Practice: Choices No Utility Function Represents

1. Finding 1. The gamble is worth $0.5(250)+0.5(0)=\pounds 125$, against a certain £$100$. This establishes nothing about expected utility theory. Taking the certainty requires , which any sufficiently concave utility function satisfies, that is what risk aversion means in this framework. A theory that accommodates every risk attitude is not embarrassed by one, and a single choice can never contradict a representation theorem, because a function can always be built to reproduce it. The $71\mathrm{\%}$ figure tells the insurer something commercially useful about demand for certainty. It is not evidence of irrationality. 2. Finding 2. This is the Allais pair. Normalise $u(0)=0$ and $u(50\text{k})=1$, and let $u=u(10\text{k})$. The first preference requires

The second requires

The two thresholds are the same number approached from opposite sides, so no utility function whatever represents both preferences. Unlike finding 1, this is a genuine violation, and it is exact rather than statistical: it would hold if a single customer gave both answers. The axiom is independence. Both options in the first problem contain a common $0.89$ chance of £$\mathrm{10,000}$; both in the second contain a common $0.89$ chance of nothing. Independence says a common component cancels, so stripping it leaves the identical residual comparison, £$\mathrm{10,000}$ at $0.11$ against £$\mathrm{50,000}$ at $0.10$, in both problems, which must therefore be answered alike. Completeness, transitivity and continuity are all intact. 3. Finding 3. No single concave utility function produces both. Paying above expected loss for cover indicates risk aversion; paying above expected value for a lottery ticket indicates risk seeking, and one function cannot be concave and convex over the same range. A non-linear probability weighting function produces both at once. Under the Prelec form with parameter $0.65$, the $1$-in-$1000$ fire probability of $0.001$ enters with weight $0.0298$, almost thirty times its true value, and a probability of $0.01$ enters with $0.0673$. Small chances are overweighted whether the outcome is a jackpot or a fire, while moderate and large probabilities are underweighted, with the crossover at exactly $1/e\approx 0.367879$. Overweighting the small chance of catastrophe drives the insurance purchase and overweighting the small chance of a jackpot drives the ticket, from one mechanism. This is the insurer's most commercially relevant finding, and it is a description of demand rather than a diagnosis of error. 4. Finding 4. Irrelevant to the claim. Expected utility is a representation theorem: if preferences satisfy the axioms, then a function exists whose maximisation reproduces the choices. It makes no assertion about mental process, so evidence that customers do not compute utilities bears on a claim the theory never made. Discovering that nobody performs the calculation is as uninformative as discovering that planets do not solve differential equations. 5. The conclusion. What the evidence supports: customers value certainty highly (findings 1 and 2), their choices in finding 2 cannot be represented by any utility function, and their treatment of small probabilities is non-linear in a way one weighting function captures (finding 3). These are usable descriptions of demand, and finding 3 in particular has direct product implications. What it does not support: the word "irrationally". Finding 2 establishes that the axioms fail, which is a technical result. Calling the failure irrational requires arguing that independence ought to govern choice, and that has been disputed since Allais constructed the example precisely to dispute it. There is no money-pump argument here of the kind that makes a preference cycle indefensible; nobody can be systematically exploited for preferring certainty. To justify the term the paper would need either a demonstration that customers can be exploited by their pattern, or a normative argument for independence that customers would accept on reflection. It has neither, and it does not need either: the commercial case rests on describing demand accurately, which findings 2 and 3 do. I would delete "irrationally" and keep the findings, which are stronger without it. A final caution on product design: modelling a pattern is not licence to exploit it. Designing around overweighted small probabilities means charging more for cover than the risk warrants because customers overvalue it, and whether that is legitimate is a question about the firm rather than about decision theory.