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Complex Decision Trees [Decision Trees
Posted on August 21, 2013 @ 11:13:00 AM by Paul Meagher

In my last blog I offered up a video tutorial on how to construct simple decision trees and analyze them using expected values. It is easy to object that these binary decision trees are too simple to represent the complex decision problems that we are confronted with each day. Before you object too loudly, you should examine what a more complex decision tree might look like and the issues that arise when we add more complexity to our decision trees.

A decision tree can become more complex in two basic ways. We can add more intermediate acts or we can add more intermediate events. In simple decision trees we have a binary set of Actions (apply 90 lb nitrogen, apply 110 lb nitrogen) leading to a binary set of Events (e.g., probability of low rainfall, probability of high rainfall) and each combination of Actions and Events lead to an Outcome. See my blog, Representing Decisions with Graphviz, for more details.

So one way we can add complexity to a decision tree, beyond just adding more than 2 branches for each action and event node, is to add intermediate actions and/or events to our decision tree. So, for example, our decision problem might involve the act of either applying 90lbs or 110lbs of Nitrogen per acre to our wheat crop. We might also have to choose between the actions of applying the Nitrogen at time X or at time Y. The combination of these actions can then lead into a season with either a low summer rainfall event or a high summer rainfall event. We can represent a fragment of this decision tree generically with the following diagram:

The diagram was constructed using Graphviz and the dot file I used to construct it looks like this:

digraph MultiStep {

  rankdir=LR;

  Decision -> Action_Step_1A;
  Decision -> Action_Step_1B;

  Action_Step_1A -> Action_Step_2A;
  Action_Step_1A -> Action_Step_2B;

  Action_Step_2A -> Event_1;
  Action_Step_2A -> Event_2;

  Action_Step_2B -> Event_3;
  Action_Step_2B -> Event_4;

}

This is just a fragment of a multistep decision problem. As you can see, the number of terminal branches in this decision problem explodes as we add more intermediate action or event nodes. This does not prevent us from using decision trees to help us make better decisions, but it does give us advance warning that we should be very sure that it is necessary to introduce intermediate actions or events into our decision tree before we do so as they add considerable complexity to the decision tree. Decision trees are not meant to capture the minute details of a decision problem, just the high level actions and events that impact upon the decision. The choice of action and event nodes, just like the assignment of probabilities to event nodes, involves alot of subjective judgement. The process of formalizing it all into a decision tree, however, brings the whole exercise out of subjective reality into consensus reality where others can comment, disagree, or agree with the manner in which you have framed the decision problem.

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