Certainty Risk And Uncertainty

Decision-making falls into three categories: certainty, risk, and uncertainty. Decisionmaking under certainty is the easiest case to work with. With certainty, we assume that all of the necessary information is available to assist us in making the right decision, and we can predict the outcome with a high level of confidence.

Decision-Making Decision-making under certainty implies that we know with 100 per-

under Certainty cent accuracy what the states of nature will be and what the expected payoffs will be for each state of nature. Mathematically, this can be shown with payoff tables.

To construct a payoff matrix, we must identify (or select) the states of nature over which we have no control. We then select our own action to be taken for each of the states of nature. Our actions are called strategies. The elements in the payoff table are the outcomes for each strategy.

A payoff matrix based on decision-making under certainty has two controlling features.

• Regardless of which state of nature exists, there will be one dominant strategy that will produce larger gains or smaller losses than any other strategy for all the states of nature.

• There are no probabilities assigned to each state of nature. (This could also be stated that each state of nature has an equal likelihood of occurring.)

Example 17-1. Consider a company wishing to invest \$50 million to develop a new product. The company decides that the states of nature will be either a strong market demand, an even market demand, or a low market demand. The states of nature shall be represented as N1 = a strong (up) market, N2 = an even market, and N3 = a low market demand. The company also has narrowed their choices to one of three ways to develop the product: either A, B, or C. There also exists a strategy S4, not to develop the product at all, in which case there would be neither profit nor loss. We shall assume that the decision is made to develop the product. The payoff matrix for this example is shown in Table 17-1. Looking for the controlling features in Table 17-1, we see that regardless of how the market reacts, strategy S3 will always yield larger profits than the other two strategies. The project manager will therefore always select strategy S3 in developing the new product. Strategy S3 is the best option to take.

Table 17-1 can also be represented in subscript notation. Let Pi,j be the elements of the matrix, where P represents profit. The subscript i is the row (strategy), and j is the column (state of nature). For example, P2 3 = the profit from choosing strategy 2 with N3 state of nature occurring. It should be noted that there is no restriction that the matrix be square (i.e., the number of states of nature need not equal the number of possible strategies).

Decision-Making under Risk

In most cases, there usually does not exist one dominant strategy for all states of nature. In a realistic situation, higher profits are usually accompanied by higher risks and therefore higher probable losses. When

TABLE 17-1. PAYOFF MATRIX (PROFIT IN MILLIONS)

States of Nature

there does not exist a dominant strategy, a probability must be assigned to the occurrence of each state of nature.

Risk can be viewed as outcomes (i.e., states of nature) that can be described within established confidence limits (i.e., probability distributions). These probability distributions are often estimated or defined from experimental data.

Consider Table 17-2, in which the payoffs for strategies 1 and 3 of Table 17-1 are interchanged for the state of nature N3.

From Table 17-2, it is obvious that there does not exist one dominant strategy. When this occurs, probabilities must be assigned to the possibility of each state of nature occurring. The best choice of strategy is therefore the strategy with the largest expected value, where the expected value is the summation of the payoff times the probability of occurrence of the payoff for each state of nature. In mathematical formulation,

where E, is the expected payoff for strategy i, Pt,j is the payoff element, and Pj is the probability of each state of nature occurring. The expected value for strategy S1 is therefore

Repeating the procedure for strategies 2 and 3, we find that E2 = 55, and E3 = 20. Therefore, based on the expected value, the project manager should always select strategy

TABLE 17-2. PAYOFF TABLE (PROFIT IN MILLIONS)

States of Nature

51 5G 4G 9G

52 5G 5G 6G

*Numbers are assigned probabilities of occurrence for each state of nature.

S1. If two strategies of equal value occur, the decision should include other potential considerations (time to impact, frequency of occurrence, resource availability, etc.). (Note: Expected value calculations implicitly assume that a risk neutral utility relationship exists. If the decision-maker is not risk neutral, such calculations may still be useful, but the results should be evaluated to see whether or not they are affected by differences in risk tolerance.)

