Choices No Utility Function Represents

Declining a favourable gamble. Offered $\mathrm{1,000,000}$ certain against a prospect worth $\mathrm{1,390,000}$, most people take the certainty. This violates nothing: it requires only , which any sufficiently concave utility function supplies. Risk aversion is a taste the theory represents, not an anomaly.

Preferring a cycle. Someone who prefers $P$ to $Q$, $Q$ to $R$ and $R$ to $P$ violates transitivity, and no utility function represents that either, but for a different reason, and one most people accept as a genuine error, since the cycle can be exploited by an intermediary who trades around it repeatedly. The Allais pattern has no such exploit, which is part of why it is contested.

The Allais pair. Both preferences are individually unremarkable and jointly unrepresentable, with the two demands meeting at exactly $10/11$. Completeness, transitivity and continuity all survive; independence does not.

The same prospect described two ways. A treatment presented as saving 200 of 600 lives, against one presented as leaving 400 of 600 to die, produces different choices though the outcomes are identical. Expected utility is defined over outcomes and cannot distinguish the two descriptions, so this is a violation of a different kind: the theory does not merely predict the wrong preference, it has no place for the distinction that drives it. A reference-dependent value function does.

Buying insurance and a lottery ticket at once. Paying a premium above expected loss indicates risk aversion; buying a ticket worth less than its price indicates risk seeking. One concave utility function cannot do both, but a weighting function that overweights small probabilities can, since $w(0.01)=0.0673$ against a true $0.01$ makes an unlikely jackpot loom larger than it is while an unlikely catastrophe does the same.

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The first is permitted, the second is a violation nearly everyone regards as an error, the third is a violation many defend, the fourth is outside what the theory can express at all, and the fifth is a pattern that one model forbids and another explains. "Violates expected utility" covers all four of the latter and means something different in each.

A single choice against a pattern.

No single choice can contradict expected utility, because a utility function can always be built to accommodate it. Only a pattern can.

Problem 1 against problem 2.

They differ by exactly one substitution: a common $0.89$ chance of $1\text{M}$ becomes a common $0.89$ chance of nothing. Strip the common part from both options in each problem and the identical residual remains, $1\text{M}$ at $0.11$ against $5\text{M}$ at $0.10$. Independence therefore requires the same answer twice.

Independence against transitivity.

Both make preferences unrepresentable. The first is almost universally regarded as an error because it can be turned into a money pump; the second is defended by some on the grounds that certainty is a legitimate thing to value. The distinction matters when deciding whether to correct behaviour or model it.

Probability against decision weight.

A decision-maker may know the probability exactly and still act on the weight. That is why the distortion is not a belief error and is not removed by better information, and why the fixed point sits at $1/e$ rather than at $0.5$, which a symmetric distortion would give.

A descriptive model against a normative one.

Prospect theory predicts the Allais pattern; expected utility says the pattern cannot be represented. Neither settles whether somebody choosing that way has erred. That question needs an argument about whether independence ought to govern, and the arithmetic is silent on it.

That choosing in a way expected utility cannot represent shows the chooser is irrational or has made a mistake. What a violation establishes is narrower: that no single utility function reproduces the whole pattern, so the theory's axioms do not hold for that decision-maker. Whether the pattern is an error depends on an argument that the axioms ought to govern, and the independence axiom in particular has been disputed on its own terms since Allais proposed the example. A person who prefers certainty in the first problem and the larger gamble in the second may be reflecting something real about how a guaranteed outcome differs from a near-guaranteed one. Prospect theory was built to describe such choices, not to correct them, and treating a descriptive model's predictions as diagnoses of failure confuses the two roles.

That declining a gamble whose expected value exceeds the certain alternative is evidence against expected utility theory. It is not evidence about the theory at all. Taking $\mathrm{1,000,000}$ with certainty over a prospect worth $\mathrm{1,390,000}$ in expectation is represented by any concave utility function, and concavity is exactly what risk aversion means in this framework. Expected utility constrains the pattern of choices across problems rather than any single choice, so a violation must be demonstrated by showing that two or more preferences impose contradictory requirements on the same utility function. In the Allais pair this is what happens: one preference requires and the other requires , and no function satisfies both.