Decision-making under uncertainty for capital budgeting and hiring

Decision-making under Uncertainty for Capital Budgeting and Hiring

EDWARD MILLER Professor of Economics and Finance, University of New Orleans, New Orleans, Louisiana, USA

Most business decisions are not merely risky but are subject to uncertainty. Different individuals estimate different measures of merit. It is shown that the traditional procedure of ranking the alternatives by their estimated measure of merit and then choosing the highest ranked alternative frequently gives the wrong answer even where the estimates are unbiased. The argument will be made using a simple decision tree. The effect will be shown to be especially serious where there is competition, direct or indirect, between the alternatives.

Making decisions in the presence of uncertainty has traditionally been considered a simple task conceptually even if hard in practice (at least if one was neutral regarding risk). One merely ranked the alternatives by a measure of merit (profitability, performance, cost, etc.), estimated the measure of merit for each alternative and then chose the highest ranked alternative. It was presumed that if the measure of merit was unbiased (i.e. its expected value was equal to the expected value of the actual figure of merit), this would give the correct answers. However, this presumption has recently been challenged in the capital budgeting literature (Miller, 1978a; Brown, 1978; Smidt, 1979).

This paper will show how the problem of selecting the best of a number of alternatives is affected by errors in estimates. This will be done using Bayesian techniques and decision trees. The statistical basis for the argument is that the expected value of the extreme of several observations does not equal the mean of the distribution. However, the claim of this paper to originality is not its recognition of this fact, or its proposal to use decision trees and Bayesian techniques, but the argument that the existence of competitive markets (including bureaucratic ‘markets’) implies that prior probabilities are not equal. Instead, there is ‘prior’ information that can be deduced from the nature of the market or the decision-making process being used which should (and can be) incorporated into the decision-making process. Typically, good alternatives are scarce; poor ones abundant.

There is an extensive literature on decision- making in the presence of risk. However, the traditional formulation is that the outcome of a decision is a random variable drawn from a distribution of known characteristics. While some problems meet this formulation (such as casino

gambling), most do not. More often, the problem is not that events are random but that there are errors in our estimates of the benefits from alternative courses of action. This paper presents a theory of decision-making with such errors. The concept of uncertainty used here is uncertainty about the true value for the measure of merit. It is realized that the true value is unknown and that different individuals make different estimates. It will be shown that standard procedures for decision- making with such uncertainty are incorrect.

To keep the terminology simple, each course of action under consideration will be referred to as a proposal. In some cases, proposals will be physical projects such as building a new factory. However, in other problems a proposal is merely a course of action that could be chosen. For instance, in selecting personnel each candidate can be considered a proposal.

For concisenses in exposition, the measure of merit of the relevant alternatives will be referred to as the benefit. If the relevant measure of merit is gross benefits divided by a measure of cost (i.e. a benefit-to-cost ratio), such division will be assumed to have been performed. In slightly more technical terms , the estimates are themselves random num- bers drawn from a distribution about the true value (which is itself unknown).

DEMONSTRATION OF BIAS

The standard procedure for making decisions is simple: the proposals are ranked by their estimated benefits, the proposal with the highest estimated benefit is selected and this proposal is adopted if its benefits equal or exceed the required minimum.

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MANAGERIAL AND DECISION ECONOMICS, VOL. 6, NO. 1, 1985 11

However, the above procedure contains a built-in bias likely to cause the decision-maker to accept a proposal which should be rejected, and thus to select the wrong alternative. Imagine that ten proposals of identical merit (although this is not known to the organization) are received and each is assigned to a different analyst. The analysts’ benefit estimates are random drawings from the same distribution. The proposal with the highest estimate is chosen. A little thought shows that the estimate for the chosen proposal is the highest of ten drawings from a random distribution. The expected value of the extreme of several observations is not equal to the mean of the distribution. (For instance the highest of ten drawings from a normal distribution will be 1.54 standard deviations above the mean). Thus the benefit from a proposal selected in this manner. will probably prove to be overestimated. If the distribution of estimates is normal and there are ten proposals, the expectation for the actual performance is 1.54 standard deviations below the estimate made at the time of the decision. This illustrates why proposals that should have been rejected will be accepted by the standard procedure (i.e. accepting all proposals whose estimated benefits equal or exceeded the required level). In this example the magnitude of the bias will be proportional to the estimate’s uncertainty (measured here by the standard deviation). A new method for utilizing the analysts’ work to make decisions is required.

