The applicant need not answer the insurer’s question. He can answer the insurer’s offer. Put two policies before him, one dear and generous, the other cheap and thin, and the private knowledge he would not disclose may govern which he takes. The uninformed party has changed the means of inquiry: instead of asking for a statement, it has arranged a choice. This is the screen. Where the signal gives the informed a way to distinguish themselves, the screen gives the other side a way to make distinction worth their while. The customer is offered a choice, and the choice speaks for him. But it speaks only if the terms have been carefully composed, and only about the differences those terms actually sort.
The menu must survive being read by the person it is meant to exclude. A contract marked “low risk” is available to the high risk too, unless some other means of identification has already solved the problem. The insurer must therefore compare each applicant’s satisfaction under every offer. The low risk must prefer the contract intended for him to the high-risk contract and to going uninsured; the high risk must prefer his own to the cheaper alternative and to staying out. These are separate requirements. A policy can attract the people it was designed for and also attract people whose claims make it unprofitable. Self-selection requires more than a plausible story about the intended customer. It requires the unintended customer to find another choice better.
The insurance example can be made exact without pretending that its numbers describe a market. Imagine wealth of one hundred, a possible loss of sixty-four, and two applicants whose loss probabilities are one-tenth and four-tenths. Each values wealth by its square root and chooses the policy with the highest expected value on that scale. The probabilities are fixed: the higher risk need not be less careful, less honest, or less deserving. Offer complete cover for a premium of twenty-five point six, and cover of only eight for a premium of point eight. Under the first policy either applicant is left with seventy-four point four in either state. Under the second he keeps ninety-nine point two if nothing happens and forty-three point two if the loss occurs. The cheaper policy offers a very small shelter against a large remaining risk.
These two offers do sort the stipulated applicants. The low risk obtains expected utility of about 9.621 from the thin policy, against about 8.626 from the full one and 9.600 from going uninsured. The high risk obtains about 8.605 from the thin policy, against 8.626 from the full one and 8.400 from going uninsured. Each strictly prefers the intended policy. Each policy also breaks even on the type that chooses it: expected claims are point eight on the small indemnity and twenty-five point six on the large. The calculation establishes a separating, individually acceptable pair. It does not establish that this is the best pair, that a competing insurer cannot defeat it, or that the applicants’ probabilities could be inferred so neatly outside the assumptions.
The revealing restriction is also a real deprivation. If the low risk could be recognized directly, full cover at his own fair premium would cost six point four and leave wealth of ninety-three point six in both states. He would prefer that to the thin policy. But the high risk would prefer it too. The cheap complete policy cannot be reserved for its intended customer merely by writing his type upon it. To keep the types apart, the menu gives the low risk a reason to accept exposure that a fully informed insurer could remove. This is the price of separation. It is paid in a worse distribution of wealth across possible futures, even though no extra invoice names it.
A menu that sorts its customers has not yet settled the market around it. Rothschild and Stiglitz asked whether a set of contracts could survive entry when each firm took the other firms’ offers as given. In their basic model, a pooling contract cannot rest undisturbed: a new offer can attract the low risks on terms that would lose their attraction for the high risks. Yet the sole separating candidate can also be vulnerable. When the population contains sufficiently few high risks, a pooling offer can provide enough additional protection to attract both groups and still cover its expected claims. The result is possible nonexistence of equilibrium under that definition. It is a failure of a proposed resting point, not a proof that every real insurance market must churn forever, or a description of the path by which it would do so.
The rules around the menu matter as much as the comparisons within it. The small policy in our example is meant to be bought once. If an applicant could secretly assemble eight copies at the same terms, he would obtain full cover for six point four; the screen would disappear. Nor is a market in single contracts necessarily equivalent to one in which a firm can commit to a package, subsidizing one policy from another. Rothschild and Stiglitz explicitly reconsidered that distinction. Allowing bundles and cross-subsidies changes the possible deviations and the welfare comparisons. What looks like an innocent detail of market organization can determine whether the inference the menu invites remains valid.
The same inquiry reaches beyond insurance without making every difference in a price list a revelation of character. A flexible fare and a restricted fare may sort travelers by their willingness to pay for flexibility; they do not identify the purpose of every journey. A loan menu may trade a lower rate against a larger pledge of assets; the customer who declines the pledge may lack assets rather than confidence in the project. Even within insurance, a deductible can reflect liquidity, dislike of risk, beliefs about loss, or understanding of the terms. When those dimensions vary together, one observed choice will not generally identify one hidden probability. The screen has elicited a preference among its offers. The designer still needs a defensible model to translate that preference into knowledge of the person.
The choice reveals what the menu makes worth revealing, and the cost of that revelation can remain with the person who has least to hide.
It matters, finally, who wants the information and what the terms are designed to accomplish. A competitive insurer attempting to cover expected costs faces a different objective from a monopolist arranging versions to extract a larger payment. Both may employ self-selection. The presence of a screen alone does not establish which arrangement benefits the buyer, whether separation is socially desirable, or whether the costs imposed upon one group serve a purpose worth their burden. The question is not answered by calling the options voluntary. A person chooses among the contracts made available to him, and the making of those contracts is itself an exercise of knowledge and power. To understand the verdict of the choice, one must inspect its alternatives.
The uninformed can sometimes design terms under which private knowledge finds its way into public action. That is the achievement of the screen: an answer extracted through incentives where an unsupported assertion would not suffice. Its limit is that the answer costs something, depends on the alternatives, and is only as specific as the structure that interprets it. The low premium is no longer simply a price for a known risk; it belongs to an arrangement that both discovers and allocates risk. And the inquiry changes again when the applicant can alter the probability after choosing. The screen asks what sort of person has entered. The next plate asks what the bargain gives that person reason to do.