Optimal decision

If there is uncertainty as to what the outcome will be but one has knowledge about the distribution of the uncertainty, then under the von Neumann–Morgenstern axioms the optimal decision maximizes the expected utility (a probability–weighted average of utility over all possible outcomes of a decision). Sometimes, the equivalent problem of minimizing the expected value of loss is considered, where loss is (–1) times utility. Another equivalent problem is minimizing expected regret.

"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 and rules of thumb 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 many available decision options and a complex decision–outcome relationship.

Formal mathematical description

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

${\displaystyle U_D(d)=U_O(f(d)).}$

We can then define an optimal decision ${\displaystyle d_{\mathrm{o}\mathrm{p}\mathrm{t}}}$ as one that maximizes ${\displaystyle U_D(d)}$ :

${\displaystyle d_{\mathrm{o}\mathrm{p}\mathrm{t}}=\arg \underset{d\in D}{max}{U}_D(d).}$

Solving the problem can thus be divided into three steps:

1. predicting the outcome ${\displaystyle o}$ for every decision ${\displaystyle d;}$
2. assigning a utility ${\displaystyle U_O(o)}$ to every outcome ${\displaystyle o;}$
3. finding the decision ${\displaystyle d}$ that maximizes ${\displaystyle U_D(d).}$