Notes on Golub, Chapter One-A

Golub points out that we need to define decision problems. Defining a decision problem is not the same as listing a set of alternatives. The alternatives are not the problem; they are not what makes the present situation unsatisfactory. Golub says that "a well-formed problem statement should clearly indicate at least three components: the nature of the current situation, the nature of the preferred situation, and a central objective that distinguishes the two."

In practice, the distinction between "the nature of the preferred situation" and the "central objective" can be a little foggy. For example, Golub lists a number of "solutions in search of a problem":

None of these questions can be answered without knowing something about the objectives that might guide an answer. And knowing that also depends on understanding what might be wrong. So Golub's first suggestion: if you find yourself trying to formulate a decision problem but instead find yourself asking about a solution, try asking why the solution is being considered:
    Why should welfare recipients be fingerprinted?
    Why should Gavin buy a new car?
    Why should the XYZ plant be closed.
    Why should some criminals be executed?
NOTE: by asking these questions, we aren't assuming that the solution implicit in them is the right one. We are asking what might lead someone to think such a solution was called for. In fact, I would suggest a different formulation than Golub suggests:
    Why might someone think welfare recipients be fingerprinted?
    Why might someone think Gavin buy a new car?
    Why might someone think the XYZ plant be closed.
    Why might someone think some criminals be executed?
For these examples, he suggests the following possible problem statements: Golub writes "These restated problems clearly imply what is the matter with the current situation and what situation would be preferred." Perhaps so. However, my problem is that "imply" is all they do. Golub later on stresses the need for a detailed formulation of a problem. And none of these come close to measuring up. Let's see how each could be made more explicit. These formulations make the central concerns clearer. However, as Golub would point out himself, such initial problem statements are not full analyses of the problem. We will want to be sure we have our facts straight. (For example: how great is the problem of unrepentant murderers being set free and then committing further crimes? We also want to be sure that we are focusing on the right issue. (For example: Why are production costs at XYZ higher?) And we want to be sure that we ask what other values and objectives we care about. (For example: while we want to enforce fairness in distributing welfare checks, we also want to do so at a reasonable cost. If the problem is small and the cost of fixing it high, we might think it wiser to set this problem aside, focusing instead on the fact -- if it is one -- that such errors are few.)

Indeed, Golub goes on to stress various ways in which we might want to re-focus our attack on the problem. We may want to widen our analysis. Often one problem is intertwined with many others. If we ignore the related problems, we may leave the problem we began with essentially untouched. And sometimes, the problem we have focused on is only a symptom of a deeper problem. We may need to go beneath the surface and attack the underlying causes. So we can ask: (i) why does this problem exist? and (ii) what other problems and situations affect the likelihood of resolving it?

Sometimes we need to narrow our focus. Attacking the whole of a problem (world hunger, to take an extreme example) may be beyond our means. But some real good may come of finding some relatively isolable aspect of the larger problem and putting our resources there.

In general, it's important to describe the problem in detail. Ask yourself the sorts of questions we have just pointed to.

In formal decision analysis, part of the detailed description is likely to be a decision tree or an influence diagram. Decision trees get complex very quickly and in the initial stages, they may not be the best way to represent the situation to yourself. Ultimately, however, a decision tree or similar structure may turn out to be necessary; without it, it may be impossible to do the calculations that forecasting and comparison require.

We will look at this by following especially closely the details of the fictitious Empire Remote Telephones decision from Golub's text. Empire has developed new cellular phone technology that will provide for clearer transmission and less breaking up of the signal. They are trying to decide on an initial pricing strategy. The units cost them $120 apiece. We will assume the analysis done and (implausibly, perhaps) assume that they have selected two alternatives: a price of $180 per unit of $250 per unit. So they have a choice to make about price. In a tree diagram, we represent this choice as follows:

Boxes represent choices. Here there is just one relevant choice: price.

However, it should be clear that there is an uncertainty that complicates this decision: the volume of sales. Sales will be affected by price. In fact, the range of sales is a variable that can take on a large number of values, but for this example, we suppose that Empire focuses on three roughly-delineated possibilities: high, medium and low sales volume. And we suppose that marketing estimates probabilities for each of these possibilities for a price of $250:

Sales represents an uncertain event. Uncertain events are represented by circles or ovals. Before we create the whole tree diagram, we will focus on one part of it: the possibilities just described. Here is the diagram for these possibilities:

Of course, if we did a complete analysis, we would need to include the probabilities for sales if the lower price were selected. And it wouldn't make any sense to decide on the price before working through that part of the analysis. These probabilities would be crucial to making the calculations that would drive the decision in the first place. In the next chapter, Golub will have us working with more elaborate diagrams, but we will anticipate here. What follows is a hypothetical diagram that includes all the relevant considerations up to this point.

Diagrams get awkward quickly. In fact, however, the same information can be represented in table form:

Price Sales Prob.
$250 Low .35
Medium .40
High .25
$180 Low .25
Medium .30
High .45

Here "Prob." means "probability This captures the same relationships in a more tractable format. In a real treatment of this problem, we would add further information about outcomes.

Influence Diagrams

Golub suggests an approach to structuring a decision problem:

  1. Create three lists: decisions, uncertain events, objectives
  2. Sort lists in order of importance
  3. Create initial influence diagram: decision --> objective
  4. Add elements one at a time until initial lists are exhausted
  5. Evaluate the overall structure, and revise if necessary
  6. Ask others to evaluate the overall structure, and revise if necessary
Notice that this approach assumes that you already know what is to be decided, which typically means knowing what the alternatives are.

In the case of Empire phones, two of the lists are short: there is one decision: price, and one uncertain event: sales. The influence diagram, unlike the decision tree, doesn't require us to decide what prices are the options, nor does it force us to decide just what range of sales we will consider. To construct the diagram, we need to introduce the company's objectives:

Now we need to interpose these objectives between the initial decision and the objective of maximizing total satisfaction. To do that, we also need to think about what influences what.

The price of the new units influences profit directly -- other things being equal, the higher the price the greater the profit -- but it also influences sales, the uncertain event, which in turn influence profits: other things being equal, the more sales, the more profit.

Market share is affected by sales. The effect of price on market share, however, is entirely by way of sales.

Price influences reputation. Low prices enhance the company's reputation for providing good value. Do sales affect reputation? I would have thought so. The more people buy a product, the more people will come to think of it as the product to buy. But Golub leaves this consideration aside, and so we will too.

What about product line? Is this affected by the price or the sales of this particular product? Not really. There might be an indirect connection: if this product does very well, there will be more capital available to develop new products. But that point, we will assume, is remote from this decision. So we can drop considerations of completeness of product line out of our analysis of this problem.

The end result is an influence diagram:

As before, rectangles represent choices and ovals represent uncertain events. Boxes with round corners represent objectives. Here "Total Satisfaction" is a term that simply functions in all such diagrams to represent the overall result of all the scores on all the objectives.

Notice what the diagram doesn't tell us. It doesn't tell us what the alternatives choices are. It doesn't tell us what the alternative possible outcomes of "Sales" are. It only speaks in terms of overall categories.

This is a simple example of an influence diagram. In the next chapter, we encounter influence diagrams that are more complicated. The basic principles are the same, however.

-Allen Stairs stairs@glue.umd.edu