Decision Theory

When linear programming models are used, decisions are made based on the results of the models. However, these LP models were all formulated under the assumption that certainty exists. It is assumed that all of the model coefficients, constraint values, and solution values are known with certainty and do not vary. Decisions are often made within an environment of risk, uncertainty, or conflict. Decision theory is concerned with decision making under conditions of risk and uncertainty. Game theory, which will not be discussed, is concerned with decision making under conflict. Decision situations can be categorized into two classes: situations where probabilities cannot be assigned to future occurrences, and situations where probabilities can be assigned. In this module you will learn:

the logic of decision-making situations with and without probabilities;

decision-making criteria and the degree to which they reflect risky decision making;

how to use probabilistic information on decision making

We will discuss each of the decision situations separately, and demonstrate the decision making criterion most commonly associated with each.

Steps of Decision Making:

A decision-making situation includes a couple of steps. The first step in the decision analysis is to identify the alternatives considered by the decision maker. These alternatives are called decision alternatives, denoted by d1, d2, d3, ....

d1 = invest in the apartment building,

d2 = invest in the office building,

d3 = invest in the warehouse,

d4 = don't invest.

Of course, the selection of the best decision alternative depends on how much profit the investor will make from each decision. Profit will depend on the future events associated with a decision situation. A decision maker may have an idea of possible future events, but it is uncertain which particular event will occur. Thus the second step in the analysis is to identify the future events that might occur. These events are referred to as the states of nature, denoted by s1, s2, s3,... The states of nature are defined so that one of the listed states of nature will occur.

s1 = good economic conditions

s2 = poor economic conditions.

Given the three decision alternatives and the two states of nature, which investment opportunity should the decision maker choose? In order to answer this question, we need information on the profit associated with each combination of a decision alternative and a state of nature. For example, how much profit would the investor experience if she/he decides to purchase the apartment building (d1) and economic conditions are good (s1)? And so on. The outcome resulting from making a certain decision and the occurrence of a particular state of nature is referred to as payoff. We will organize these types of decision situations into payoff tables. Entries in a payoff table can be stated in terms of profits, costs, or any other measure of output that may be appropriate for the particular situation. Below is an example:

Decision Making Without Probabilities:

In this case, we consider decision-making approaches without knowledge of the probabilities of the states of nature. Once a payoff table is organized, several criteria are available for decision-making. They will each yield different decisions. The decision maker must select the criterion or combination of criteria that best suits his or her needs. The criteria are:

1. The Maximax Criterion

2. The Maximin Criterion

3. The Minimax Regret Criterion

4. The Hurwicz Criterion

5. The Laplace Criterion

When a compound word like "maximax" is given, the first portion "max" chooses the maximum among the maxima of the second portion "max." The "maximin" means that we select the maximum among the minima.

The Maximax Criterion:

The maximax criterion is based on the assumption of an optimistic decision maker. With this criterion, the decision maker selects the alternative that represents the maximum of the maximum payoff. For the above example, the maximum payoff for each of the four investment plans is highlighted:

Then the maximum among the maximum values is 550,000, so the decision is to construct a "large-sized facility."

The Maximin Criterion:

The reverse approach to the maximax criterion is the maximin criterion , which is based on the assumption that the decision maker is conservative (pessimistic) about the future. With this criterion, the minimum payoffs for each alternative are compared, and the alternative that produces the maximum of the minimum payoffs must be chosen. In the above example, this would be:

So the decision maker would choose "No facility" because a payoff of 0 is the maximum among the minimum payoffs. The maximin is designed for the decision maker who wants to avoid risk.

The Minimax Regret Criterion:

We feel regret if we did not get what we wanted, or we discovered that what we chose produced less than expected. We define the regret function under each state of nature as

As you see, there is no regret when you get the best payoff under each state of nature. We construct a regret (opportunity loss) table:

Each of the highlighted payoffs is the maximum for each alternative. The minimum among the highlighted maximum is 250,000 for a "medium-sized facility." The decision should be to construct a medium-sized facility.

