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Unformatted text preview: DECISION ANALYSIS: APPLIED DECISION THEORY Analyse des Decisions: The’orz'e Applique’e ’
des De’cz'sions RONALD A. HOWARD Institute in EngineeringEconomic Systems
Stanford University, Cali orm'a
United States of America I. INTRODUCTION Decision theory in the modern sense has existed for more than a decade. Most
of the effort among the present developers of the theory has been devoted to
Bayesian analysis of problems formerly treated by classical statistics. Many
practical management decision problems, hoWever, can be handled by formal
structures that are far from novel theoretically. The world of top management
decision making is not often structured by simple Bernoulli, Poisson, or normal
models. Indeed, Bayes's theorem itself may not be so important. A statistician for
a major company wrote a report in which he commented that for all the talk
about the Bayesian revolution he did not know of a single application in the
company in which Bayes’s theorem was actually used. The observation was
probably quite correct—but what it shows by implication is that the most sig
niﬁcant part of the revolution is not Bayes's theorem or conjugate distributions
but rather the concept of probability as a state of mind, a 200yearold concept.
Thus the real promise of decision theory lies in its ability to provide a broad
logical basis for decision making in the face of uncertainty rather than in any
speciﬁc models. The purpose of this article is to outline a formal procedure for the analysis
of decision problems, a procedure that I call “decision analysis." We shall also
discuss several of the practical problems that arise when we attempt to apply
the decision analysis formalism. 2. DECISION ANALYSIS To describe decision analysis it is ﬁrst necessary to define a decision. A decision
is an irrevocable allocation of resources, irrevocable in the sense that it is im
possible or extremely costly to change back to the situation that existed before
making the decision. Thus for our purposes a decision is not a mental commit—
ment to follow a course of action but rather the actual pursuit of that course of
action. This deﬁnition often serves to identify the real decision maker within a
loosely structured organization. Finding the exact nature of the decision to be 97 RONALD A. HO\VA RD made, however, and who will make it, remains one of the fundamental problems
of the decision analyst. Having deﬁned a decision, let us clarify the concept by drawing a necessary
distinction between a good decision and a good outcome. A good decision is a
logical decision—one based on the uncertainties, values, and preferences of the
decision maker. A good outcome is one that is proﬁtable or otherwise highly
valued. In short, a good outcome is one that we wish would happen. Hopefully,
by making good decisions in all the situations that face us we shall ensure as
high a percentage as possible of good outcomes. We may be disappointed to
find that a good decision has produced a bad outcome or dismayed to learn
that someone who has made what we consider to be a bad decision has enjoyed
a good outcome. Yet, pending the invention of the true clairvoyant, we ﬁnd no
better alternative in the pursuit of good outcomes than to make good decisions. Decision analysis is a logical procedure for the balancing of the factors that
inﬂuence a decision. The procedure incorporates uncertainties, values, and
preferences in a basic structure that models the decision. Typically, it includes
technical, marketing, competitive, and environmental factors. The essence of
the procedure is the construction of a structural model of the decision in a
form suitable for computation and manipulation; the realization of this model
is often a set of computer programs. 2.1. The Decision Analysis Procedure
Table 1 lists the three phases of a decision analysis that are worth distinction:
the deterministic, probabilistic, and postmortem phases. TABLE 1
The Decision Analysis Procedure I. Deterministic phase Deﬁne the decision Identify the alternatives Assign values to outcomes Select state variables Establish relationship at state variables
. Specify time preference amewwr An lysis: (a) Determine dominance to eliminate alternatives
(b) Measure sensitivity to identify cnicial state variables II. Probabilistic phase 1. Encode uncertainty on crucial state variables
Analysis: Develop profit lottery 2. Encode risk preference
Analysis: Select best alternative III. Postmortem phase Analysis: (a) Determine value of eliminating uncertainty in crucial state
variables (b) Develop most economical informationgathering program 98 DECISION ANALYSIS: APPLIED DECISION THEORY 2.1.1. The Deterministic Phase The ﬁrst step in the deterministic 'phase is to answer the question, ” What
decision must be made?” Strange as it may seem, many people with what
appear to be decision problems have never asked themselves that question.
\Ve must distinguish between situations in which there is a decision to be made
and situations in which we are simply worried about a bad outcome. If we have
resources to allocate, we have a decision problem, but if we are only hand
wringing about circumstances beyond our control no formal analysis will help.