Likewise, the standard procedure of ranking by the estimated measure of merit can easily lead to error if the degree of uncertainty differs between proposals. Imagine that our organization has several groups of analysts, each handling proposals about which there is the same degree of uncertainty. For instance, one might handle new items in existing lines, while another would consider proposed new products, while a third would consider entry into new businesses. Again consider the case where all proposals just happen to have the same actual merit (a fact that is of course unknown to the company), It was shown that the expected value of benefit estimates for the highest ranked proposal in each uncertainty class will exceed the actual benefit by an amount proportional to the uncertainty. The higher the uncertainty the greater the overestimation.

Unbiased estimates of the merits of proposals in each uncertainty class can be obtained by subtracting the expected error from the estimate (this can be calculated using order statistics). Since the amount to be subtracted depends on the uncertainty class, a ranking of proposals without this subtraction will be in error. The firm will most likely find the highest estimated benefits are from entering a new business, even if expanding in the known (low uncertainty) lines is the best course.

Of course the assumption of ten proposals which just happen to have identical benefits is made only for expositional convenience. Actual situations are

Estimated henefits

True benefits Esthation 300 error I CIo0

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Figure 1. Decision tree example

not that simple. For a more realistic situation consider Figure 1.

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DECISION TREE ANALYSIS

The decision tree in Fig. 1 shows the essence of the decision-making problem under uncertainty. The actual benefits are unknown, and all decisions must be made from estimates. There are three types of possible proposals, good ones yielding $200 benefits, medium ones with $100 benefits and bad proposals yielding no benefits. All proposals have a $160 cost, and prior to analysis they are considered to have an equal probability of being the proposal with the highest benefits. The benefits are received immediately after the expenditures so that there is no discounting needed. The analysts have no information about the proportion of the proposals they analyze that are good or bad. The analysts are quite skilled and produce unbiased estimates. Thus the expected value of a decision’s estimated benefits equals the benefit’s actual value.

In spite of the analysts’ skill, errors occur and in only one third of the cases are estimates exactly correct. In another third the benefits are overestimated by $100. In the remaining third the benefits are underestimated by $100. Of course, there is no way of knowing whether errors have been made on any particular estimate.

For instance, an estimate of $200 could be from a proposal with a true benefit of $200. However, it could equally as well have a true benefit of $100, and an overestimate of $100. The decision-maker does not know which of these circumstances he is dealing with. This illustrates the essence of decision-making under uncertainty.

Suppose that the goal is to accept all proposals with benefits in excess of $160. (Such a goal might arise if all alternatives had a cost of $160.) The standard procedure would be to accept all proposals

12 MANAGERIAL AND DECISION ECONOMICS, VOL. 6, NO. 1, 1985

whose estimated benefits exceed $160. In particular, any proposal with estimated benefits of $200 or more would be accepted. However, inspection of the decision tree shows that for the set of proposals with estimated benefits of $200, the true benefits expected value is only $150. (Half of the projects with $200 estimated benefits actually have such benefits and half have $100 benefits, which have been overestimated.) Thus, proposals with $200 estimated returns should not be accepted. Only proposals whose estimated returns exceed $200 should be accepted.

A similar error can occur in ranking proposals if the uncertainty about the benefits from different proposals differs. To visualize this error add to the problem discussed above a proposal about whose benefits there is no uncertainty. (This might be the current method of doing things.) Suppose these benefits are known to be $175. By the traditional methods this would be added to the list of proposals with estimated benefits of $175. It would rank after the proposals with estimated benefits of $200. Yet, as we have seen, the expected benefit from proposals estimated to have benefits of $200 is only $150. It would be an error to select such proposals over the proposal which assures benefits of $175. A similar effect occurs if the estimates for one proposal involve substantially less uncertainty than other proposals. Thus, errors are to be expected in ranking if estimates differ in their uncertainty. Inspection of the decision tree shows that if there are more poor proposals than good ones, the expected error will overstate benefits by an amount that increases with uncertainty about the proposal.

