Criteria of choice under uncertainty

Topic 7. Decision making under uncertainty

Definition of uncertainty conditions

Criteria of choice under uncertainty

Definition of uncertainty conditions

In the condition of uncertainty, the decision maker recognizes different potential states of nature but cannot confidently estimate the probabilities of their occurrence. This is undesirable but often unavoidable situation. It may occur when one faces a completely new phenomenon or when a completely new product, process, or state of nature is under consideration and it is impossible to make comparative judgments, and when no historical data are available from which to infer (делать вывод о) probabilities. In many such situations even prominent experts cannot agree on the chances of the various states of nature occurring.

Dealing with uncertainty is an important facet (аспект) of the jobs of many managers. Managers, teams, and other professionals often need to absorb uncertainty by using their intuition, creativity, and all available information to make a judgment regarding the course of action (decision) to take.Let us consider a specific problem related to decision making under uncertainty. Example (problem): The Palm Tree Hotel is considering the construction of an additional wing (крыло). Management is evaluating the possibility of adding 30, 40, or 50 rooms. The success of the addition depends on a combination of local government legislation (местное законодательство) and competition in the field; four states of nature are being considered. They are shown together with the anticipated payoffs (in percent of yearly return on investment) in Table 1. Table 1. Payoff table (percent of return on investment)

Alternatives	States of nature
Positive legislation and low competitionS1	Positive legislation and strong competition S2	Bad legislation and low competitionS3	Bad legislation and strong competitionS4
A1 = 30 rooms	10	5	4	-2
A2 = 40 rooms	17	10	1	-10
A3 = 50 rooms	24	15	-3	-20

Management cannot agree on the probabilities of the states of nature. The problem is: how many rooms to build in order to maximize the return on investment. This problem will be used in the following paragraph to illustrate the application of different choice criteria for making a decision.

Criteria of choice under uncertainty

At the present time, decision theory does not provide a single best criterion for selecting an alternative under conditions of uncertainty. Instead, there are a number of different criteria, each with its justifications (обоснование) and limitations. The choice among these is determined by organizational policy, the attitude of the decision maker toward risk, or both.

We will consider five different criteria.

1. The criterion of equal probabilities (the Laplace criterion) ( критерий равных вероятностей / критерий недостаточного основания / критерий Лапласа)

The user of this criterion assumes that all states of nature are equally likely to occur. Thus, equal probabilities are assigned to each. The expected values are then computed and the alternative with the highest expected payoff is selected.

Using the above example, probabilities ¼ are assigned to each of the four states of nature. The expected payoffs are:

E(A₁) = ¼ × 10 + ¼ × 5 + ¼ × 4 + ¼ × (-2) = 17/4

E(A₂) = ¼ × 17 + ¼ × 10 + ¼ × 1 + ¼ × (-10) = 18/4 (the largest expected payoff)

E(A₃) = ¼ × 24 + ¼ × 15 + ¼ × (-3) + ¼ × (-20) = 16/4

Thus, the best alternative under this criterion is A₂, with an expected payoff of 18/4. The major argument against this criterion is that there is absolutely no reason to assume the probabilities are all equal. Such an assumption may be as erroneous (ошибочный) as assuming one outcome in particular will occur.

2. Criterion of pessimism (maximin or minimax) / The Wald criterion ( критерий пессимизма / максимин или минимакс / критерий Вальда) )

The user of this criterion is completely pessimistic, since he or she assumes that the worst will happen, no matter which alternative is selected. To provide protection, the decision maker should select the alternative that will give as large a payoff as possible under this pessimistic assumption (best of the worsts).

Let us reproduce Table 1 and show the application of the pessimism criterion.

Table 2. Pessimistic approach to selecting an alternative

Alternatives	States of nature	Worst	Best of worst(maximum of minimum)
S1	S2	S3	S4	(minimum)
A1 = 30 rooms	10	5	4	-2	-2	-2
A2 = 40 rooms	17	10	1	-10	-10
A3 = 50 rooms	24	15	-3	-20	-20

Assume that the decision maker selects A₁; then the worst that can happen is a loss of 2 percent when S₄ occurs. Similarly, the worst for A₂ is -10, and for A₃ is -20. This information is entered into a new column labeled Worst. From this column, the best entry is then selected (-2 in the example). The decision maker maximizes the minimum payoffs, and that is why this criterion is labeled maximin.

In the case of cost minimization, the decision maker will minimize the maximum possible costs; that is the decision criterion will be called minimax.

One drawback ( недостаток ) of this criterion (which is also a drawback of all the remaining criteria) is that the decision is based on only a small portion of the available information. Thus, valuable information is completely disregarded, as shown in Table 3, which illustrates a deliberately exaggerated (утрированный) case (which is not related to the above example).