The difference is that between selecting a surgeon to operate on a member of
your family and waiting for the result of the operation. We may be in a state of
anguish throughout, but decision analysis can help only with the ﬁrst question. The next step is to identify the alternatives that are available, to answer the
question, “ What courses of action are open to us? " Alternative generation is the
most creative part of the decision analysis procedure. Often the introduction
of a new alternative eliminates the need for further formal analysis. Although
the synthesis of new alternatives necessarily does not fall within the province of
the decision analysis procedure, the procedure does evaluate alternatives and
thereby suggests the defects in present alternatives that new alternatives might
remedy. Thus the existence of an analytic procedure is the ﬁrst step toward
synthesis. We continue the deterministic phase by assigning values to the various
outcomes that might be produced by each alternative. We thus answer the
question, “How are you going to determine which outcomes are good and
which are bad? " In business problems this will typically be a measure of proﬁt.
Military and governmental applications should also consider proﬁt, measured
perhaps with more difﬁculty, because these decision makers are also allocating
the economic resources of the nation. Even when we agree on the measure of
proﬁt to be assigned to each outcome, it may be difﬁcult to make the assignment
until the values of a number of variables associated with each outcome are
speciﬁed. We call these variables the state variables of the decision. Their
selection is the next step in the deterministic phase. A typical problem will have state variables of many kinds: costs of manu
facture, prices charged by competitors, the failure rate of the product, etc. We
select them by asking the question, “If you had a crystal ball, what numerical
questions would you ask it about the outcome in order to specify your proﬁt
measure?" At the same time that we select these variables we should assign
both nominal values for them and the range over which they might vary for
future reference. Next we establish how the state variables are related to each other and to
the measure of performance. We construct, in essence, a proﬁt function that
shows how proﬁt is related to the factors that underlie the decision. The con
struction of this proﬁt function requires considerable judgment to avoid the twin
difﬁculties of excessive complexity and unreal simplicity. If the results of the decision extend over a long time period, it will be neces
sary to have the decision maker specify his time preference for profit. We must 99 RONALD A. HOWARD ask, ” How does proﬁt received in the future compare in value to proﬁt received
today? ” or an equivalent question. In cases in which we can assume a perfect
ﬁnancial environment the present value of future proﬁt at some rate of interest
will be the answer. In many large decision problems, however, the nature of the
undertaking has an effect on the basic ﬁnancial structure of the enterprise. In
these cases a much more realistic modeling of the time preference for proﬁt
is necessary. Now that we have completed the steps in the deterministic phase we have a
deterministic model of the decision problem. We next perform two closely
related analyses. We perform them by setting the state variables to their
nominal values and then sweeping each through its range of values, individually
and jointly, as judgment dictates. Throughout this process we observe which
alternative would be best and how much value would be associated with each
alternative. We often observe that regardless of the values the state variables
take on in their ranges one alternative is always superior to another, a condition
we describe by saying that the ﬁrst alternative dominates the second. The
principle of dominance may often permit a major reduction in the number of
alternatives that need be considered. As a result of this procedure we have performed a sensitivity analysis on
the state variables. \Ve know how much a 10 percent change in one of the
variables will affect proﬁt, hence the optimum alternative. Similarly, we know
how changes in state variables may interact to affect the decision. This sensi
tivity analysis shows us where uncertainty is important. We identify those state
variables to which the outcome is sensitive as “crucial” state variables. Deter
mining how uncertainties in the crucial state variable inﬂuence the decision is
the concern of the probabilistic phase of the decision analysis. 2.1.2. Probabilistic Phase The probabilistic phase begins by encoding uncertainties on each of the
crucial state variables; that is, gathering priors on them. A subset of the crucial
state variables will usually be independent—for these only a single probability
distribution is necessary. The remainder will have to be treated by collecting
conditional as well as marginal distributions. We have more to say on this
process later. The next step is to ﬁnd the uncertainty in proﬁt for each alternative implied
by the functional relationship of proﬁt to the crucial state variables and the
probability distribution on those crucial state variables for the alternative.
