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1 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 descriptive or positive 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.

2 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

2.1 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.

2.2 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.

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 theoryThe Dempster-Shafer theory is a mathematical theory of evidence that was introduced in the late 1970s by Glenn Shafer. It is a way of representing epistemic plausibilities. It developed from a sequence of works of Arthur Dempster, who was Shafer's advisor 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 CoxRichard Threlkeld Cox ( 1898 May 2, 1991) was a professor of physics at Johns Hopkins University, known for Cox's theorem relating to the foundations of probability. He was born in Portland, Oregon the son of attorney Lewis Cox and Elinor Junkin Cox. for justification of the probability axioms,

• the Dutch bookIn gambling a Dutch book or lock is a set of odds and bets which guarantees a profit, no matter what the outcome of the gamble. It is associated with probabilities implied by the odds not being coherent. One example is where a bookmaker has offered odds a paradoxes of Bruno de FinettiBruno de Finetti ( Innsbruck, June 13, 1906 Rome, July 20, 1985) was an Italian probabilist and statistician, noted for the "operational subjective" conception of probability. The classic exposition of his distinctive theory is the 1937 "La prevision: ses 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.