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Decision theory in mathematics and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision and the resulting optimal decision.
## Normative and descriptive decision theory

## What kinds of decisions need a theory?

### Choice between incommensurable commodities

### Choice under uncertainty

### Intertemporal choice

### Competing decision makers

### Complex decisions

## Paradox of choice

Observed in many cases is the paradox that more choices may lead to a poorer decision or a failure to make a decision at all. It is sometimes theorized to be caused by analysis paralysis, real or perceived, or perhaps from rational ignorance. A number of researchers including Sheena S. Iyengar and Mark R. Lepper have published studies on this phenomenon. A popularization of this analysis was done by Barry Schwartz in his 2004 book, The Paradox of Choice.
## Statistical decision theory

Several statistical tools and methods are available to organize evidence, evaluate risks, and aid in decision making. The risks of Type I and type II errors can be quantified (estimated probability, cost, expected value, etc) and rational decision making is improved.

## Alternatives to probability theory

## See also

## References

## Further reading

Most of decision theory is normative or prescriptive, i.e. it is concerned with identifying the best decision to take, assuming an ideal decision maker who is fully informed, able to compute with perfect accuracy, and fully rational. The practical application of this prescriptive approach (how people should make decisions) is called decision analysis, and aimed at finding tools, methodologies and software to help people make better decisions. The most systematic and comprehensive software tools developed in this way are called decision support systems.

Since it is obvious that people do not typically behave in optimal ways, there is also a related area of study, which is a positive or descriptive discipline, attempting to describe what people will actually do. Since the normative, optimal decision often creates hypotheses for testing against actual behaviour, the two fields are closely linked. Furthermore it is possible to relax the assumptions of perfect information, rationality and so forth in various ways, and produce a series of different prescriptions or predictions about behaviour, allowing for further tests of the kind of decision-making that occurs in practice.

This area represents the heart of decision theory. The procedure now referred to as expected value was known from the 17th century. Blaise Pascal invoked it in his famous wager (see below), which is contained in his Pensées, published in 1670. The idea of expected value is that, when faced with a number of actions, each of which could give rise to more than one possible outcome with different probabilities, the rational procedure is to identify all possible outcomes, determine their values (positive or negative) and the probabilities that will result from each course of action, and multiply the two to give an expected value. The action to be chosen should be the one that gives rise to the highest total expected value. In 1738, Daniel Bernoulli published an influential paper entitled Exposition of a New Theory on the Measurement of Risk, in which he uses the St. Petersburg paradox to show that expected value theory must be normatively wrong. He also gives an example in which a Dutch merchant is trying to decide whether to insure a cargo being sent from Amsterdam to St Petersburg in winter, when it is known that there is a 5% chance that the ship and cargo will be lost. In his solution, he defines a utility function and computes expected utility rather than expected financial value.

In the 20th century, interest was reignited by Abraham Wald's 1939 paper pointing out that the two central concerns of orthodox statistical theory at that time, namely statistical hypothesis testing and statistical estimation theory, could both be regarded as particular special cases of the more general decision problem. This paper introduced much of the mental landscape of modern decision theory, including loss functions, risk functions, admissible decision rules, a priori distributions, Bayes decision rules, and minimax decision rules. The phrase "decision theory" itself was first used in 1950 by E. L. Lehmann.

The rise of subjective probability theory, from the work of Frank Ramsey, Bruno de Finetti, Leonard Savage and others, extended the scope of expected utility theory to situations where only subjective probabilities are available. At this time it was generally assumed in economics that people behave as rational agents and thus expected utility theory also provided a theory of actual human decision-making behaviour under risk. The work of Maurice Allais and Daniel Ellsberg showed that this was clearly not so. The prospect theory of Daniel Kahneman and Amos Tversky placed behavioural economics on a more evidence-based footing. It emphasized that in actual human (as opposed to normatively correct) decision-making "losses loom larger than gains", people are more focused on changes in their utility states than the states themselves and estimation of subjective probabilities is severely biased by anchoring.

Castagnoli and LiCalzi (1996), Bordley and LiCalzi (2000) recently showed that maximizing expected utility is mathematically equivalent to maximizing the probability that the uncertain consequences of a decision are preferable to an uncertain benchmark (e.g., the probability that a mutual fund strategy outperforms the S&P 500 or that a firm outperforms the uncertain future performance of a major competitor.). This reinterpretation relates to psychological work suggesting that individuals have fuzzy aspiration levels (Lopes & Oden), which may vary from choice context to choice context. Hence it shifts the focus from utility to the individual's uncertain reference point.

Pascal's Wager is a classic example of a choice under uncertainty. The uncertainty, according to Pascal, is whether or not God exists. Belief or non-belief in God is the choice to be made. However, the reward for belief in God if God actually does exist is infinite. Therefore, however small the probability of God's existence, the expected value of belief exceeds that of non-belief, so it is better to believe in God. (There are several criticisms of the argument.)