We call this derived probability distribution of proﬁt the proﬁt lottery of the
alternative. In a few cases the proﬁt lottery can be derived analytically and in
many by numerical analysis procedures. In any case it may be approximated by
a Monte Carlo simulation. Regardless of the procedure used, the result is a
probability distribution on proﬁt (or perhaps on discounted proﬁt) for each of
the alternatives that remain in the problem. Now we must consider how to choose between two alternatives with different
proﬁt lotteries. In one case the choice is easy. Suppose that we plot the proﬁt
lottery for each alternative in complementary cumulative form; that is, plot the 100 DECISION ANALYSIS: APPLIED DECISION THEORY
Alternative .41 Profit lottery
(density function) Profit Alternative A2 Proﬁt lottery
(probability of
profit exceeding x) Figure 1. Stochastic dominance. probability of proﬁt exceeding .7: for any given x. Suppose further, as shown
in Figure 1, that the complementary cumulative for alternative A2 always lies
above that for alternative A1. This means that for any number .1: there is a
higher probability of proﬁt exceeding that number with alternative A2 than
with alternative A1. In this case we would prefer alternative A2 to alternative
A1 , provided only that we liked more proﬁt better than less proﬁt. We describe
this situation by saying that the proﬁt from alternative A2 is stochastically
greater than the proﬁt from alternative A1 or equivalently by saying that alter
native A2 stochastically dominates alternative A1. Stochastic dominance is a
concept that appeals intuitively to management; it applies in a surprising
number of cases. Alternative A1 Alternative A2 Proﬁt lottery
(density function) Profit Alternative A; Alternative A2 Profit lottery
(probability of
profit exceeding x) Figure 2. Lack of stochastic dominance. lOl RONALD A. HOWARD Figure 2, however, illustrates a case in which stochastic dominance does not
apply. When faced with a situation like this, we must either abandon formal
methods and leave the selection of the best alternative to judgment or delve into
the measurement of risk preference. If we choose to measure risk preference,
we begin the second step of the probabilistic phase. We must construct a
utility function for the decision niaker that will tell us whether or not, for
example, he would prefer a certain 4 million dollars proﬁt to equal chances of
earning zero or 10 million dollars. Although these questions are quite foreign
to management, they are being asked increasingly often with promising results.
Of course, when risk preference is established in the form of a utility function,
the best alternative is the one whose proﬁt lottery has the highest utility. 2.1.3. PostMorten: Phase The postmortem phase of the procedure is composed entirely of analysis.
This phase begins when the best alternative has been selected as the result of
the probabilistic phase. Here we use the concepts of the clairvoyant lottery to
establish a dollar value of eliminating uncertainty in each of the state variables
individually and jointly. Being able to show the impact of uncertainties on
proﬁt is one of the most important features of decision analysis. It leads directly
to the next step of the post—mortem, which is ﬁnding the most economical
informationgathering program, if,_in fact, it would be proﬁtable to gather more
information. The informationgathering program may be physical research, a
marketing survey, or the hiring of a consultant. Perhaps in no other area of its
operations is an enterprise in such need of substantiating analysis as it is in the
justiﬁcation of informationgathering programs. Of course, once the informationgathering scheme, if any, is completed, its
information modiﬁes the probability distributions on the crucial state variables
and consequently aﬁ'ects the decision. Indeed, if the informationgathering
program were not expected to modify the probability distributions on the
crucial state variables it would not be conducted. We then repeat the proba
bilistic phase by using the new probability distributions to find the proﬁt lotteries
and then enter the postmortem phase once more to determine whether further
information gathering is worthwhile. Thus the decision analysis is a vital
structure that lets us compare at any time the values of such alternatives as
acting, postponing action and buying information, or refusing to consider the
problem further. We must remember that the analysis is always based on
the current state of knowledge. Overnight there can arrive a piece of infor
mation that changes the nature of the Conclusions entirely. Of course, having
captured the basic structure of the problem, we are in an excellent position to
incorporate any such information. Finally, as the result of the analysis the decision maker embarks on a course
of action. At this point he may be interested in the behavior of several of the
state variables for planning purposes; for example, having decided to introduce
a new product, he may want to examine the probability distributions for its
sales in future years to make subsidiary decisions on distribution facilities or 102 DECISION ANALYSIS: APPLIED DECISION THEORY on the size of the sales force. The decisionanalysis model readily provides
such planning information. 2.2. The Advantages of Decision Analysis Decision analysis has many advantages, of which we have described just
a few, such as its comprehensiveness and vitality as a model of the decision and
its ability to place a dollar value on uncertainty. We should point out further
that the procedure is relevant to both one of a kind and repetitive decisions.
Decision analysis offers the operations research profession the opportunity to
extend its scope beyond its traditional primary concern with repetitively
veriﬁable operations. One of the most important advantages of decision analysis lies in the way it
encourages meaningful communication among the members of the enterprise
because it provides a common language in which to discuss decision problems.
