The Certainty — The Bias, Plate i

Every volume of this table has leaned, somewhere, on a single assumption: that a person confronted with a choice ranks the options by what they are worth to him and takes the best, and that when the options are gambles he ranks them by their EXPECTED UTILITY — each prize weighted by its probability, the weighted sum compared. The assumption is powerful, and for most of a century it was treated as very nearly a law of thought, given its definitive form by John von Neumann and Oskar Morgenstern, who showed that anyone whose choices obeyed a few reasonable-sounding axioms must behave exactly as though maximizing such an expectation. This volume takes up what happens at the edges of that assumption — the systematic, repeatable ways that real choosers part company with the model — and it opens where the first rigorous crack appeared, in a deceptively simple pair of questions posed by the French economist Maurice Allais. His demonstration did not show that people are foolish or that they cannot calculate. It showed something more unsettling: that they knowingly, stubbornly, and in unison violate the most innocent-looking of the axioms, because a certain thing is worth more to them than its mere probability can account for.

The axiom in question is the one that makes expected utility tick, called independence, or, in a homelier phrase, the sure-thing principle. It says this: if two gambles contain a common component — some identical outcome that occurs with the same probability under both — then that shared part cannot rightly affect which of the two you prefer. It is, in effect, a rule of cancellation. Suppose two lottery tickets both pay a thousand dollars if a certain coin comes up heads, and differ only in what they pay on tails; then in choosing between the tickets you should ignore the heads outcome entirely, since it is the same either way, and decide wholly on the tails outcome where they differ. The principle has enormous intuitive appeal — what is identical on both sides should wash out of the comparison, exactly as a common term cancels from both sides of an equation — and it is precisely this appeal that Allais set out to test, by constructing a case where the cancellation our reason endorses runs headlong into the choices our gut insists upon.¹

1. The axiom that makes expected utility tick is independence, or the sure-thing principle: if two gambles contain a common component — an identical outcome occurring with the same probability under both — that shared part cannot affect which you prefer. A rule of cancellation. If two tickets both pay a thousand dollars on heads and differ only on tails, choose on the tails outcome alone, since heads is the same either way. The appeal is exactly that of cancelling a common term from both sides of an equation. Allais set out to test it where the cancellation our reason endorses collides with the choice our gut insists upon.

Here is his case, in round numbers. First, choose between two prospects. The first is a certain one million dollars, yours for sure. The second is a gamble: an eighty-nine-percent chance of that same million, a ten-percent chance of five million, and one slim chance in a hundred of walking away with nothing at all. Faced with this, the great majority of people take the sure million; the one-in-a-hundred risk of ending empty-handed, when a certain fortune was right there for the taking, is more than they can stomach. Now a second choice, between two further prospects. The first offers an eleven-percent chance of one million and otherwise nothing; the second offers a ten-percent chance of five million and otherwise nothing. Here the great majority go the other way and take the second — for both are long shots, the odds of winning barely differ, eleven against ten, and if you are probably getting nothing either way you may as well chase the larger prize. Each choice, taken alone, feels not just reasonable but obvious. Taken together, they are impossible — there is no way to assign worth to the sums of money that makes both of them the act of a single expected-utility maximizer.²

2. Allais's case, in round numbers. First: a certain million, or a gamble of eighty-nine percent for that million, ten percent for five million, one percent for nothing. Most take the sure million — the one-in-a-hundred chance of ending empty-handed, with a fortune there for the taking, is unbearable. Second: an eleven-percent chance of a million, or a ten-percent chance of five million, otherwise nothing either way. Most take the second — both are long shots, the odds barely differ, so chase the larger prize. Each choice alone feels obvious; together they are impossible.

The impossibility is the whole point, and it is the sure-thing principle that exposes it. Look again: the two experiments are built from the same parts. In each, there is an eighty-nine-percent slice that is common to both options on offer — in the first experiment that slice pays a million either way, in the second it pays nothing either way — and the sure-thing principle says that common slice must cancel, leaving the real choice to the remaining eleven percent. But once you strip away the common slice, the leftover choice is identical in the two experiments: in both, you are deciding between a plain shot at a million and a slightly longer shot at five million with a sliver of risk of nothing. The principle therefore demands that you choose the same way in both. Yet people flip. They guard the certain million in the first experiment and chase the five million in the second, choosing opposite sides of what is, once the common part is removed, one and the same gamble. No utilities for nothing, a million, and five million can be written down that honor both choices at once; the choices contradict each other as squarely as a set of numbers that claims one quantity is both larger and smaller than another.³

3. The two experiments are built from the same parts. Each has an eighty-nine-percent slice common to both its options — paying a million either way in the first, nothing either way in the second — and the sure-thing principle says that slice must cancel, leaving the choice to the remaining eleven percent. Stripped of the common slice, the leftover choice is identical across the two experiments: a plain shot at a million against a longer shot at five million with a sliver of risk. The principle demands the same choice in both. Yet people flip — guarding the million in the first, chasing the five in the second. No utilities can honor both at once.

