Decision Making without Experimentation

A a∈

2. Nature would choose one of the possible states of nature:

Θ θ∈

3. Each combination of an action and state of nature would result in a payoff/decision, which is given as one of the entries in a payoff/decision table.

C a

c( ,θ)∈

4. This payoff/decision table should be used to find an optimal action for the decision maker according to an appropriate criterion.

An additional element needs to be incorporated into this framework – prior probabilities.

Many decision makers would generally want to incorporate additional information to account for the relative likelihood of the possible states of nature. This information is translated into a probability distribution where the state of nature is considered to be a random variable - making up the prior probability distribution.

In general terms, the decision maker must choose an action from a set of possible actions.

The set contains all the feasible alternatives under consideration for how to proceed with the problem of concern. This choice of an action must be made in the face of uncertainty, because the outcome will be affected by random factors that are outside the control of the decision maker. These random factors determine what situation is referred to as a possible state of nature. For each combination of an action and a state of nature, the decision maker knows what the resulting payoff (or consequence) would be. The payoff is a quantitative measure of the value to the decision maker of the consequences of the outcome. Although in many instances the payoff is a monetary gain, there are other measures that can be used (as explained in utility theory). If the consequences of the outcome do no become completely certain even when the state of nature is given, then the payoff becomes an expected value (in the statistical sense) of the measure of the consequences. The payoff table can be used to determine an optimal action according to an appropriate criterion which suits the beliefs of the decision maker.

The decision problem faced by the oil company can now be placed within the decision framework just presented. This can be summarized in the following decision⁷ table:

Table 14.2: Formulation of the problem within the framework of decision analysis

Status of Land State of Nature Θ

Alternative A θ1 θ2

a1

a2

c11=700

c21=90 c12= -100

c22=90

Prior Probability 0.25 0.75

Section 14.3: Decision Making without Experimentation

7 Payoff table and decision table can be used interchangeably.

This section illustrates the decision making process without experimentation. A formulation of the problem being assessed has two possible actions under consideration:

drill for oil or sell the land. The possible states of nature are that the land contains oil and that it is does not. Since the consulting geologist has estimates that there is a 1 in 4 chance of oil, the prior probabilities of the two states of nature are 0.25 (for oil) and 0.75 (for no oil). Table 14.1 has been redesigned in Table 14.2 with the payoff units in thousands of dollars of profit. This payoff table will be used to determine the optimal action under the three main criterions discussed here.

1. Minimax Criterion:

The minimax criterion was described in section 2. Its rationale is that it provides the best guarantee of the payoff that will be obtained. That is, one determines, for each action a, the maximum loss over the various possible states of nature:

) , ( max )

(a l a

M θ

= Θ ,

and this provides an ordering among the possible actions. In words, for each possible action, find the minimum payoff over all possible states of nature. Next find the maximum of these minimum payoffs. Choose the maximum of the minimum payoff gives the maximum.

The application of this criterion to the prototype example suggests that selling the land is the optimal action to take. Regardless of what the true state of nature turns out to be for the problem, the payoff from selling the land cannot be less than 90, which provides the best available guarantee. Thus, this criterion provides the pessimistic viewpoint that regardless of which action is selected, the worst state of nature for that action is likely to occur, so one should choose the action which provides the best payoff with its worst state of nature.

2. Maximum Likelihood Criterion:

The maximum likelihood function focuses on the most likely state of nature. Recall a simple general definition of the likelihood function: For the observed data, x, the function )l(θ)= f(x|θ , considered as a function of θ, is called the likelihood function.

The intuitive reason for the name “maximum likelihood function” is that a θ for which )

| (x θ

f is small, in that x would be more plausible occurrence if f(x|θ)were large. In simplicity, the steps in this approach begin by identifying the most likely state of nature (largest prior probability). For this state of nature, find the action with the maximum.

Choose this decision.

The application of this criterion to the prototype example indicates that the dry state has the largest prior probability. In Table 10.2, in the dry cell column, the sell alternative has the maximum payoff, so the choice is to sell the land.

The appeal of this criterion is that the most important state of nature is most likely one, so the action chosen is the best one for this particularly important state of nature. Basing the decision on the assumption that this state of nature will occur tends to give a more, the criterion does not rely on questionable subjective estimates of the probabilities of the respective states of nature other than identifying the most likely state. However, the major drawback of this criterion is that it completely ignores some relevant information.

No state of nature besides the most likely one is considered. Therefore, a problem with many possible states of nature, the probability of the most likely one may be quite small and or show little difference between “quite likely” states of nature.

3. Bayes’ Decision Rule (Expected Monetary Value Criterion):

The Bayes decision rule was introduced in Section 2. In this instance, the probability weight assigned to each state of nature θ, the loss incurred for a given action incurs the expected value:

This approach uses the best available estimates of the probabilities of the respective states of nature (current prior probabilities) and calculates the expected value of the payoff for each of the possible actions. Choose the action with the maximum expected payoff.

In this application of the criterion, it can be easily determined that the optimal action is to drill. The expected payoffs can be calculated from Table 14.2 directly as follows:

90

Since 100 > 90, the alternative action to drill is selected. Note that this choice differs from the other two preceding criteria.

The advantage to the Bayes’ decision rule is that it incorporates all the available information, including payoffs and the best available estimates of the probabilities of the respective states of nature. It is sometimes argued that these estimates of the probabilities are largely subjective and so are too shaky to be trusted. Nevertheless, under many circumstances, past experience and current evidence enables one to develop reasonable estimates of the probabilities. The methodology of including such information was described in Section 3. Before applying this approach to the problem at hand, we will consider the use of sensitivity analysis to assess the effect of possible inaccuracies in the prior probabilities.

Section 13 briefly discussed the role and importance of sensitivity analysis in various