Thus engineers and marketing planners with quite different jargons can appreci
ated one another’s contributions to a decision. Both can use the decisionanalysis
language to convey their feelings to management quickly and effectively. A phenomenon that seems to be the result of the decision—analysis language
is the successive structuring of staff groups to provide reports that are useful
in decisionanalysis terms. Thus, if the decision problem being analyzed starts
in an engineering group, that group ultimately seeks inputs from marketing,
product planning, the legal staff, and so on, that are compatible with the proba
bilistic analysis. Soon these groups begin to think in probabilistic terms and to
emphasize probabilistic thinking in their reports. The process seems irrever
sible in that,once the staff of an organization becomes comfortable in dealing
with probabilistic phenomena they are never again satisﬁed with deterministic
or expected value approaches to problems. Thus the existence of decision
analysis concepts as a language for communication may be its most important advantage. 2.3. The Hierarchy of Decision Analysis It is informative to place decision analysis in the hierarchy of techniques
that have been developed to treat decision problems. “'e see that a decision
analysis requires two supporting activities. One is a lower order activity that we
call alternative evaluation; the second, a higher order activity that we call goal
setting. Performing a decision analysis requires evaluating alternatives according
to the goals thathave been set for the decision. The practitioners of operations
research are quite experienced in alternative evaluation in both industrial and
military contexts. In fact, in spite of the lip service paid to objective functions,
only rare operations researchers have had the scope necessary to consider the
goal—setting problems. All mankind seems inexpcrt at goal setting, although it is the most important
problem we face. Perhaps the role of decision analysis is to allow the discussion
of decisions to be carried on at a level that shows the explicit need for goals or
criteria for selection of the best alternative. We need to make goals explicit only 103 RONALD A. HOWARD if the decision maker is going to delegate the making of the decision or if he is
unsure of his ability to be consistent in selecting the best alternative. \Ve shall
not comment on whether there is a trend toward more or less delegation of
decision making. However, it is becoming clear to those with decisionmaking
responsibilities that the increasing complexity of the operations under their
control requires correspondingly more formal approaches to the problem of
organizing the information that bears on a decision if inconsistent decisions are
to be avoided. The history of the analysis of the procurement of military weapons systems
points this out. Recent years have shown the progression of procurement
thinking from effectiveness to cost effectiveness. In this respect the military
authorities have been able to catch up in their decision—making apparatus to
what industry had been doing in its simpler problems for years. Other agencies
of government are now in the process of making the same transition. Now all
must move on to the inclusion of uncertainty, to the establishment of goals that
are reflected in risk and time preferences. These developments are now on the horizon and in some cases in sight;
for example, although we have tended'to think of the utility theory as an
academic pursuit, one of our major companies was recently faced with the
question, ” Is 10 million dollars of proﬁt sufﬁcient to incur one chance in 1 mil
lion of losing 1 billion dollars? " Although the loss is staggering, it is realistic
for the company concerned. Should such a large company be riskindifferent
and make decisions on an expected value basis? Are stockholders responsible
for diversifying their risk externally to the company or should the company be
riskaverting on their behalf? For the ﬁrst time the company faced these ques
tions in a formal way rather than deciding the particular question oh its own
merits and this we must regard as a step forward. Decision analysis has had its critics, of course. One said, “In the ﬁnal
analysis, aren't decisions politically based? " The best answer to that came from
a high official in the executive branch of our government who said, “ The better
the logical basis for a decision, the more difﬁcult it is for extraneous political
factors to hold sway.” It may be discouraging in the short run to see logic over
ridden by the tactical situation, but one must expect to lose battles to win
the war. Another criticism is, “If this is such a good idea, why haven’t I heard of it
before?” One very practical reason is that the operations we conduct in the
course of a decision analysis would be expensive to carry out without using
computers. To this extent decision analysis is a product of our technology.
There are other answers, however. One is that the idea of probability as a state
of mind and not of things is only now regaining its proper place in the world of
thought. The opposing heresy lay heavy on the race for the better part of a
century. We should note that most of the operations research performed in
World War II required mathematical and probabilistic concepts that were
readily available to Napoleon. One wonders about how the introduction of
formal methods for decision making at that time might have affected the
course of history. 104 DECISION ANALYSIS: APPLIED DECISION THEORY 3. THE PRINCIPLES OF THE DECISION ANALYST Next we turn to the principles of the decision analyst, the professional who
embarks on preparing a decision analysis. His ﬁrst principle is to identify and
isolate the components of the decision—the uncertainty, risk aversion, time
preference, and problem structure. Often arguments over which is the best
decision arise because the participants do not realize that they are arguing on
different grounds. Thus it is possible for A to think that a certain alternative is
riskier than it is in B’s opinion, either because A assigns different probabilities
to the outcomesthan B but both are equally risk—averting, or because A and B
assign the same probabilities to the outcomes but differ in their risk aversion.
If we are to make progress in resolving the argument, we must identify the
nature of the difﬁculty and bring it into the open. Similar clariﬁcations may be
made in the areas of time preference or in the measurement of the value of
outcomes. One aid in reducing the problem to its fundamental components is restricting
the vocabulary that can be used in discussing the problem. Thus we carry on
the discussion in terms ofevents, random variables, probabilities, density functions,
expectations, outcomes, and alternatives. We do not allow fuzzy thinking about
the nature of these terms. Thus “The density function of the probability”
and “The conﬁdence in the probability estimate” must be nipped in the bud.