⁂

What drives the flip is a force the model has no room for, which Allais's successors would name the CERTAINTY EFFECT: an outcome that is certain commands a premium, a weight in the choosing far beyond what its probability, treated arithmetically, would grant it. In the first experiment, the sure million is exactly that — certain — and to trade it for the gamble is to surrender certainty itself, accepting a one-in-a-hundred chance of the bitterest outcome imaginable, ending with nothing when a fortune had been guaranteed. The mind recoils not at the one percent as a number but at what it threatens: the loss of a sure thing, and the sharp regret that would follow. In the second experiment there is no certainty anywhere to be had; both prospects are overwhelmingly likely to pay nothing, and the gap between an eighty-nine and a ninety-percent chance of disappointment is felt as nothing at all, so the choice falls, sensibly enough, to the size of the prize. The same one-percent difference in the odds of nothing is decisive in the first case and negligible in the second — not because the arithmetic changes, but because in the first case it stands between the chooser and a certainty, and certainty, to the living mind, is a thing apart. People do not weight probabilities the way the theory says; they treat the move from merely-probable to certain as worth more than an equal move between two improbabilities.⁴

4. The force the model has no room for is the CERTAINTY EFFECT: a certain outcome commands a premium far beyond its arithmetical probability. In the first experiment the sure million is certain, and trading it for the gamble surrenders certainty itself, risking the bitterest outcome — nothing, when a fortune was guaranteed. The mind recoils not at the one percent as a number but at the loss of a sure thing and the regret to follow. In the second, no certainty is available, and the gap between eighty-nine and ninety percent disappointment is felt as nothing. The same one percent is decisive in the first case, negligible in the second — because in the first it stands between the chooser and a certainty.

The force of Allais's finding lay in three features that together made it impossible to wave away. It was SYSTEMATIC: the violations did not scatter randomly, as honest mistakes would, but tilted overwhelmingly in one direction, the same direction for nearly everyone. It was PERSISTENT: posed again and again, in laboratories and lecture halls across the decades, the pattern came back unchanged. And — most tellingly — it was ROBUST TO REFLECTION. When the contradiction was laid bare for people, when they were walked through the cancellation and shown that their two choices could not both be right, many of them looked it over and declined to change their minds, insisting that the certain million was genuinely, defensibly worth more to them than the gamble, whatever the axiom said. There is a famous report that Leonard Savage, who had done as much as anyone to axiomatize the theory, was given the Allais questions, chose in the forbidden pattern himself, and only afterward, on reflection, revised his answers to bring them back into line with the rule he had helped to build. The episode cuts both ways, and Allais and Savage drew opposite lessons from it: to Savage, his first answers were simply an error, a slip his considered judgment corrected; to Allais, they were data — evidence that the theory, whatever its merits as a counsel of prudence, was false as a description of how human beings actually choose.⁵

5. Three features made the finding impossible to wave away. SYSTEMATIC: the violations tilted overwhelmingly one way, not scattering as honest mistakes would. PERSISTENT: posed for decades, the pattern returned unchanged. ROBUST TO REFLECTION: shown the contradiction, many declined to change, insisting the certain million was genuinely worth more. Famously, Leonard Savage, who had helped axiomatize the theory, chose in the forbidden pattern himself, then revised on reflection. Allais and Savage drew opposite lessons — to Savage a corrected error, to Allais data that the theory was false as description.

That disagreement is the seam this whole volume works. Behind it lies a distinction Allais pressed hard and that everything to come depends on: the difference between a NORMATIVE theory, which tells us how an ideally rational agent ought to choose, and a DESCRIPTIVE one, which tells us how people in fact do. Expected utility might be impeccable as the first and hopeless as the second; the sure-thing principle might be a fine rule to live by and a poor prediction of how anyone lives. Allais insisted on taking the descriptive failure seriously rather than explaining it away as a remediable lapse, and in doing so he opened the door through which a half-century of work would pour — a body of findings, gathered in the plates that follow, mapping the regular shape of the departures: how losses are felt more keenly than gains, how the mind reaches for fast shortcuts that misfire in knowable ways, how the present swells against the future. But the deepest question is already visible in the quarrel between Allais and Savage, and the volume will not be able to dodge it. When a person reliably chooses against the rule of reason, where exactly is the fault — in the person, who has erred, or in the rule, which has mistaken a model of choosing for the thing itself? That question waits at the end. The next plate builds the description that Allais demanded.