The Prospect

The same hand signs the insurance proposal and buys the lottery ticket. In one transaction it pays to remove a chance of loss; in the other it pays to acquire a chance of gain. Calling its owner cautious explains the first act and makes the second awkward. Calling him reckless does the reverse. The difficulty is not that a person must either accept every gamble or reject every gamble: stakes, resources, and purposes differ. It is that a single, smooth attitude toward final wealth may miss the way the choices are experienced. The threatened house and the imagined jackpot occupy different sides of a reference point, and a small probability may have a weight in the decision that is quite unlike its weight in an actuarial calculation. Prospect theory makes those features part of the account. The premium then depends on the framing of the prospect as well as the distribution of the money it pays.

Daniel Kahneman and Amos Tversky began with gains and losses measured from a reference point. Consider two people who each end a transaction with one thousand. One began with eight hundred; the other expected to retain twelve hundred. Their final balances are equal, but the changes they register can differ: one has gained two hundred, while the other has lost two hundred against his expectation. The reference point gives a direction to the movement, not an additional entry in the bank account. It may be a present position or an expectation, and identifying it is part of explaining the choice.¹ This matters for an insurance offer because the same payment can be experienced as the cost of securing an ordinary life or as a fresh loss imposed to remove a hazard that has not occurred. A renewal and a first purchase need not begin from the same psychological starting place. Those possibilities do not let the analyst choose a convenient reference point after seeing the answer. They make it necessary to establish how the buyer represents the transaction if the representation is to explain anything.

Loss aversion describes a further asymmetry: a loss can count more heavily than an equal gain from that reference point. To isolate it, suppose a chooser values gains dollar for dollar and losses at twice their size, with no probability distortion. An equal chance of gaining one hundred and fifty or losing one hundred has a monetary expectation of twenty-five. Under this deliberately simple valuation its score is minus twenty-five: half of one hundred and fifty, less half of two hundred. Raising the possible gain to two hundred makes the score zero. The factor of two is an assumption for the example, not a constant of human nature.² Nor is refusal of an equal-stakes, even-chance gamble by itself evidence against expected utility: a concave utility of final wealth can also reject a gamble whose monetary expectation is zero. The distinctive inquiry concerns the role of the reference point and the asymmetry around it. One rejected bet cannot identify the function, the beliefs, and the person's other circumstances all at once. The example shows what loss aversion can do when those other elements are held fixed.

The probability weight supplies another mechanism, one that can favor insurance and a long shot at the same time. Keep the simple valuation and suppose that a one-per-cent chance receives a decision weight of four per cent. A prospect paying one thousand with that chance and zero otherwise then scores forty, above the score of its ten-dollar monetary expectation received for certain. Its reflected prospect, losing one thousand with the same chance, scores minus eighty; a certain loss of ten scores minus twenty. The chooser prefers the chance of the large gain and the certainty of the small loss. These are comparisons between prospects, with the numbers selected to display the mechanism; a purchased ticket also requires an account of how its price enters the choice. Nothing in the example says that the chooser believes the event happens four times in a hundred. A decision weight describes the event's influence on valuation. It can differ from the probability the chooser accepts as true, and it does not change the insurer's expected claim cost from ten to forty.³ Belief, psychological weight, and the frequency used to fund a pool are different quantities.

The full theory also lets sensitivity diminish as outcomes move farther from the reference point. Its familiar value curve bends downward among gains and upward among losses. That curvature is distinct from the greater steepness on the loss side: a person can have a kink between gains and losses without the curved limbs, as the preceding examples did. Together with probability weighting, these features produce a fourfold pattern. At moderate or high probabilities, the usual pattern favors a sure gain over a risky gain and a risky loss over a sure loss; at small probabilities it favors the long-shot gain and protection against the large loss. These are conditional tendencies, not instructions that every choice must reverse whenever its signs reverse. The cumulative version weights the ranked distribution, separately for gains and losses, rather than multiplying every outcome by an independently distorted probability.⁴ The machinery gives the same person several responses to risk without requiring several different personalities. What changes across the choices is the position of the outcomes and probabilities within the valuation.

That explanation has to coexist with the limits of attention. A rare event can be overweighted once it is represented in a decision, yet an event omitted from the representation has no such influence. The original prospect-theory paper recognized both overweighting and neglect of very small probabilities, and expressly stopped short of a complete theory of insurance demand.⁵ The absence of flood cover does not, by itself, identify how a household weights probabilities; the absence can have several explanations. The household may not have considered it, may misunderstand the terms, or may be unable to meet the payment. Conversely, buying cover against a loss that would destroy the household's means can be supported by ordinary risk aversion without any distorted weighting. The analytical task is to separate these cases. Otherwise the language of bias becomes a label attached to whichever purchase the observer dislikes, and the theory loses the very distinctions that made it useful.

The answer to the plate's opening question is therefore a mechanism, not a diagnosis of every person holding both a policy and a ticket. Gains and losses can be valued from different starting points; losses can weigh more than corresponding gains; and small probabilities can acquire an influence out of proportion to their numerical size. These features can make the two purchases intelligible within one account of choice. Whether the resulting terms serve the buyer is a further question, requiring the actual consequences and the buyer's purposes. A seller's ability to obtain a high payment does not establish that the protection is adequate, any more than identifying a frame establishes that the buyer should have gone uninsured. This brings the volume from the prospect as experienced to the loss as it may arrive. The Tail considers events whose severity can dominate the reckoning even when observations of them are scarce. The weight given to a catastrophe in the mind and the resources required when it occurs need not coincide. A premium must pass through both accounts.