The source of the problem was pointed out by Brown (1974). The very process of selecting alternatives introduces a correlation between the magnitude and direction of the error and the selection probability. By the definition of unbiased estimates, the expected value of the estimated value equals the true value. If we randomly select estimated values, and then determine the true values, our estimates will be unbiased estimators of the true values. However, in most real-life situations selections are not random but are based on estimated benefits (which by definition consists of the true value plus an error term). If proposals with higher estimates are being selected, the selection probability is higher if the error term is positive than if it is negative. Thus the expected value of the error is no longer zero, but some positive amount. In deciding on the minimum estimate required to accept a proposal an allowance for the error term’s expected value is necessary. To act on unadjusted estimates is to assume that the expected error is zero. In general, the error will exceed zero. Looking at it another way, treating the estimated value of the chosen project as an estimate of the expected value of choosing it is to equate the highest of several drawings to the mean of the underlying distribution, which it is not.

POSSIBLE SOLUTIONS

Having identified the problem, it is necessary to seek a solution. The simplest method is to use the decision tree to estimate the magnitude of the expected error. With an assumption about the probability distribution of the estimation errors and an estimate of the distribution of possible proposals over different true benefit levels, the expected error can be determined. It will be the difference between the expected value for the estimate and the actual value. If the uncertainty about the merits of all proposals is equally great, the simplest strategy may be to raise the benefit estimate required for project-approval. For instance, the minimum acceptable benefit cost ratio may be raised above unity. In capital budgeting, the required rate of return may be raised above the cost of capital. This allows the analysts to continue making the best estimates possible with their usual procedures, and only the decision-maker need make a correction for uncertainty.

In the more general case, where uncertainty about the merits of proposals differs, it will be necessary to make an uncertainty correction for each proposal (although it may be practical to group the alternatives and use one correction for all members of each group). Such corrected estimates can be derived in a straightforward manner using a decision tree. The detailed calculations can easily be programmed for a computer.

For many problems, it is plausible that the benefits have a symetrical, bell-shaped distribution. It is also likely that the distribution of errors will be unimodal and symetrical. If the selection standard is high enough to select less than half of the proposals (which is to say that one is trying to select the better proposals), the effect of estimation error will cause the expected estimates for the selected proposals to exceed the expected value for the actual benefits. For a normal distribution, this results from the well-known fact that the mean of the posterior distribution will always lie between the mean of the prior distribution and the ‘sample distribution’. (See the discussion below.)

SOURCES OF PRIOR INFORMATION

It is argued that there is no prior information in the typical case. In general, this is not true. When the benefits are bounded from either above or below this constitutes prior information. If the magnitude of the possible errors in estimates is not bounded, there will still be bias from uncertainty (except in special cases). The seriousness of the bias will increase as the criterion for decisions approaches the limits to the possible benefits.

In the decision tree example discussed above,


there was prior information that the distribution of benefits was bounded at $200. Thus the set of proposals with estimated benefits of $200 would contain no proposals with underestimated benefits. Since it could contain projects whose benefits had been overestimated, the expected benefits had to be less than those estimated.

Frequently a little thought will disclose prior information. For instance, consider the common problem of evaluating a proposed investment. Suppose all levels of profitability were equally probable. This implies that the population of potential investment projects contains as many projects yielding 1000000% as it does projects yielding 10%. If this is so, one only has to screen the projects until one finds a sufficiently high- yielding project, invest in it and then retire. Unfortunately, making one’s fortune is not this simple. Thus one is forced to conclude that there is some prior information about investment projects, namely, that good ones are rare. (Professor Brown suggested this argument.) Given this prior information, it follows that the standard capital budgeting procedure of ranking projects by their estimated rate of return or present value of profits per dollar invested will lead to error in the presence of uncertainty (see below).

A similar argument can be made for the problem of selecting baseball players. If a scout tries to argue that there is no prior information about the ability of a player before he has been seen, it follows that players of batting averages of 0.350 to 0.400 are just as frequent as those with averages 0.150 to 0.200. All that is required to put together a team that will win the World Series is simply to keep searching until one finds enough players with batting averages exceeding 0.350 and then hire them. Again putting together a winning team is not so simple, and there is prior information. Failure to explicitly consider uncertainty induced bias leads to error.

