Decision theory is an interdisciplinary area of study, related to and of interest to practitioners in mathematics, statistics, economics, philosophy, management and psychology. It is concerned with the optimal decisions to be taken under particular circumstances.

Contents

1 Normative and descriptive decision theory 2 What kinds of decision need a theory? 2.1 Choice between incommensurable commmodities 2.2 Choice under uncertainty 2.3 Intertemporal choice 2.4 Social decisions 3 Complex decisions 4 References

Normative and descriptive decision theory

Most of decision theory is normative or prescriptive, i.e. it is concerned with identifying the best decision to take, assuming an ideal decision taker who is fully informed, able to compute with perfect accuracy, and fully rational. However, 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.

What kinds of decision need a theory?

Decision theory is only relevant in decisions that are difficult for some reason. A few types of decision have attracted particular attention:

• riskless choice between incommensurable commodities
• choice under uncertainty
• intertemporal choice
• social decisions

Choice between incommensurable commmodities

This area is concerned with the decision whether to have, say, one ton of guns and 3 tons of butter, or 2 tons of guns and 1 ton of butter. This is the classic subject of study of microeconomics and is rarely considered under the heading of decision theory, but such choices are often in fact part of the issues that are considered within decision theory.

Choice under uncertainty

This area represents the heartland of decision theory. Daniel Bernoulli stated 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 they 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 reality people do not behave like this, at least if "value" is taken to mean "objective financial value" - otherwise no-one would either gamble or take out insurance. Within behavioural decision theory, this has led to various dilutions of the expected value theory; for example, objective probabilities can be replaced by subjective estimates, and objective values by subjective utilities, giving rise to the subjective expected utility or SEU theory, developed by Savage. The prospect theory of Daniel Kahneman and Amos Tversky is another alternative to the expected value model within behavioural decision theory.

Pascal's wager is a classic example of a choice under uncertainty. The uncertainty, according to Pascal, is whether or not God exists. And the personal 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.

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 and Dempster-Shafer 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. 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 to

• 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-probabilitic 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.

Intertemporal choice

This area is concerned with the kind of choice where different actions lead to outcomes that are realised at different points in time. If I receive a windfall of several thousand dollars, I could spend it on an expensive holiday, giving me immediate pleasure, or I could invest it in a pension scheme, giving me 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, my life expectancy, and my confidence in the pensions industry. However even with all those factors taken into account, human behaviour 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.

Social decisions

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.

Complex decisions

Other areas of decision theory are concerned with decisions that are difficult simply because of their complexity, or the complexity of the organisation that has to take 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.

References

• 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 Pearl & Shafer. (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
• Morris De Groot Optimal Statistical Decisions. Wiley Classics Library. 2004. (Originally published 1970.) ISBN 0-471-68029-X.
• J.Q. Smith Decision Analysis: A Bayesian Approach. Chapman and Hall. 1988. ISBN 0-412-27520-1
• James O. Berger Statistical Decision Theory and Bayesian Analysis. Second Edition. 1980. Springer Series in Statistics. ISBN 0-387-96098-8.