We speak of “ assigning,” not “ estimating," the probabilities of events and think
of this assignment as based on our “state of information.” These conventions
eliminate statements like the one recently made on a TV panel of doctors who
were discussing the right of a patient to participate in decision making on his
treatment. One doctor asserted that the patient should be told of “some kind
of a chance of a likelihood of a bad result." I am sure that the doctor was a
victim of the pressures of the program and would agree with us that telling
the patient the probability the doctor would assign to a bad result would be
preferable. One principle that is vital to the decision analyst is professional detachment
in selecting alternatives. The analyst must not become involved in the heated
political controversies that often surround decisions except to reduce them to a
common basis. He must demonstrate his willingness to change the recommended
alternative in the face of new information if he is to earn the respect of all con
cerned. This professional detachment may, in fact, be the analyst’s single most
valuable characteristic. Logic is often severely strained when we are personally
involved. The detachment of the analyst has another positive beneﬁt. As an observer
he may be able to suggest alternatives that may have escaped those who are
intimately involved with the problem. He may suggest delaying action, buying
insurance, or performing a test, depending on the nature of the decision. Of
course, the comprehensive knowledge of the properties of the existing alternatives
that the decision analyst must gain is a major aid in formulating new alternatives. Since it is a rare decision that does not imply other present and future
decisions, the decision analyst must establish a scope for the analysis that is 105 RONALD A. HOWARD broad enough to provide meaningful answers but not broad enough to impose
impractical computational requirements. Perhaps the fundamental question in
establishing scope is how much to spend on decision analysis. Because the
approach could be applied both to selecting a meal from a restaurant menu and
to allocating the federal budget, the analyst needs some guidelines to determine
when the analysis is worthwhile. The question of how much decision analysis is an economic problem sus
ceptible to a simpler decision analysis, but rather than pursue that road let us
pose an arbitrary and reasonable but indefensible rule of thumb: spend at least
1 percent of the resources to be allocated on the question of how they should be
allocated. Thus, if we were going to buy a 2000.dollar automobile, the rule
indicates a 20dollar analysis, whereas for a 20,000dollar house it would specify
a ZOOdollar analysis. A 1milliondollar decision would justify 10,000 dollars’
worth of analysis or, let us say, about three man—months. The initial reaction to
this guideline has been that it is conservative in the sense of not spending much
on analysis; yet, when we apply it to many decisions now made by business and
government, the reaction is that the actual expenditures on analysis are only
onetenth or onehundredth as large as the rule would prescribe. Of course,
we can all construct situations in which a much smaller or larger expenditure
than given by the rule would be appropriate, and each organization can set its
own rule, perhaps making the amount spent on analysis nonlinear in the re
sources to be allocated. Nevertheless, the 1 percent ﬁgure has served well to
illustrate where decision analysis can be expected to have the highest payoff. The professional nature of the decision analyst becomes apparent when he
balances realism in the various parts of the decisionanalysis model. Here he
can be guided only by what used to be called engineering judgment. One
principle he should follow is to’ avoid sophistication in any part of the problem
when that sophistication would not affect the result. We can describe this
informally by saying that he should strive for a constant “wince ” level as he
surveys all parts of the analysis. One indication that he has achieved this state
is that he would be torn among many possibilities for improvement if we
allowed him to devote more time and resources to the decision model. 4. THE ENCODING OF SUBJECTIVE INFORMATION One unique feature of decision analysis is the encoding of subjective infor
mation, both in the form of risk aversion and in the assignment of probabilities. 4.1. Risk Aversion and Time Preference Since we are dealing in most cases with enterprises rather than individuals,
the appropriate risk aversion and time preference should be that of the enter
prise. The problem of establishing such norms is beyond our present scope.
It is easy, however, to demonstrate to managers, or to anyone else for that.
matter, that the phenomenon of risk aversion exists and that it varies widely
from individual to individual. One question useful in doing this is, ” How much
would you have to be paid to call a coin, double or nothing, for next year's 106 DECISION ANALYSIS: APPLIED DECISION THEORY salary?" Regardless of the salary level of the individuals involved, this is a
provocative question. We point out that only a rare individual would play such
a game for a payment of zero and that virtually everyone would play for a
payment equal to next year’s salary, since then there would be nothing to lose.