In all of these cases the prior information is that there are more bad projects than good projects (i.e. a non-uniform prior) which implies that uncertainty will cause bias and the above effect should be taken into account.

APPLICATION TO CAPITAL BUDGETING

The capital budgeting case is of particular interest to teachers of managerial economics, since many textbooks (see Pappas et al., 1983) use it to discuss decision-making under uncertainty. Unfortunately, they appear unaware of the effects discussed here and hence propose incorrect solutions for the problem. Typically the student is advised to estimate his firm’s capital cost and then use it to estimate the present values of revenues and costs. If the ratio of the present value of revenue (benefits)

to costs exceeds one, the student is told to adopt the project, otherwise to reject it. There will be more accepted projects with overestimated benefits than underestimated benefits. The simplest solution is to require accepted projects have benefit-cost ratios exceeding unity by an appreciable amount. Another solution is to use a hurdle rate for projects exceeding the firm’s capital cost.

It should be noticed that real-world firms appear never to use the managerial economics’ techniques with hurdle rates equal to their cost of capital. Instead, they require that projects promise appreci- ably higher rates of return before accepting them. This behavior, which has puzzled business obser- vers, can rather easily be explained by firms having learned (probably from experience, without understanding the theory) that projects appearing mar- ginal (i.e. those promising a return only slightly exceeding the capital cost) typically failed to cover the capital cost. The wisdom used by the firms in picking a high cut-off rate suggests that their actual procedures (perhaps a short pay-out period) probably outperform the more theoretically correct procedures taught in managerial economics (see Miller, 1978a).

In their discussion of the constrained budget problem most textbooks suggest ranking the projects by the ratio of present value of expected profits to costs and then going down the list until the available funds are exhausted. As shown, this does not work if uncertainty about project benefits differs.

Managerial economics professors wishing to in- troduce the argument of this paper into their courses could do so most easily using the decision tree format shown above. Decision trees are frequently covered in existing texts, and an example similar to that above could easily be presented. Little additional instructional time would be needed.

APPLICATION TO SECURITY MARKETS

Suppose there is a competitive stock market without short selling, and there is a company with only one share of stock outstanding (an assumption used to make exposition simple). There is consider- able uncertainty about the present value of future dividends and different investors’ estimates are random drawings from a normal distribution. Suppose investors follow the textbook procedure of making their bids for the stock equal their estimate of future dividends’ present value. The investor who ends up owning the share will be the one with the highest estimate for the value. If investor’s evaluation procedures are unbiased on average, the owner of the typical stock will be disappointed, since the expected value for the highest of a series of drawings from a distribution of estimates will


normally exceed the mean of the distribution. As the author has shown elsewhere (Miller, 1977), this implies that stock markets are unlikely to be efficient under realistic conditions (divergence of opinion combined with restricted short selling). It also suggests that valuation formulas or those for estimating dividends, earnings or future prices should not be evaluated by how well they approxi- mate actual values, but by how well stock selections based on them do.

However, the investor may realize that if he holds the stock, his initial evaluation of the present value of future dividends was the highest of n drawings from a normal distribution. He will consult a textbook of order statistics to determine the expected difference between the highest of n drawings and the mean of the distribution. He will subtract this amount from his estimate of the present value. This will be his bid. With this strategy, the correction factor will be proportional to the standard deviation of the normal distribition. The uncertainty bias will be proportional to the uncertainty. If all investors follow this strategy, the market will be efficient. If some investors fail to follow this strategy, these investors will usually underperform the market. This will make it possible for the remaining investors to outperform the market (if they follow the strategy described rather than buying randomly, or purchasing an index fund) (Miller, 1978b).

