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# Decision Trees

The most basic technique in decision theory is the decision tree. Take, as an example, the question of whether to regulate air pollution to improve forest health.2

Suppose you are asked to provide an opinion on whether or not air pollution should be regulated to improve forest health.3 Three responses are possible:

1. Pollution is affecting forest health and should be regulated.

2. Pollution is not affecting forest health and should not be regulated .

3. The evidence is inconclusive (so I need some grant money to study the problem, and I'll get back to you).

Of course, regardless of your opinion about whether pollution is affecting forest health, pollution either is having an effect or it is not having an effect. Even if you regard the evidence as inconclusive, pollution either is or is not having an effect.4

In addition, regulators will either decide to regulate air pollution or not. We hope that their decision is informed by the advice we give them, but their decision on whether to regulate or not depends on whether they think that pollution is affecting forest health,5 and just as our opinion can be wrong, their decision can be wrong too.6 Notice that there are 12 possible outcomes: 3 answers we might give to the question, 2 possible states of the world, and 2 possible decisions by the regulator.

These possibilities and their consequences can be summarized in a decision tree (Figure 1).

• The squares represent points in the tree over which you have control as a decision-maker, i.e., they represent the decisions you can make. In this case our decision is what to tell the person who asked for our opinion.

• The circles represent points in the tree over which you have no control.

• Along each branch of the tree, all possibilities are considered.

The consequences of following each branch of the tree can be evaluated for several different variables. Notice that the payoffs to the scientist and the forest are not the same. More importantly, they payoffs are measured in different units. They're not easy to compare.7

Next: Statistical Decision Theory Up: Decision Making Under Uncertainty: Previous: The Framework of Statistical
Kent Holsinger 2011-11-13