strategy. What did you choose? A cross section of student responses typically
reveals a wide variety of opinion. To the nearest 10 percent, roughly 20 per-
cent of students choose not to invest, 30 percent elect to invest all the way if
necessary, and the remainder choose to stop after a certain number of fail-
ures. In fact, the most popular choice (about one-third of the responses) is to
stop after three failures. Typical reasoning is that the “prize” is worth $10 mil-
lion, so the firm should spend no more than $9 million (three failures) in
pursuing it.
However, a little reflection shows that this reasoning is faulty. Stopping after
a $9 million loss clearly is inferior to investing all the way. With the latter strat-
egy, a $10 million “success” is ensured at a cost of no more than $15 million;
the firm’s loss is $5 million at worst.
The key to a correct analysis lies in recognizing the repetitive nature of the
firm’s decision problem. Having invested and failed, the firm faces the same
decision as before under nearly the same conditions. Money already invested
is a sunk cost and so is irrelevant as far as future actions are concerned. The rel-
evant variables are the incremental investment cost, the profit from future suc-
cess, and the probability of success. The first two variables are unchanged
throughout, whereas the last increases stage by stage. This observation leads to
an important conclusion: If it is ever worth investing initially, it is worth continuing
to invest, because the odds of success get better and better (and no other facts change).
Thus, the firm can narrow its courses of action down to two: Either it should
invest all the way or it should not invest at all.
The decision tree in Figure 13.3 shows the firm’s best course of action.
Note the repetitive nature of the tree. Decisions alternate with chance nodes.
The firm continues to travel down the tree for as long as it invests and fails to
achieve success. At each branch tip, the firm’s resulting profit (net of all costs
accumulated to date) is shown. As always, the optimal decision is found by aver-
aging back the tree from right to left. Thus, the last decision encountered is the
first one analyzed. If it comes to that decision, the firm obviously should invest
a fifth time. (A $5 million loss from continuing is better than a $12 million loss
from quitting.) Similarly, a comparison of expected values at each point of deci-
sion shows that the firm should invest a fourth time, a third time, and so on.
Averaging back the tree, we find the initial investment to be profitable. Its
expected value is $1 million. Thus, investing all the way is the optimal course
of action.
Because of the ever-increasing probabilities in the preceding example, the
optimal-stopping strategy is to start and neverstop. The more the company
spends on the program, the closer it gets to ultimate success. Here, the basic
uncertainty is not whether success will come but whether it will come sooner
or later. Of course, not all R&D programs share this feature. The riskiest
research programs—those that depend on breakthroughs beyond the “current
state of the art”—may never succeed, regardless of the size of the investment.
(If it were simply a matter of spending money, the war on cancer would have560 Chapter 13 The Value of Informationc13TheValueofInformation.qxd 9/26/11 11:02 AM Page 560