The above example seems to capture the essence of investment in the presence of uncertainty. Of course, there are usually thousands of shareholders, and the holders of the shares are among the set of IZ investors out of the m potential investors who have the highest evaluation of the security. The argument is not altered if investors bid other than their estimate of future dividends’ present value. Their measure of an investment’s merit may include the risk assumed. If they believe in the capital asset pricing model the risk measure will be the security’s systematic risk. Different investors may differ in their procedures for valuing securities. Regardless of their risk measure, investors should not bid their first estimates of value but should adjust their bid for the uncertainty-induced bias. Thus the verx existence of a competitive security market with restricted short selling conveys prior information which can be used for decision-making, namely that opportunities are not as good as they appear with naive analysis.

Security purchases in a market with no short selling is an example of problems that occur in competitive bidding in general. The winning bid is the highest (for a purchase, lowest for an offer to supply) of a drawing from a large number of estimates. If the estimated value of the item, (or cost of producing it) is bid, the bidding firm is more likely to submit the winning purchase bid when it overestimates than when it underestimates. This effect (called the winner’s curse) results from the

tendency for the winner to be the firm with the highest error among those bidding.

Similar effects occur in other competitive markets. For instance, in hiring personnel the candidates presenting themselves have probably been rejected by other potential employers, thus insuring that selection will be from a population with more poor than good candidates, and this observation calls for a correction for uncertainty-induced bias. Because applicant selection is based on estimates (often subjective) of the ability of a large number of candidates, the selected candidate’s quality is likely to have been overestimated. Just as stock markets operate so that stocks end up being held by the most optimistic investors, human capital markets insure that personnel end up working for the employers with the most optimistic expectations about them. The worse the quality of the information about the candidate, the greater the expected error for the highest ranking candidate. Thus one should prefer candidates about which information is best. (The formal analysis is quite similar to that given above for the price of a company with a single share of stock (Miller, 1980).)

A BAYESIAN TREATMENT

These conclusions can be stated using Bayesian statistics. The decision tree is merely a graphic way of illustrating the use of Bayes’ theorem to calculate the probability of various outcomes, given a proposal’s estimated merit. The initial distribution of proposals is the ‘prior’ distribution (the distribution believed to exist before information-gather- ing). As pointed out, the theory of competitive markets provides the prior information that good opportunities will normally be fewer than poor ones. The distribution after obtaining and optimally using information about each proposal is the ‘posterior’ distribution. The mean of the posterior distribution is the expected outcome given the estimate of the outcome from the proposal. The procedures required to go from the prior distribution to the posterior distribution given sampling information (here, information about the opportunities) are described in standard statistical textbooks (Dyckman et al., 1969; Raiffa, 1968).

The procedure is illustrated in Fig. 2 . The top figure illustrates the distribution of the estimated outcomes e about the true outcome t. This is the error curve that would be obtained by taking proposals of known merit (presumably using histor- ical outcomes) and subjecting them to the evaluation procedure to be used on the proposals. This is the distribution of e given t, or f(e 1 t ) , where f is the distribution function. The second curve shows the distribution of true outcomes, orf(t). To get the distribution of the proposals’ true outcomes, given the estimates of their outcomes, it is merely


Dlstrlbution of errors

Distribution of investment opportunities

Mean estimated Actual return Return

Figure 2

necessary to multiply the two distributions together and then standardize so that the area under the density curve is equal to one. Thus, f(t I e) = f(e I t),f(t)/f(e), which is Bayes’ theorem.

There is one case especially common for human populations worth discussing further. Frequently, the outcome distribution in the population the proposals are drawn from will be normal, and the distribution of estimation errors normal. (The latter assumption implies a normal distribution for estimates, given that the true values are normal.) In this situation, the posterior distribution will be normal. Furthermore, formulas exist for calculating the mean and standard deviation of the posterior distribution-that is, the distribution of ‘true’ outcomes, given the predictions. For example:

Let M , = the mean outcome of the population of proposals (the prior mean).

Me = the estimated outcome for the proposal.

M , = the expected outcome for the proposals given the estimate (the posterior mean).

sp = the standard deviation for the population of proposals (the prior estimate).

s, = the standard deviation of the estimated outcomes.

S, = the outcomes’ standard deviation given the estimate.

