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Anoptimal decision is a decision that leads to at least as good a known or expected outcome as all other available decision options. It is an important concept indecision theory. In order to compare the different decision outcomes, one commonly assigns autility value to each of them.

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

"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 amathematical optimization problem. In practice, few people verify that their decisions are optimal, but instead useheuristics and rules of thumb to make decisions that are "good enough"—that is, they engage insatisficing.

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.

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${\displaystyle U_D(d).}$

In case it is not possible to predict with certainty what will be the outcome of a particular decision, a probabilistic approach is necessary. In its most general form, it can be expressed as follows:

Given a decision${\displaystyle d}$, we know the probability distribution for the possible outcomes described by theconditional probability densityFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle p(o|d)}. Considering${\displaystyle U_D(d)}$ as arandom variable (conditional on${\displaystyle d}$), we can calculate the expected utility of decision${\displaystyle d}$ as

${\displaystyle \text{E}{U}_D(d)=\int p(o|d)U(o)do}$ ,

where the integral is taken over the whole set${\displaystyle O}$ (DeGroot, pp 121).

An optimal decision${\displaystyle d_{\mathrm{o}\mathrm{p}\mathrm{t}}}$ is then one that maximizes${\displaystyle \text{E}{U}_D(d)}$, just as above:

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

An example is theMonty Hall problem.

• Decision-making
• Decision-making software
• Two-alternative forced choice

• Morris DeGrootOptimal Statistical Decisions. McGraw-Hill. New York. 1970.ISBN 0-07-016242-5.
• James O. BergerStatistical Decision Theory and Bayesian Analysis. Second Edition. 1980. Springer Series in Statistics.ISBN 0-387-96098-8.

• Optimal decisions

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