This area is concerned with the kind of choice where different actions lead to outcomes that are realised at different points in time. If someone received a windfall of several thousand dollars, they could spend it on an expensive holiday, giving them immediate pleasure, or they could invest it in a pension scheme, giving them an income at some time in the future. What is the optimal thing to do? The answer depends partly on factors such as the expected rates of interest and inflation, the person's life expectancy, and their confidence in the pensions industry. However even with all those factors taken into account, human behavior again deviates greatly from the predictions of prescriptive decision theory, leading to alternative models in which, for example, objective interest rates are replaced by subjective discount rates.

Some decisions are difficult because of the need to take into account how other people in the situation will respond to the decision that is taken. The analysis of such social decisions is the business of game theory, and is not normally considered part of decision theory, though it is closely related. In the emerging socio-cognitive engineering the research is especially focused on the different types of distributed decision-making in human organizations, in normal and abnormal/emergency/crisis situations. The signal detection theory is based on the Decision theory.

Other areas of decision theory are concerned with decisions that are difficult simply because of their complexity, or the complexity of the organization that has to make them. In such cases the issue is not the deviation between real and optimal behaviour, but the difficulty of determining the optimal behaviour in the first place. The Club of Rome, for example, developed a model of economic growth and resource usage that helps politicians make real-life decisions in complex situations.

One example shows a structure for deciding guilt in a criminal trial:

Actual condition | |||
---|---|---|---|

Guilty | Not guilty | ||

Decision | Verdict of 'guilty' | True Positive | False Positive (i.e. guilt reported unfairly) Type I error |

Verdict of 'not guilty' | False Negative (i.e. guilt not detected) Type II error | True Negative |

A highly controversial issue is whether one can replace the use of probability in decision theory by other alternatives. The proponents of fuzzy logic, possibility theory, Dempster-Shafer theory and info-gap decision theory maintain that probability is only one of many alternatives and point to many examples where non-standard alternatives have been implemented with apparent success. Work by Yousef and others advocate exotic probability theories using complex-valued functions based on the probability amplitudes developed and validated by Birkhoff and Von Neumann in quantum physics.

Advocates of probability theory point to:

- the work of Richard Threlkeld Cox for justification of the probability axioms,
- the Dutch book paradoxes of Bruno de Finetti as illustrative of the theoretical difficulties that can arise from departures from the probability axioms, and
- the complete class theorems which show that all admissible decision rules are equivalent to a Bayesian decision rule with some prior distribution (possibly improper) and some utility function. Thus, for any decision rule generated by non-probabilistic methods, either there is an equivalent rule derivable by Bayesian means, or there is a rule derivable by Bayesian means which is never worse and (at least) sometimes better.

- Paul Anand, "Foundations of Rational Choice Under Risk", Oxford, Oxford University Press (an overview of the philosophical foundations of key mathematical axioms in subjective expected utility theory - mainly normative) 1993 repr 1995 2002
- Sven Ove Hansson, " Decision Theory: A Brief Introduction", (an excellent non-technical and fairly comprehensive primer)
- Paul Goodwin and George Wright, Decision Analysis for Management Judgment, 3rd edition. Chichester: Wiley, 2004 ISBN 0-470-86108-8 (covers both normative and descriptive theory)
- Robert Clemen. Making Hard Decisions: An Introduction to Decision Analysis, 2nd edition. Belmont CA: Duxbury Press, 1996. (covers normative decision theory)
- D.W. North. "A tutorial introduction to decision theory". IEEE Trans. Systems Science and Cybernetics, 4(3), 1968. Reprinted in Shafer & Pearl. (also about normative decision theory)
- Glenn Shafer and Judea Pearl, editors. Readings in uncertain reasoning. Morgan Kaufmann, San Mateo, CA, 1990.
- Howard Raiffa Decision Analysis: Introductory Readings on Choices Under Uncertainty. McGraw Hill. 1997. ISBN 0-07-052579-X
- Lev Virine and Michael Trumper. Project Decisions: The Art and Science, Vienna, VA: Management Concepts, 2008. ISBN 978-1567262179
- Morris De Groot Optimal Statistical Decisions. Wiley Classics Library. 2004. (Originally published 1970.) ISBN 0-471-68029-X.
- Khemani , Karan, Ignorance is Bliss: A study on how and why humans depend on recognition heuristics in social relationships, the equity markets and the brand market-place, thereby making successful decisions, 2005.
- J.Q. Smith Decision Analysis: A Bayesian Approach. Chapman and Hall. 1988. ISBN 0-412-27520-1
- Akerlof, George A. and Janet L. YELLEN, Rational Models of Irrational Behavior
- Arthur, W. Brian, Designing Economic Agents that Act like Human Agents: A Behavioral Approach to Bounded Rationality
- James O. Berger Statistical Decision Theory and Bayesian Analysis. Second Edition. 1980. Springer Series in Statistics. ISBN 0-387-96098-8.
- Miller, L. (1985). Cognitive risk taking after frontal or temporal lobectomy I. The synthesis of fragmented visual information. Neuropsychologia, 23, 359 369.
- Miller, L., & Milner, B. (1985). Cognitive risk taking after frontal or temporal lobectomy II. The synthesis of phonemic and semantic information. Neuropsychologia, 23, 371 379.
- Anderson, Barry F. The Three Secrets of Wise Decision Making. Single Reef Press. 2002. ISBN 0-9722177-0-3.

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