We want to discuss making a decision.1 To discuss decisions, we first must speak of the choices to be made.
We have a set which includes all possible choices. The set is called the actions (or acts, decisions, choices or designs).
For each decision, we have a set which includes all possible outcomes. Often the outcomes are uncertain, and associated with the future (see Uncertain Outcomes).
Since we select an action prior to observing the outcome, we want to talk about which actions and outcomes are preferable to others. We call an action-outcome pair a history. We refer to the set of action-outcome pairs the histories. A preference is an order on histories.
A simple decision problem is a triple $(A, \set{O_a}_{a \in A}, \preceq)$ in which $A$ is a set of actions, $\set{O_a}$ is a family of sets of outcomes, and $\preceq$ is a preference on the histories $\Set*{(a, o)}{a \in A, o \in O_a}$.
Consider deciding whether to host a party
indoors or outdoors.
We are unsure of the weather.
We have a set of two actions $A = $
{In,
Out} and a set of
two outcomes (the same for each action) $O_a =
W =$ {Rain,
Shine} for $a \in A$.
Although many orders on $A \times W$ exist,
one such order is
\[
(\textsc{Out}, \textsc{Shine}) \prec (\textsc{In}, \textsc{Rain})
\prec (\textsc{In}, \textsc{Shine}) \prec (\textsc{Out},
\textsc{Rain}).
\]
Let $(A, \set{O_a}, \preceq)$ be a simple decision problem in which $O_a = S$ for each $a \in A$. Let $o \in S$. An action $a \in A$ is best for outcome $o$ if $(a, o) \preceq (a', o)$ for all $a' \in A$. An action $a \in A$ is best for all outcomes (or uniformly best) if, for all $o \in O$, $(a, o) \preceq (a', o')$ for all $a' \in A, o' \in O$.2
In the party example, the best action for Rain is In and the best action for Shine is Out. On the other hand, there is no uniformly best action. For the action Out, there is an outcome Rain, and action-outcome pair (In, Rain) so that (In, Rain) $\prec$ (Out, Rain). In other words, Out is not uniformly best, because there is an outcome, Rain for which the action In is preferred. Similarly for In.