16 Decision Problems

The basic building blocks of a decision problem consist of three sets:

A set $A$ of acts (also called alternatives).
A set $O$ of outcomes (also called consequences).
A set $S$ of states.

Each act, $a \in A$, produces a specific outcome when paired with a state. More formally, an act $a$ is defined as a function $a: S \rightarrow O$, mapping each state in $S$ to an outcome in $O$. For any act $a \in A$, state $s \in S$, and outcome $o \in O$, we use the notation $a(s) = o$ to signify that the combination of act $a$ and state $s$ results in outcome $o$. In rational choice theory, it is typically assumed that decision makers choose among the available acts in $A$ based on their preferences for certain outcomes and their beliefs about which states are more likely to occur.

To illustrate these concepts, imagine you are faced with a choice between two bets:

Bet 1: You receive $\$100$ if it rains tomorrow at noon.
Bet 2: You receive $\$200$ if it does not rain tomorrow at noon.

This decision problem can be represented in the following table, where the columns represent the possible states (whether it rains or does not rain), the rows represent the available acts (bet 1 or bet 2), and each cell indicates the outcome that results from the combination of a chosen act and a 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 (most likely, you prefer more money over less) and your beliefs about the likelihood of rain tomorrow at noon.

One complication when representing a decision problem using sets of acts, outcomes, and states is that there are often multiple ways to define the states. For example, instead of a single state labeled “rain at noon tomorrow,” we might refine this into more specific conditions, such as one state where it rains between 11 a.m. and 1 p.m. and another state where it rains between 11:30 a.m. and 1:30 p.m. Similarly, the state “does not rain at noon tomorrow” could be split into one state where it rains between 1 p.m. and 2 p.m. and another state where it does not rain at all. This refined description leads to the following representation of the decision problem:

	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 define the states in a decision problem. The key assumption is that each state should resolve all remaining uncertainty, so that when a state is combined with an act, it leads to a single, well-defined outcome. For instance, using a state like ‘it is cloudy at noon tomorrow’ would be inadequate because it does not provide enough information to fully determine the outcome associated with each act.

To summarize, a decision problem is defined as a tuple $(S, O, A)$, where $S$ is a non-empty set of states, $O$ is a non-empty set of outcomes, and $A$ is a set of acts, each being a function that maps each state in $S$ to an outcome in $O$. In this course, we will focus on two main types of decision problems:

Decisions under certainty: In this scenario, the decision maker knows which state will be realized (or, equivalently, there is only one state to consider). This allows us to simplify the decision problem by assuming that the decision maker chooses directly from the set of possible outcomes, without uncertainty about which state will occur.
Decisions under uncertainty: Here, the decision maker is unsure about which state will be realized. Consequently, decision making under uncertainty requires considering both the decision maker’s preferences over the possible outcomes and their beliefs about the likelihood of different states being realized.