Thereafter we are merely haggling over the price. Payments in the range of
60 percent to 99 percent of next year's salary seem to satisfy the vast majority
of professional individuals. The steps required to go from a realization of personal risk aversion and time
preference to corporate counterparts and ﬁnally to a reward system for managers
that will encourage them to make decisions consistent with corporate risk
aversion and time preference remain a fascinating area of research. 4.2. Encoding of Uncertainty When we begin the probabilistic phase of the decision analysis, We face the
problem of encoding the uncertainty in each of the crucial state variables.
We shall want to have the prior probability distributions assigned by the people
within the enterprise who are most knowledgeable about each state variable.
Thus the priors on engineering variables will typically be assigned by the
engineering department; on marketing variables, by the marketing department,
and so on. However, since we are in each case attempting to encode a probability
distribution that reﬂects a state of mind and since most individuals have real
difﬁculty in thinking about uncertainty, the method we use to extract the priors
is extremely important. As people participate in the priorgathering process,
their attitudes are indicated successively by, “This is ridiculous,” “ It can't be
done," “ I have told you what you want to know but it doesn’t mean anything,"
“ Yes, it seems to reflect the way I feel," and “ \Vhy doesn’t everybody do this? ”
In gathering the information we must be careful to overcome the defenses the
individual develops as a result of being asked for estimates that are often a
combination of targets, wishful thinking, and expectations. The biggest difﬁ—
culty is in conveying to the man that you are interested in his state of knowledge
and not in measuring him or setting a goal for him. If the subject has some experience with probability, he often attempts to
make all his priors look like normal distributions, a characteristic we may
designate as “bellshaped” thinking. Although normal distributions are appro
priate priors in some circumstances, we must avoid making them a foregone
conclusion. Experience has shown certain procedures to be effective in this almost
psychoanalytic process of prior measurement. The ﬁrst procedure is to make
the measurement in a private interview to eliminate group pressure and to over
come the vague notions that most people exhibit about matters probabilistic.
Sending around forms on which the subjects are supposed to draw their priors
has been worse than useless, unless the subjects were already experienced in
decision analysis. Next we ask questions of the form, " What are the chances that x will exceed
10,” because people seem much more comfortable in assigning probabilities to
events than they are in sketching a density function. As these questions are 107 RONALD A. HOWARD asked, we skip around, asking the probability that x will be “greater than 50,
less than 10, greater than 30,” often asking the same question again later in the
interview. The replies are recorded out of the view of the subject in order to
frustrate any attempt at forced consistency on his part. As the interview pro
ceeds, the subject often considers the questions with greater and greater care,
so that his answers toward the end of the interview may represent his feelings
much better than his initial answers. We can change the form of the questions by
asking the subject to dine the domain of the random variable into n mutually
exclusive regions with equal probability. (Of course, we would never put the
question to him that way.) We can use the answers to all these questions to
draw the complementary cumulative distribution for the variable, a form of
representation that seems easiest to convey to people without formal prob
abilistic training. The result of this interview is a prior that the subject is willing to live with,
regardless of whether we are going to use it to govern a lottery on who buys coffee
or on the disposal of his life savings. \Ve can test it by comparing the prior with
known probabilistic mechanisms; for example, if he says that a is the median
of the distribution of x, then he should be indifferent about whether we pay him
one hundred dollars if x exceeds a or if he can call the toss of a coin correctly.
If he is not indifferent, then we must require him to change (1 until he is. The
end result of such questions is to producea prior that the subject is not tempted to
change in any way, and we have thus achieved our ﬁnal goal. The prior—gathering
process is not cheap, but we perform it only on the crucial state variables. In cases in which the interview procedure is not appropriate, the analyst
can often obtain a satisfactory prior by drawing one himself and then letting the
subject change it until the subject is satisﬁed. This technique may also be useful
as an educational device in preparation for the interview. If two or more variables are dependent, we must gather priors on conditional
as well as marginal distributions. The procedure is generally the same but
somewhat more involved. However, we have the beneﬁt of being able to apply
some checks on our results. Thus, if we have two dependent variables x and y,
we can obtain the joint distribution by measuring the prior on x and the con—
ditional on y, given x, or, alternatively, by measuring the prior onyand the con
ditional on 9:, given y. If we follow both routes, we have a consistency check on
the joint distribution. Since the treating ofjoint variables is a source of expense,
we should formulate the problem to avoid them whenever possible. To illustrate the nature of prior gathering we present the example shown
in Figure 3. The decision in a major problem was thought to depend primarily
on the average lifetime of a new material. Since the material had never been
made and test results would not be available until three years after the decision
was required, it was necessary to encode the knowledge the company now had
concerning the life of the material. This knowledge resided in three professional
metallurgists who were experts in that field of technology. These men were
interviewed separately according to the principles we have described. They
produced the points labeled “ Subjects 1, 2, and 3 ” in Figure 3. These results
have several interesting features. We note, for example, that for l: 17 Subject 108 DECISION ANALYSIS: APPLIED DECISION THEORY 1.0
+ Subject 1
0 Subject 2
E. 0.8 0 Subject 3
g 0 6
E '
E
1.; o 4 Consensus
E
:3
0.2 o 5 10 15 20
I Figure 3. Priors on lifetime of material. 2 assigned probability 0.2 and 0.25 at various points in the interview. On the
whole, however, the subjects were remarkably consistent in their assignments.
