Optimal decision

An optimal decision is a decision such that no other available decision options will lead to a better outcome. 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 in what the outcome will be, the optimal decision maximizes the expected (average) utility.

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.

In case the decision outcome is subject to uncertainty, an optimal decision is maximizing the expected utility.

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 more intuitive approaches to make decisions that are "good enough."

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.

Formal mathematical description

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

$U_D(d) \ = \ U_O(f(d)) \,$

We can then define an optimal decision d_opt as one that maximizes U_D(d) :

$d_\mathrm{opt} = \arg\max \limits_{d \in D} U_D(d) \,$

Solving the problem can thus be divided into three steps:

1. predicting the outcome o for every decision d
2. assigning a utility U_O(o) to every outcome o
3. finding the decision d that maximizes U_D(d)

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 d, we know the probability distribution for the possible outcomes described by the conditional probability density p(o | d). We can then calculate the expected utility of decision d as

$U_D(d)=\int{p(o|d)U(o)do}\,$    ,

where the integral is taken over the whole set O (DeGroot, pp 121)

An optimal decision d_opt is then one that maximizes U_D(d), just as above

$d_\mathrm{opt} = \arg\max \limits_{d \in D} U_D(d) \,$

Example

The Monty Hall problem.

See also

• Decision making
• Decision making software

References

• Morris DeGroot Optimal Statistical Decisions. McGraw-Hill. New York. 1970. ISBN 0070162425.
• James O. Berger Statistical Decision Theory and Bayesian Analysis. Second Edition. 1980. Springer Series in Statistics. ISBN 0-387-96098-8.