To quantify potential payoffs, we must identify the strategy we are willing to take, the expected outcome (element of the payoff table), and the probability that the outcome will occur. In the previous example, we should accept the risk associated with strategy S1, since it gives us the greatest expected value. If the expected value is positive, then this risk should be considered. If the expected value is negative, then this risk should be proactively managed.

An important factor in decision-making under risk is the assigning of the probabilities for each of the states of nature. If the probabilities are erroneously assigned, different expected values will result, thus giving us a different perception of the best strategy to take. Suppose in Table 17-2 that the assigned probabilities of the three states of nature are 0.6, 0.2, and 0.2. The respective expected values are:

In this case, the project manager would always choose strategy S3.

Decision-Making The difference between risk and uncertainty is that under risk there are under Uncertainty assigned probabilities, and under uncertainty meaningful assignments of probabilities are not possible. As with decision-making under risk, uncertainty also implies that there may exist no single dominant strategy. The decisionmaker, however, does have at his disposal four basic criteria from which to make a management decision. The decision about which criterion to use will depend on the type of project as well as the project manager's tolerance to risk.

The first criterion is the Hurwicz criterion, often referred to as the maximax criterion. Under the Hurwicz criterion, the decision-maker is always optimistic and attempts to maximize profits by a go-for-broke strategy. This result can be seen from the example in Table 17-2. The maximax criterion says that the decision-maker will always choose strategy S3 because the maximum profit is 100. However, if the state of nature were N3, then strategy S3 would result in a maximum loss instead of a maximum gain. The use of the maximax, or Hurwicz, criterion must then be based on how big a risk can be undertaken and how much one can afford to lose. A large corporation with strong assets may use the Hurwicz criterion, whereas the small private company might be more interested in minimizing the possible losses.

A small company would be more apt to use the Wald, or maximin, criterion, where the decision-maker is concerned with how much he can afford to lose. In this criterion, a pessimistic rather than optimistic position is taken with the viewpoint of minimizing the maximum loss.

In determining the Hurwicz criterion, we looked at only the maximum payoffs for each strategy in Table 17-2. For the Wald criterion, we consider only the minimum payoffs. The minimum payoffs are 40, 50, and -50 for strategies S1, S2, and S3, respectively. Because the project manager wishes to minimize his maximum loss, he will always select strategy S2 in this case. If all three minimum payoffs were negative, the project manager would select the smallest loss if these were the only options available. Depending on a company's financial position, there are situations where the project would not be undertaken if all three minimum payoffs were negative.

The third criterion is the Savage, or minimax, criterion. Under this criterion, we assume that the project manager is a sore loser. To minimize the regrets of the sore loser, the project manager attempts to minimize the maximum regret; that is, the minimax criterion.

The first step in the Savage criterion is to set up a regret table by subtracting all elements in each column from the largest element. Applying this approach to Table 17-2, we obtain Table 17-3.

The regrets are obtained for each column by subtracting each element in a given column from the largest column element. The maximum regret is the largest regret for each strategy, that is, in each row. In other words, if the project manager selects strategy S1 or S2, he will only be sorry for a loss of 50. However, depending on the state of nature, a selection of strategy S3 may result in a regret of 140. The Savage criterion would select either strategy S1 or S2 in this example.

The fourth criterion is the Laplace criterion. The Laplace criterion is an attempt to transform decision-making under uncertainty into decision-making under risk. Recall that the difference between risk and uncertainty is a knowledge of the probability of occurrence of each state of nature. The Laplace criterion makes an a priori assumption based on Bayesian statistics, that if the probabilities of each state of nature are not known, then we can assume that each state of nature has an equal likelihood of occurrence. The procedure then follows decision-making under risk, where the strategy with the maximum expected value is selected. Using the Laplace criterion applied to Table 17-2, and thus assuming that P1 = P2 = P3 = 1/3, we obtain Table 17-4. The Laplace criterion would select strategy S1 in this example.

The important conclusion to be drawn from decision-making under uncertainty is the risk that the project manager wishes to incur. For the four criteria previously mentioned, we have shown that any strategy can be chosen depending on how much money we can afford to lose and what risks we are willing to take. 