Uncertainty and Decision Making

Since the probability of outcomes under uncertainty is not known, decision making under it is largely subjective. Two specific decision rules are discussed here. 

The Maximin Criterion

The maximin criterion postulates that the decision maker should determine the worst possible outcome of each strategy and then pick the strategy that provides the best of the worst possible outcomes. This criterion is appropriate when the firm has a very strong aversion to risk, for instance, when the survival of a small firm depends on avoiding losses. The maximin criterion is also applicable in the case of oligopoly, when the actions of one firm affect the others. If one firm lowers its price, it can expect the others to soon lower theirs, thus reducing profits of all.

The maximin criterion can be illustrated by applying it to the example in Table 9.2, where the firm could follow the strategy of introducing a new product that would provide a return of Rs 20,000 if it succeeded or lead to a loss of Rs 10,000 if it failed or choose not to invest in the venture, with zero possible return of loss. This matrix is shown in Table 9.3. Since we assume that the manager does not know and cannot estimate the probability of success and failure of investing in the new' product, he cannot calculate the expected payoff or return and risk the investment.

Table 9.2: Expected Return of Project

To apply the maximin criterion to this investment, the manager first determines the worst possible outcome of each strategy (row). This is -Rs 10,000 in the case of failure for the investment strategy and 0 for the strategy of not investing. These worst possible outcomes are recorded in the last column of the table. Then he picks up the strategy that provides the best (maximum) of the worst (minimum) possible outcomes (i.e., maximin). This is the strategy of not investing, which is indicated by the asterisk next to its zero return or loss in the last column. Thus, the maximin criterion picks the strategy of not investing, which has the maximum of the minimum payoffs.

The Minimax Regret Criterion

According to this criterion, the decision maker should select the strategy that minimises the maximum regret or the opportunity cost of a wrong decision, whatever the state of nature that actually occurs. Regret is measured by the difference between the payoff of a given strategy and the payoff of the best strategy under the same state of nature.

To apply the minimax regret criterion, the decision maker must first construct a regret matrix from the payoff matrix. For example, Table 9.4 presents the payoff and regret matrices for the investment problem of Table 9.2 that we have been examining. The regret matrix is constructed by determining the maximum payoffs for each state of nature (column) and then subtracting each payoff in the same column from that figure. These differences are the measures of regrets. For example, if the manager chooses to invest in the product and the state of nature that occurs is the one of success, he or she has no regret because this is the correct strategy. Thus, the regret value of zero is appropriately entered at the top of the first column in the regret matrix in Table 9.4. On the other hand, if the firm had chosen not to invest, so that it had a zero payoff under the same state of nature of success, the regret is Rs 20,000. This regret value is entered at the bottom of the first column of the regret matrix. Moving to the state of failure column in the payoff matrix, we see that the best strategy (i.e., the one with the largest payoff) is not to invest. This has a payoff of zero. Thus, the regret value of this strategy is zero (the bottom of the second column in the regret matrix). If the firm undertook the investment under the state of nature of failure, it would incur a loss and a regret of Rs 10,000 (the top of the second column of the regret matrix).

Table 9.4: Payoff and Regret Matrices for the Maximum Regret Criterion

States of Nature              Regret Matrix

*        The strategy with the minimum regret value

Note that the regret value of the best strategy under each state of nature is always zero and that the regret values in the regret matrix necessarily be positive since we are always subtracting smaller payoffs from the largest payoff under each state of nature (column).

After constructing the regret matrix (with the maximum regret for each strategy under each state of nature), the decision maker then chooses the strategy with the minimum regret value. In our example, this strategy of investing, which has the minimum regret value of Rs 10,000 (indicated by the asterisk in the maximum regret column of Table 9.4). This compares with the maximum regret of Rs 20,000 resulting from the strategy of not investing. Thus, while the best strategy for the firm according to the maximin is not to invest, the best strategy according to minimax regret is to invest.

The Hurwics Alpha Index

The Hurwics alpha criterion tries to provide a more acceptable approach to make decisions under uncertainity. It compromises between the maximin and minimin criterion. An index is developed that is based on the coefficient of optimism. Maximum and minimum payoffs from every action are accounted for and weighed in accordance to subjective valuation. The value of pursuing an action is determined by an index:

Hi = µ C max + (1 - µ) C min

The outcome with maximum Hi is selected. The equation shows that the more optimistic the decision maker the larger will be Hi value, and vice versa. 

The Maximax Criterion

In contrast to minimin criterion, the maximax criterion emphasises on the selection of the 'best of the best" alternative. The decision maker makes the extreme assumption that the probable outcome will always be best. It takes a positive viewpoint, thereby, it is more suitable for extreme risk takers.

The La Place (Bayes') Criterion

The Bayes theorem expresses the conditional probability required for the decision in terms of the reverse conditional probability and the prior probability. Generally it expresses the way new information affects a decision maker's probability assessments. The decision maker begins with a prior probability assessment that is revised in light of new information. The revised probability depends directly on the prior probability. Other things being equal, the larger the prior probability, the larger will be the revised probability. If the revised probability is greater than prior probability then it indicates a better outcome. 

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