Table 3. Profits under two alternatives

Alternatives	States of nature	Minimum worst
S1	S2	S3
A1	40,000	20,000
A2

According to the criterion of pessimism, A₂ should be selected. The decision is based on the “minimum” column, which includes only one entry from each row. The rest of the data is ignored. In reality, most decision makers will pay attention to the remaining information and consequently not use this approach and select A₁. The pessimistic decision maker acts in a super conservative manner, paying attention only to the risks and completely neglecting the opportunities.

3. Criterion of optimism (maximax or minimin) ( критерий оптимизма / максимакс или минимин )

An optimistic decision maker assumes that the very best outcome will occur and selects the alternative with the best possible payoff.

To do so, the decision maker searches for the best possible payoff for each alternative (best of the bests). These are placed in a new column to the right of the decision table.

Reproducing the data of Table 1, in Table 4, the “best” column is created. According to the maximax criterion, alternative A₃ would be selected.

Table 4. Maximax choice

Alternatives	States of nature	Best
S1	S2	S3	S4
A1 = 30 rooms	10	5	4	-2	10
A2 = 40 rooms	17	10	1	-10	17
A3 = 50 rooms	24	15	-3	-20	24

If the data were costs, then the optimistic decision maker would select as best the lowest cost payoff for each alternative and then select the lowest of these lowests. Such an approach is labeled minimin.

Notice again that no attention is paid to most of the available information; only the highest payoff is considered. Thus, an optimistic decision maker is a gambler who disregards the risks and looks forward only to the opportunities.

4. Coefficient of optimism (the Hurwicz criterion) ( коэффициент оптимизма /критерий Гурвица )

Most decision makers are not completely optimistic or completely pessimistic. Therefore, it was suggested by Hurwicz that a degree of optimism labeled a be measured on a 0 to 1 scale (0 is completely pessimistic, 1 is completely optimistic). Hurwicz suggested that the best alternative is the one with the highest (in the case of maximization) weighted value, where the weighted value, WV, for each alternative (row in the decision table) is expressed by:

(WV)_i= a[best O_ij] + (1-a)[worst O_ij],

where O_ij is the payoff. Then, the best (WV)_i is selected.

Example: Examining Table 4 with a given as 0.7 we get:

WV(A₁) = 0.7 × 10 + (1- 0.7) × (-2) = 6.4

WV(A₂) = 0.7 × 17 + (1- 0.7) × (-10) = 8.9

WV(A₃) = 0.7 × 24 + (1- 0.7) × (-20) = 10.8 maximum

Thus, alternative A₃ is the best.

Note: in the case of minimization, such as with costs, where the best is lowest, select the alternative with the lowest WV.

At a = 0, the Hurwicz criterion coincides with the Wald criterion, while at a = 1, with the maximax criterion. The major difficulty in applying the Hurwicz criterion is the measurement of a. The coefficient α is derived from a subjective estimate of the decision situation by the decision maker. If there are no assumptions regarding the value of α, it is usually set at 0,5.

Note that when applying this criterion, more information is used (than in minimax), yet only the two extreme payoffs are considered and the remaining information is ignored.

5. The criterion of regret (Savage’s criterion) ( критерий потерь / критерий Сэвиджа)

The concept of regret is equivalent to the determination of opportunity loss, discussed earlier. Both concepts represent the important economic concept of opportunity cost, which indicates the magnitude of the loss incurred by not selecting the best alternative. The value of regret, by definition, can never be negative.

Savage argued that the decision maker should attempt to minimize the largest anticipated regret. That is, employ a minimax approach to the regret data (in a basically pessimistic manner).

Let us use the hotel example again and reproduce Table 1.

Table 1. Payoff table (percent of return on investment)

Alternatives	States of nature
Positive legislation and low competitionS1	Positive legislation and strong competition S2	Bad legislation and low competitionS3	Bad legislation and strong competitionS4
A1 = 30 rooms	10	5	4	-2
A2 = 40 rooms	17	10	1	-10
A3 = 50 rooms	24	15	-3	-20

Now let us build a regret (opportunity loss) table (Table 5).

Table 5. Regret table (таблица потерь / упущенных выгод)

Alternatives	S1	S2	S3	S4	Largest regret (worst)
A1	24-10=14	15-5=10	4-4=0	-2-(-2)=0
A2	24-17=7	15-10=5	4-1=5	-2-(-10)=8
A3	24-24=0	15-15=0	4-(-3)=7	-2-(-20)=18

Now we should minimax the regret. This is done by finding the worst (largest) regret in each row, and then selecting the lowest regret in the newly formed column.

In our example the lowest regret is for alternative A₂. This selection guarantees that regardless of what happens, the decision maker will never have a regret larger than 8. Note again that only small portion of the available information is used.

Decision making under uncertainty is more difficult that it is for risk or certainty. All five different criteria presented above have some deficiencies and will usually point to different selections of alternatives. In management, decision making under uncertainty should be avoided since the results can be unpredictable. Instead, enough information should be acquired so that decisions can be made at least under calculated risk or, at best, under certainty.

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