course of action.

Decision trees can be simple or complex, spare or “bushy,” small enough

to evaluate by hand or large enough to require a computer. To illustrate the

method, we start with a concise example.

An Oil Drilling Decision

An oil wildcatter must decide whether to drill at a given site before his option

period expires. The cost of drilling is $200,000. This sum will be completely

lost if the site is “dry,” that is, contains no oil. The wildcatter estimates that, if

he strikes oil, the total profit (before drilling costs) over the well’s life will be

$800,000. Thus, if there is a strike, the wildcatter will earn a $600,000 profit.

Figure 12.1 shows the decision tree for the wildcatter’s problem. The tree

depicts the sequence of events in the decision, reading from left to right. The

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