Optimal decision

An optimal decision is a decision that leads to at least as good an outcome as all other available decision options. It is an important concept in decision theory. In order to compare the different decision outcomes, one commonly assigns a relative utility to each of them. If there is uncertainty as to what the outcome will be, then under the von Neumann–Morgenstern axioms the optimal decision maximizes the expected utility (utility averaged over all possible outcomes of a decision).

Sometimes, the equivalent problem of minimizing loss is considered, particularly in financial situations, where the utility is defined as economic gain.

"Utility" is only an arbitrary term for quantifying the desirability of a particular decision outcome and not necessarily related to "usefulness." For example, it may well be the optimal decision for someone to buy a sports car rather than a station wagon, if the outcome in terms of another criterion (e.g., effect on personal image) is more desirable, even given the higher cost and lack of versatility of the sports car.

The problem of finding the optimal decision is a mathematical optimization problem. In practice, few people verify that their decisions are optimal, but instead use heuristics to make decisions that are "good enough"—that is, they engage in satisficing.

A more formal approach may be used when the decision is important enough to motivate the time it takes to analyze it, or when it is too complex to solve with more simple intuitive approaches, such as with a large number of available decision options and a complex decision–outcome relationship.

Each decision ${\displaystyle d}$ in a set ${\displaystyle D}$ of available decision options will lead to an outcome ${\displaystyle o=f(d)}$. All possible outcomes form the set ${\displaystyle O}$. Assigning a utility ${\displaystyle U_{O}(o)}$ to every outcome, we can define the utility of a particular decision ${\displaystyle d}$ as

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