We observe that Subject 3 was more pessimistic than Subject 1. At the conclusion of the three interviews the three subjects were brought
together and shown the results. At this point a vigorous discussion took place.
Subjects 1 and 3, in particular, brought forth information of which the other two
members of the group were unaware. As the result of this information exchange,
the three group members drew the consensus curve—each subject said that this
curve represented the state of information about the material life at the end of the
meeting. It has been suggested that the proper way to reconcile divergent priors is
to assign weights to each, multiply, and add, but this experiment is convincing
evidence that any such mechanistic procedure misses the point. Divergent
priors are an excellent indicator of divergent states of information. The ex
perience just described not only produced the company's present encoding of
uncertainty about the lifetime of the material but at the same time encouraged
the exchange of information within the group. 5. A DECISIONANALYSIS EXAMPLE To illustrate the ﬂavor of application let us consider a recent decision analysis
in the area of product introduction. Although the problem was really from
another industry, let us suppose that it was concerned with the development
and production of a new type of aircraft. There were two major alternatives:
to develop and sell a new aircraft (A2) or to continue manufacturing and selling 109 RONALD A. HOWARD Deterministic
business
model Sensitivity
analysis Market
priors Physical Operating “3:123:23” Profit
priors cost function model lottery Production Capital
priors cost function Figure 4. Decision analysis for new product introduction. the present product (A1). The decision was to be based on the present value of
future expected proﬁts at a discounting rate of 10 percent per year. Initially,
the decision was supposed to rest on the lifetime of the material for which we
obtained the priors in Figure 3; however, a complete decision analysis was
desired. Since several hundred million dollars in present value of proﬁt were at
stake, the decision analysis was well justiﬁed. The general scheme of the analysis appears in Figure 4. The ﬁrst step was
to construct a model of the business, a model that was primarily a model of the
market. The proﬁt associated with each alternative was described in terms of
the price of the product, its operating capital costs, the behavior of its competi
tors, and the national characteristics of customers. The actual proﬁt and dis
counted proﬁt were computed over a 22year time period. A suspicion grew
that this model did not adequately capture the regional nature of demand.
Consequently a new model was constructed that included the market charac
teristics, region by region and customer by customer. Moving to the more
detailed basis affected the predictions so much that the additional reﬁnement
was clearly justiﬁed. Other attempts at reﬁnement, however, did not affect the
results sufﬁciently to justify a still more reﬁned model. Now, the sensitivity
analysis was performed to determine the crucial state variables, which turned
out to be the operating cost, capital cost, and a few market parameters. Because
of the complexity of the original business model, an approximate business model
essentially quadratic in form was constructed to show how proﬁt depended on
these crucial state variables in the domain of interest. The coefficients of the
approximate business model were established by runs on the complete business
model. The market priors were directly assigned with little trouble. However,
because the operating and capital costs were the two most important variables llO DECISION ANALYSIS: APPLIED DECISION THEORY in the problem, these priors were assigned according to a more detailed pro
cedure. First, the operating cost was related to various physical features of the
design by the engineering department. This relationship was called the oper
atingcost function. One of the many input physical variables was the average
lifetime of the material whose priors appear in Figure 3. All but two of the
12 physical input variables were independent. The priors on the whole set of
input variables were gathered and used with the operating—cost function in a
Monte Carlo simulation that produced a prior for the operating cost of the
product. The capitalcost function was again developed by engineering but was
much simpler in form. The input certainties were the production costs for
various parts of the product. Again, a Monte Carlo analysis produced a prior
on capital cost. . Once we had established priors on all inputs to the approximate business
model, we could determine the proﬁt lottery for each alternative, in this case
by using numerical analysis. The presentvalue proﬁt lotteries for the two alternatives looked very
much like those shown in Figure 1. The new product alternative A2 sto
chastically dominated the alternative A1 of continuing to manufacture the present
product. The result showed two interesting facets of the problem. First, it
had been expected that the proﬁt lottery for the new product alternative would
