Choice under uncertainty

This area represents the heartland 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 emphasised 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.