Using this notation,

- - M p n + M e - S,’ s,‘ se +s, s,2 + s,2

The above equations give the mean and the standard deviation of the normal density function for the outcomes, given the estimates. These two parameters specify the normal curve of the posterior distribution. For selection, proposals should be ranked by M , or the expected outcomes. With one exception, where a proposal’s estimated outcome is equal to the population’s mean outcome (where M e = M,), one should expect proposals to have different outcomes on average than they were estimated to have, even though the estimating procedure is unbiased. In theory, this difference between expected performance and the estimates could be either positive or negative. Typically, the primary interest is in the highest-ranked proposals. These proposals will have estimated figures that exceed the potential for all proposals ( M e exceeds M p ) . For such proposals the expected outcome will be less than the estimated. Thus for the proposals that might be selected, future performance will be overestimated.

If the expected error in the estimated outcome E defined to be M , - MI:

Expected error =

This shows the expected error is directly proportional to the difference between the population mean and the estimate for the proposal. In other words, the better the proposal appears, the greater the expected disappointment. Also the size of the expected error goes down as the variance (sp2) in the population increases. Finally, the expected error increases as the variance (3:) in the estimates of future outcomes increases. As uncertainty about the proposal’s merit increases, the correction that must be made to the estimates to arrive at the proposal’s expected merit goes up.

If it is necessary to rank proposals drawn from different populations or about which there are different degrees of uncertainty, it is necessary to reduce the estimates for each proposal by the amount shown in Eqn ( 3 ) . Once the estimates have been corrected by subtracting the uncertainty premium, the highest-ranked proposal can then be selected. This proposal will be one whose selection maximizes the expected outcome (an implicit assumption is that maximization of expected outcome is the correct criterion). If risk enters into the decision criterion one can use a weighted average or


other function of the expected outcome ( M , ) and the expected standard deviation (st).

In general, if the proposal comes from a population with a low mean, a higher score on the selection criteria should be required. The intuitive reason for this is that the procedure used for evaluating proposals sometimes gives good rankings to proposals whose actual outcome will be poor. Therefore, if there is a highly ranked proposal in a population known to contain few high performers, the probability is high that the superior estimated performance is a result of an overestimate in the evaluation process rather than that the proposal is truly superior. The solution for this problem is to require higher standards for proposals from the population with a low mean than is required for proposals from populations with higher means. This applies even if the evaluation process itself is unbiased and produces the same error distribution in evaluating proposals from all populations. Some- times, especially if evaluating proposals is expen- sive, optimum policy may exclude projects from certain populations because these contain too few qualified proposals to justify the expense of evaluation. Of course, where proposals are being drawn from populations with different means or standard deviations, the size of the expected error will differ. This, in turn, implies that the required scores in the evaluation procedures should depend on which population the proposal is being drawn from.

The above discussion has dealt with the case where distribution of proposals was normal, and the estimation errors were normally distributed. With normal distributions, the posterior distribution is normal and an analytic solution is possible. Howev- er, the relevant distribution often will not be normal. And even where the initial population distribution is normal, other firms seeking the same investment or employee characteristics may already have selected the best investment opportunities or employees, leaving a distribution (that may not be normal) with fewer good ones than there were initially. With modern computers, non-normal distributions can be handled easily if an analytic solution is not available.

variances) of the prior distribution and the information provided by the evaluation procedure. This is shown by Eqn (2), which shows M, to be a weighted average of the prior mean (weightse’/[s2 + s ’I)

This, of course, conflicts with anti-discrimination laws that forbid taking such information into account in personnel decisions.

Likewise, the standard deviations in different populations will in general be relevant. If only a small fraction of candidates are selected the uncertainty correction will be greatest for groups about which there is the greatest inaccuracy in the estimates.

Representatives of minority groups frequently contend that standarized tests are less accurate for members of their groups. They usually argue that recognizing this would benefit members of their groups. However, the above analysis shows that the lower the test’s precision, the higher the weight that should rationally be given to the prior distribution. If an employer believes that the minority is less qualified than other groups, his rational response to being convinced that the test (or other evaluation instrument) is inaccurate should be to give less weight to the test and more to his prior information (which is what is often called prejudice). Members of groups widely believed to have lower abilities will normally benefit from having additional information available which permits those hiring to pick the higher-ability individuals out of the group with an unfavorable prior distribution.

