An Introduction to Decision Theory

Before you make a decision you have to determine what to decide about. Or, to put it differently, you have to specify what the relevant acts, states and outcomes are. Suppose, for instance, that you are thinking of taking out fire insurance on your home. Perhaps it costs $100 to take out insurance on a house worth $100,000 and you ask, “Is it worth it?” Before you decide, you have to get the formalization of the decision problem right. In this case, it seems that you face a decision problem with two acts, two states, and four outcomes. It is helpful to visualize this information in a decision matrix; see Table 2.1.

To model one's decision problem in a formal representation is essential in decision theory, because decision rules are only defined relative to a formal representation. For example, it makes no sense to say that the principle of maximizing expected value recommends one act rather than another unless there is a formal listing of the available acts, the possible states of the world and the corresponding outcomes. However, instead of visualizing information in a decision matrix it is sometimes more convenient to use a decision tree. The decision tree in Figure 2.1 is equivalent to the matrix in Table 2.1.

The square represents a choice node, and the circles represent chance nodes. At the choice node the decision maker decides whether to go up or down in the tree. If there are more than two acts to choose from, we simply adds more lines. At the chance nodes nature decides which line to follow. The rightmost boxes represent the possible outcomes. Decision trees are often used for representing sequential decisions, i.e. decisions that are divided into several separate steps. (Example: In a restaurant, you can either order all three courses before you start to eat, or divide the decision-making process into three separate decisions taken at three points in time. If you opt for the latter approach, you face a sequential decision problem.) To represent a sequential decision problem in a tree, one simply adds new choice and chance nodes to the right of the existing leaves.