be considerably broader than it was for the old product. The image was that of
a proﬁtable and risky new venture compared with a less proﬁtable but less risky
standard venture. In fact, the results showed that the uncertainties in proﬁt
were about the same for both alternatives, thus showing how initial concepts
may be misleading. The second interesting facet was that the average lifetime of the material
whose priors appear in Figure 3 was actually of little consequence in the de
cision. It was true enough that proﬁts were critically dependent on this lifetime
if the design were ﬁxed, but if the design were left flexible to accommodate to
different average material lifetimes proﬁts would be little affected. Furthermore,
leaving the design ﬂexible was not an expensive alternative; therefore another
initial conception had to be modiﬁed. However, the problem did not yield so easily. Figure 5 shows the present
value of proﬁts through each number of years t for each alternative. Note that
if we ignore returns beyond year 7 the new product has a higher present value
but that if we consider returns over the entire 22year period the relationship
reverses, as we have already noted. When management saw these results, they
were considerably disturbed. The division in question had been under heavy
pressure to show a proﬁt in the near future—alternative A2 would not meet that
requirement. 'Thus the question of time preference that had been quickly
passed off as one of present value at 10 percent per year became the central issue
in the decision. The question was whether the division was interested in the
quick kill or the long pull. At last report the division was still trying to convince
the company to extend its proﬁt horizon. This problem clearly illustrates the use of decision analysis in clarifying the ill RONALD A. HOWARD 500 § Alternative A2 § ‘8” ._.
O
0 Expected present value of profit through year t in millions of dollars 0 2 4 6 81012141618202224
Year: Figure 5. Expected present value of proﬁt. issues surrounding a decision. A decision that might have been made on the
basis of a material lifetime was shown to depend more fundamentally on’ the
question of time preference for proﬁt. The nine man—months of effort devoted
to this analysis were considered well spent by the company. The review com
mittee for the decision commented, “ We have never had such a realistic analysis
of a new business venture before." The company is now interested in insti
tuting decisionanalysis procedures at several organizational levels. ll2 DECISION ANALYSIS: APPLIED DECISION THEORY 6. CONCLUSION Decision analysis offers operations research a second chance at top manage
ment. By foregoing statistical reproducibility we can begin to analyze the
oneof—akind problems that managers have previously had to handle without
assistance. Experience indicates that the higher up the chain of management
we progress the more readily the concepts we have outlined are accepted. A
typical reaction is, “ I have been doing this all along, but now I see how to reduce
my ideas tonumbers." Decision analysis is no more than a procedure for applying logic. The
ultimate limitation to its applicability lies not in its ability to cope with problems
but in man’s desire to be logical. ANALYSE DES DECISIONS: THEORIE
APPLIQUEE DES DECISIONS RESUME Au cours de ces derniéres anne'es, la the’orie de decision a été de plus en plus
acceptée en tant que cadre conceptuel pour la prise de décision. Ccpendant,
cette the’orie a surtout affecté les statisticiens plutét que les personnes qui en
ont le plus bcsoin: les responsables dc decisions. Cctte étude décrit un proce’dé
qui permet de replaccr dcs problemes de decision réels dans la structure de la
the'orie de décision. Le proeédé d’analyse de décision englobe chaque e'tape,
du mesurage des choix de risques et des jugemcnts portant sur des facteurs
critiques par l'établissement de structures des facteurs rclatifs a la technique,
au marche’, a la rivalitc’ commercials et a l'cnvironnement, jusqu’au mesurage
des preferences subjectives et dc la valeur dc la prediction. L’analyse de décision
met en perspective les nombrcux instruments de simulation, d’analyse nu
mérique, et de transformations de probabilitc’s qui deviennent de plus en plus
commodcs depuis le développement des systemes d’ordinateurs électroniques
dont les diﬂ'e'rentes “stations” dependent d'une “centrale” unique. Le procédé est applique a un probleme de de'cision re'ellc qui s’e'tend sur dcs
dizaines d'anne’s et dont la valeur actuelle est de plusieurs centaines de millions
de dollars. Cette étude analyse le probleme de la determination des dépenses
consacrées a l’analyse de decisions. L’une des plus importantes propriétés
de ce proce'dé tient au nombre des bénéﬁccs auxiliaircs créés au cours de l’élabor—
ation de ce genre d'étude. L’expérience montre que ces bénéﬁces peuvant
excéder en valeur le coﬁt des de’penses consacrées a l’e'laboration dc la decision. H3 ...
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This note was uploaded on 09/28/2010 for the course MS&E 252 taught by Professor Howard during the Fall '08 term at Stanford.
 Fall '08
 HOWARD

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