It should be noticed that some discrimination against minority groups may be based on difficulty in evaluating them rather a belief that the group lacks ability. Difficulty in evaluation makes the standard deviation of the estimates and the uncertainty premium greater. As a simple illustration, consider the problem of selecting candidates from groups that the hirer lacks personal contacts in and who hence cannot rely on references. (This situation is obviously more common for minorities.) One may have no reason to believe such candidates that move in different circles are inferior, but it would be rational to require higher paper qualifica- tions from them than from those about whom more information was available.

and the estimated mean (weight sP2/[se2 + sp 4 I).

IMPLICATIONS FOR HUMAN RESOURCE DECISION

ORGANIZATIONAL IMPLICATIONS

The above analysis has interesting implications when the problem is to select candidates for jobs and there are several populations (age, sex, race, religion, etc.) known to contain different distributions of abilities for candidates. This is prior information that in general should be taken into account by applying different standards to candidates from different populations. The final evaluation is a weighted average (weights are the

If there is decentralized decision-making those making benefit estimates for different alternatives normally lack the information required to correct their estimates for uncertainty bias. How uncertainty-induced bias affects estimates depends on the characteristics of the proposals it is competing with. Thus, for the analyst to adjust his own estimates requires that he know the merits of all other


proposals being considered. In large organizations, analysts may not even know what the other proposals being considered are, much less know anything about their quality. Thus, it will generally be best to have the analysts continue making their estimates in the traditional manner. The adjustment for uncertainty bias will have to be done separately.

Let us be somewhat more technical. The uncertainty bias is the difference between the expected value of the variable, given that it is the highest of a number of variables (each drawn from a different distribution), and the expected value of that distribution. To determine this is a complicated problem in order statistics (although one that is solvable conceptually). If it is to be solved at all, it is necessary to know what other variables are in the ranking and the parameters of their distributions. This information is typically not possessed by the analysts examining individual proposals. Even if they had the required information it would be simplest to do the computations centrally. Decision- making in the presence of uncertainty is not easy to decentralize.

An important implication is that the decision- maker’s job is more difficult than the standard managerial economics or finance textbook discussion implies. The final decision-making process is often depicted as a simple matter of collecting estimates of a measure of merit (perhaps a rate of return for capital budgeting), ranking the proposals by these estimates, and then choosing the proposal at the top of the list. This is a simple clerical process, which would not justify the high salaries paid most decision-makers. However, the above analysis shows that the process is much more complex.

While the examples used might imply that all that is involved is a mathematical process, in actuality estimates of the uncertainty of different estimates and the benefit distribution in the population of proposals will not be available. The decision-maker will have to draw on his judgement and experience for these. Frequently, he will not explicitly use the procedure described, but understanding the theory should still help him to make better decisions.

CONCLUSIONS

The textbook procedure for selecting the best of several alternatives (as in capital budgeting or personnel hiring) when there is uncertainty about the merits of the different alternatives is to estimate an unbiased measure of merit for each alternative, rank the alternatives by this measure of merit and then select the highest ranked alternative. Unfortu- nately, this procedure is not correct. Selecting the highest ranked from a number of uncertain estimates involves a positive expected error. A selection procedure should take this effect into account.

Frequently, the selection process can be modeled to estimate how important this effect is. A decision tree is useful in setting out the structure of the problem. With this procedure an estimate of the expected error for each estimated value can be calculated using the distribution of errors in the estimates of merit and the underlying distribution. This correction for uncertainty bias will involve reducing estimated benefits by an amount that increases with the uncertainty in the estimation procedure (for the normal case where only a minority of the alternatives are being selected). A rational decision procedure is to estimate this expected error, and then subtract it from the estimates, before ranking the alternatives. The amount to be subtracted will typically be greatest for the alternatives about which information is the worse, and uncertainty greatest.

Problems of this type can often be structured in a Bayesian manner. Frequently, the structure of the problem provides some prior information. For instance, if there are competitors for opportunities (firms trying to purchase the same securities or hire the same employees) the better ones will have been selected in disproportionate number. One thus should assume that there are many poor alternatives and only a few good ones.

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Decision-making under uncertainty for capital budgeting and hiring