The Hurwicz Criterion:

This criterion strikes a compromise between the maximax and maximin criteria. The decision maker is neither optimistic nor pessimistic (conservative). With the Hurwicz criterion, the decision payoffs are weighted by a coefficient of optimism (realism), a measure of the decision maker's optimism:

The Hurwicz criterion requires that, for each decision alternative, the maximum payoff be multiplied by the coefficient of optimism, and the minimum payoff be multiplied by the coefficient of pessimism. In the above example:

For example, assume that the coefficient of optimism is 0.6. Then each of the weighted values is computed as follows:

The Hurwicz criterion specifies the selection of the decision alternative corresponding to the maximum weighted value. Therefore the decision should be a "large-sized facility" with $206,000.

However, if the coefficient of optimism is 0.4, then the values are:

The decision is now a "Small-sized facility."

The LaPlace Criterion:

The LaPlace criterion is also called the equal likelihood criterion, which equally weights the possibility of occurrence of each state of nature. For example, if there are three states of nature, each has a 1/3 chance of occurrence. Then, each decision alternative will be assigned an average payoff value. In the above example, we have:

So, the decision is a "Large-sized facility."

Summary of Criteria Results:

The decision indicated by the decision criteria examined in the above example can be summarized as follows:

 Criterion  Decision

Maximax Large-sized facility

Maximin No facility

Minimax Regret Medium-sized facility

Hurwicz Small-sized facility

LaPlace Large-sized facility

 

The use of several decision criteria often results in a mix of decisions, with no one decision being selected more than the others. Hence, the appropriate criterion is dependent on the "risk" personality and philosophy of the decision maker.

Decision Making With Probabilities:

The Expected Value Criterion:

It is possible to obtain probability estimates for each state of nature in decision-making situations. We use the expected value criterion (used in Statistics) to identify the best decision alternative. The expected value EV is calculated by multiplying each decision outcome (payoff value) for each state of nature by the probability of its occurrence. Let's use the same example as we used above. This time each of the states of nature is assigned a probability:

We calculate each expected value as:

The best decision is the one with the largest expected value. In this case, $116,000 is the largest value, which gives us the "large-sized facility" as the best decision.

The Expected Opportunity Loss Criterion:

An alternative to the above approach is the expected opportunity loss criterion (EOL). This utilizes regrets (opportunity losses) to minimize the expected regret. From the regret table with each state of nature assigned probability:

We calculate the expected opportunity loss (EOL) for each decision alternative:

The best decision results from minimizing the regret. In this case, the decision is a " large-sized facility." The expected value and expected opportunity loss criteria result in the same decision. You may wonder why you need two separate approaches to reach the same conclusion. This will be discussed in the next section.

The Expected Value of Perfect Information:

If we examine the best two EV's from the above example,

EV(d1) = $116,000,

EV(d2) = $111,600,

The difference is only $4,400. But if we know there is a poor market condition, a choice of d2 would be a better choice than d1. Then, how can we figure out if such a condition may occur? One way is to hire a consultant to do some research on future market conditions. But it would be foolish for us to pay a consultant fee more than what we could gain in extra profit from the consultant's information. The information has some maximum value that represents the limit of what we would be willing to spend. This value of information can be computed as an expected value, referred to as the expected value of perfect information (EVPI). "Perfect information" leads a decision maker to no regret (the right decision) under each state of nature.

Using perfect information, the expected value of the decision is

$550,000(0.30) + 129,000(0.40) + 0(0.30)

= $216,600.

This is NOT the expected value of perfect information. The expected value of perfect information is the maximum amount that would be paid to gain information that would result in a decision better that the one made without perfect information. The expected value decision without perfect information is to construct a large-sized facility

EV(d1) = $116,000.

The expected value of perfect information is calculated as follows:

EVPI = $216,600 - 116,000 = 100,600.

That is, EVPI is equal to the expected value given perfect information minus the expected value without perfect information. In this case, $100,600 is the maximum amount that we would pay to purchase perfect information from other sources like a consultant and marketing surveys. It is rare for us to obtain perfect information in general.

It is noted that the expected value of perfect information is the same as the expected opportunity loss for the decision alternative selected. That is,

EVPI = best EOL.

This is always the case because if we use perfect information, there is no regret.

Summary

In this module, we have demonstrated the concepts and basics of decision making with or without probabilities. We have used both a payoff table approach and a decision tree approach. Although the examples presented here are rather simple, and actual decision-making situations are more complicated, the methods are the ones most decision makers use in order to make a decision.

The text compiled by S.R.ROY for references only