Decision Theory

What it asks

You must choose, the outcomes depend on facts you do not have, and you cannot wait. Almost every consequential decision has that shape.

The standard answer is expected utility: assign a value to each outcome, weight it by how likely it is, add them up, take the largest. It is simple, it can be derived from a handful of assumptions each of which looks undeniable, and it is the backbone of economics, policy analysis and artificial intelligence.

It also gives the wrong answer in cases where people's contrary judgement is stable, considered and shared. That gap is the subject.

The recurring problems

The Allais paradox. Presented with two pairs of gambles, most people choose in a way that no assignment of utilities can make consistent. They do not change their minds when the inconsistency is explained, which is what makes it a problem for the theory rather than for them.

Ambiguity. The Ellsberg cases show that people will pay to avoid unknown probabilities as distinct from unfavourable ones. Standard theory has no room for the distinction; ordinary judgement treats it as obvious.

Newcomb's problem. A predictor who has never been wrong has already filled two boxes. Taking one box has done far better historically; taking both dominates whatever is in them. The two most natural principles of rational choice give opposite answers, which split the field into causal and evidential decision theory and has kept it split.

Where the numbers come from. The theory takes utilities and probabilities as inputs. Getting them out of a real person, without assuming the theory, is unsolved.

How it developed

The St Petersburg problem in the eighteenth century showed that expected monetary value cannot be the criterion, since a gamble with infinite expected value is worth very little to anyone. Bernoulli's answer, that what matters is utility rather than money and that utility rises more slowly than wealth, is the origin of the whole framework.

Von Neumann and Morgenstern axiomatised it in 1944, and Savage extended it in 1954 to cases where the probabilities are themselves the agent's. Both proved representation theorems: anyone whose preferences satisfy a few structural conditions can be described as maximising expected utility, whether or not they are doing any arithmetic.

Allais produced his counterexample in 1953 and Ellsberg his in 1961, both aimed at exactly those conditions. Nozick set out Newcomb's problem in 1969.

From the 1970s the empirical challenge became systematic, as work on how people actually decide documented departures too regular to dismiss as error - which left the theory in the awkward position of being clearly normative and clearly not descriptive, without a settled account of how something can be the first and not the second.

The positions within it

Causal and evidential decision theory divide over Newcomb cases. Expected utility theory and its rivals - prospect theory, regret theory, maximin - divide over the paradoxes. Bayesian approaches treat probability as credence throughout. All are in the Schools section.

Where it stands now

The field has real institutional weight: cost-benefit analysis, clinical decision-making, insurance and the objective functions of automated systems all rest on it.

The active philosophical work is on cases the framework handles badly and that now matter - very small probabilities of very large harms, decisions whose outcomes are not comparable on any scale, and choices affecting who will exist, where the person the decision is for does not yet exist to have preferences.