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Optimal Sequential Decisions

In addition to optimal-stopping problems, managers face a variety of sequen-

tial decisions involving the order of actions. Chapter 12’s R&D decision is one

such example. Although such decisions usually are more complicated than

stopping problems, many have simple enough structures that they can be

solved without decision trees. Here is an example.

SEQUENCING R&D INVESTMENTS Suppose a firm can choose one of several

programs to develop a new product. Regardless of which method it uses, the

firm earns a predictable profit (call this ) upon successful development. How-

ever, the methods have differing investment costs (c) and probabilities of suc-

cess (p). In what order should the firm pursue the methods?

The surprisingly simple answer is that the methods should be pursued in

order of their probability-to-cost ratios, p/c. The program with the greatest p/c

ratio should be tried first. If it succeeds, the firm’s search is over; if it fails, the

program with the next highest ratio should be tried next; and so on. To check

this result, consider the case of two programs, A and B. If the firm pursues A

first, its expected net benefit is

[13.5]

Here, the firm’s expected gross profit is pApBpApB. The cost of

the two programs is cAcB. However, if A is successful, the firm saves cB.

This happens with a probability of pAand accounts for the last term in the

second line of Equation 13.5. If program B is pursued first instead, the

firm’s expected net benefit is identical to the second line of Equation 13.5

except that the last term is pBcA. Therefore, pursuing A first is more prof-

itable than pursuing B first if and only if pAcBpBcA, or, equivalently, pA/cA

pB/cB. Thus, the programs should be pursued in order of their probability-

to-cost ratios. When there are more than two programs, the demonstration

is analogous.

It is interesting to note that this solution also applies to the classic prob-

lem of searching in one of a number of locations for a lost object. Suppose the

goal is to find the object in the fewest number of searches on average. It is

assumed that all locations have the same costof search. With equal search costs,

the preceding solution instructs us to begin the search in the location with

the highest likelihood of success and, if the object isn’t there, to try the next

most likely spot, and so on. Of course, this is exactly what we would expect. If

search costs differ, the way to minimize expected search costs is to search in

order of p/c. This is the best we can do, but it cannot change the fact that

pApBpApBcAcBpAcB.

pAcA(1pA)(pBcB)

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