A Theory About Consistency

The most common objection to economics is that people do not sit down and calculate expected utilities before deciding anything.

This is true and it is not an objection, because expected utility theory has never claimed otherwise. What von Neumann and Morgenstern proved in 1944 is a representation result: if a person's preferences satisfy a small set of consistency conditions, then those preferences can be described as if they were maximising the expected value of some function. The person need not know the function exists.

The analogy usually offered is a skilled billiards player, who does not solve differential equations and behaves as though they had. The prediction is about behaviour, not about mental process, and it is the consistency conditions rather than the arithmetic that carry the content.

Which means the interesting evidence is not that people are bad at sums. It is that the axioms fail.

Before getting to that, the idea the whole framework is built on deserves its own paragraph, because it explains an enormous amount.

Daniel Bernoulli posed a puzzle in 1738. A coin is flipped until it lands heads; if that happens on flip n, you are paid two to the power of n. The expected monetary value of this game is infinite: a half chance of two, a quarter chance of four, an eighth chance of eight, forever.

Almost nobody will pay more than a few coins to play. Bernoulli's resolution was that the value of money is not linear in money: each additional pound adds less satisfaction than the last, so the astronomical prizes far out in the tail contribute almost nothing to the game's value.

Diminishing marginal utility is the entire explanation for risk aversion. It explains why insurance exists; you will pay more than the expected loss to avoid a large one. It explains why a hundred pounds matters more to someone with nothing than to someone with a million, which is the basis of every argument for redistribution that rests on efficiency rather than on fairness. And it follows from one observation about a coin-flipping game.

Now the failures, and they are failures of the axioms rather than of arithmetic.

Maurice Allais constructed a pair of choices in 1953. In the first, most people prefer a certain million to a gamble with a small chance of nothing and a large chance of more. In the second, where a common component has been added to both options, most of the same people switch to the gamble.

Independence, the axiom saying that a component shared by two options should not affect the ranking between them, has been violated. The switch is robust, replicates, and persists after people have the inconsistency explained to them, which is what distinguishes it from a mistake. Told they have added wrongly, people correct; told they have violated independence, many shrug and keep their preferences.

That is the interesting kind of result: not an error to be fixed but a preference the theory cannot represent.

Herbert Simon attacked from a different direction and arguably did more damage. Optimising over a set of options requires knowing what the options are and what each is worth. For almost any real decision (which job, which flat, which supplier) the set is not enumerable and the values are not known, and finding out costs time and money.

His alternative was satisficing: set an aspiration level, search until something clears it, take it. This is not a degraded version of optimisation; it is a different procedure, and once search costs are counted it is frequently the better one. Simon's term, bounded rationality, gets misread as "not very rational". It means rationality that accounts for the cost of thinking, which is the only kind anybody has.

None of this leaves economists defending perfect rationality, whatever the caricature says. The standard model is used as a tractable benchmark, practitioners say "as if" about it, and the productive research of the past fifty years (information economics, behavioural economics, bounded rationality) has been substantially about where it breaks.

The caricature persists for a structural reason worth naming: introductory courses teach the benchmark for a year or more before teaching its limits, so anybody who takes one course and stops leaves with the benchmark and none of the corrections.

One last case, because it shows the theory straining in public. A great many people buy insurance and also buy lottery tickets. Insurance implies risk aversion; lottery tickets imply risk seeking; a single concave utility function cannot produce both.

Friedman and Savage proposed a utility curve with a convex segment in the middle, which fits and feels like a patch. Prospect theory, which the next chapter is about, handles the combination naturally: people overweight small probabilities, and both the jackpot and the house fire are small probabilities.

When a theory needs a specially shaped curve to accommodate the most common financial behaviour in the population, that is a signal about the theory rather than about the population.