# 20 Decision Matrices

The basic building blocks of a decision problem are the following three sets:

- the set of acts (also called the alternatives);

- the set of outcomes (also called the consequences); and
- the set of states.

Suppose that \(A\) is the set of acts, \(O\) is the set of outcomes, and \(S\) is the set of states in some decision problem. An act together with a state leads to an outcome. More formally, each act \(a\in A\) is a function \(a: S\rightarrow O\) associating states with outcomes. For \(a\in A\), \(s\in S\) and \(o\in O\), we write \(a(s)=o\) when act \(a\) and state \(s\) results in outcome \(o\). The standard assumption in rational choice theory is that a decision maker chooses an element from the set \(A\) of acts and that this choice depends on which outcome is desired and beliefs about the states.

To illustrate the above ideas, suppose that you are offered a choice between two bets:

- bet 1: you receive $100 if it rains tomorrow at noon, and
- bet 2: you receive $200 if it does not rain tomorrow at noon.

This decision problem can be visualized using the following table, where the columns are labeled by the states, the rows are labeled by the acts and each cell of the table is the outcome that results from the chosen act and the realized state.

rain at noon tomorrow | does not rain at noon tomorrow | |

bet 1 | win $100 | receive nothing |

bet 2 | receive nothing | win $200 |

The act you will choose (either bet 1 or bet 2) depends on your preferences over the outcomes (presumably you prefer more money to less) *and* your beliefs about whether or not it will rain tomorrow at noon.

Typically, when describing a decision problem it is straightforward to write down the set of acts and the set of outcomes. However, there are often multiple ways to describe the states in a decision problem. For example, one might split the state “rain at noon tomorrow” into two states: the first state is that it rains between 11am and 1pm and the second state is that it rains between 11:30am and 1:30pm. Similarly, the state “does not rain at noon tomorrow” may be split into two states: the first state is that it rains between 1pm and 2pm and the second state is that it does not rain at all. This way of describing the decision problem leads to the following table:

rain 11am-1pm | rain 11:30am-1:30pm | rain 1:00pm-2:00pm | does not rain | |

bet 1 | win $100 | win $100 | receive nothing | receive nothing |

bet 2 | receive nothing | receive nothing | win $200 | win $200 |

In general, there is no single best way to describe the states in a decision problem. There are two important assumptions about states. The first assumption is that a state resolves all remaining uncertainty, so that a state together with an act results in a single outcome. So, for example, “it is cloudy at noon tomorrow” cannot be used as a state since it does not specify the outcome associated with each act. The second assumption is that the player’s choice of act does not influence which state is realized. For instance, consider the following representation of the above decision problem:

choose the correct bet | choose the wrong bet | |

bet 1 | win $100 | receive nothing |

bet 2 | win $200 | receive nothing |

This representation of the decision problem implies that bet 2 is clearly better than bet 1. The problem is that which of the two states is realized depends on whether it rains tomorrow at noon *and* the act chosen by the decision maker.

In these notes, we study two types of decision problems.

- Decisions under certainty: The decision maker knows which state is realized (or, equivalently, there is only one state). In this case, we can simplify the description of the decision problem by assuming that the decision maker directly chooses from the set of outcomes.

- Decisions under uncertainty: The decision maker is uncertain about the which state is realized. Thus, both the decision maker’s preferences over the outcomes and the decision maker’s beliefs about the states are represented in a decision under uncertainty.