An optimal 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 in
decision theory
Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
. In order to compare the different decision outcomes, one commonly assigns a
utility
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...
value to each of them.
If there is uncertainty as to what the outcome will be but knowledge about the distribution of the uncertainty, then under the
von Neumann–Morgenstern axioms the optimal decision maximizes the
expected utility The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on the ...
(a probability–
weighted average
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of utility over all possible outcomes of a decision). Sometimes, the equivalent problem of minimizing the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of
loss
Loss may refer to:
Arts, entertainment, and media Music
* ''Loss'' (Bass Communion album) (2006)
* ''Loss'' (Mull Historical Society album) (2001)
*"Loss", a song by God Is an Astronaut from their self-titled album (2008)
* Losses "(Lil Tjay son ...
is considered, where loss is (–1) times utility. Another equivalent problem is minimizing expected
regret
Regret is the emotion of wishing one had made a different decision in the past, because the consequences of the decision were unfavorable.
Regret is related to perceived opportunity. Its intensity varies over time after the decision, in regard ...
.
"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
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
problem. In practice, few people verify that their decisions are optimal, but instead use
heuristics
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...
to make decisions that are "good enough"—that is, they engage in
satisficing
Satisficing is a decision-making strategy or cognitive heuristic that entails searching through the available alternatives until an acceptability threshold is met. The term ''satisficing'', a portmanteau of ''satisfy'' and ''suffice'', was introduc ...
.
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
in a set
of available decision options will lead to an outcome
. All possible outcomes form the set
.
Assigning a utility
to every outcome, we can define the utility of a particular decision
as
:
We can then define an optimal decision
as one that maximizes
:
:
Solving the problem can thus be divided into three steps:
# predicting the outcome
for every decision
# assigning a utility
to every outcome
# finding the decision
that maximizes
Under uncertainty in outcome
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
, we know the probability distribution for the possible outcomes described by the
conditional probability density . Considering
as a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
(conditional on
), we can calculate the expected utility of decision
as
:
,
where the integral is taken over the whole set
(DeGroot, pp 121).
An optimal decision
is then one that maximizes
, just as above:
:
An example is the
Monty Hall problem
The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show ''Let's Make a Deal'' and named after its original host, Monty Hall. The problem was originally posed (and solved) ...
.
See also
*
Decision-making
In psychology, decision-making (also spelled decision making and decisionmaking) is regarded as the Cognition, cognitive process resulting in the selection of a belief or a course of action among several possible alternative options. It could be ...
*
Decision-making software Decision-making software (DM software) is software for computer applications that help individuals and organisations make choices and take decisions, typically by ranking, prioritizing or choosing from a number of options.
An early example of DM so ...
*
Two-alternative forced choice Two-alternative forced choice (2AFC) is a method for measuring the sensitivity of a person, child or infant, or animal to some particular sensory input, stimulus, through that observer's pattern of choices and response times to two versions of the s ...
References
* Morris DeGroot ''Optimal Statistical Decisions''. McGraw-Hill. New York. 1970. .
* James O. Berger ''Statistical Decision Theory and Bayesian Analysis''. Second Edition. 1980. Springer Series in Statistics. .
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