Screening Theory

Screening Theory in Health Insurance and Payment Design

Screening theory studies how an insurer or payer that cannot see a person's risk, or a provider's costs, can learn them by offering several contracts and seeing which one each party picks. In health economics it explains why insurers and employers offer plans that trade a lower premium for a higher deductible, and why a choice of payment terms can sort hospitals or suppliers by their costs. The theory also shows that sorting has a price, since the contracts that keep high risks away leave lower risks with less cover than they would buy if risk were observable. In competitive insurance markets with hidden risk there may be no stable set of contracts at all, which bears on the case for compulsory pooled cover. Screening is the uninformed side's response to adverse selection, whereas in signalling theory the informed side moves first.

Self-selection as the uninformed party's tool

The scientific background to the 2001 Nobel prize in economics, shared by Akerlof, Spence and Stiglitz, credits Stiglitz with showing that poorly informed agents can extract information from better-informed ones by offering a menu of contracts, which it calls screening through self-selection. Its example is an insurer that sorts clients into risk classes by letting them exchange a lower premium for a higher deductible. The background links this to the Vickrey and Mirrlees analyses of optimal income taxation, where a tax authority that cannot observe productivity gives earners incentives to choose the intended work effort.

The background also notes that Stiglitz used the word screening in 1975 for what is now called signalling.

Risk types, contracts and the single-crossing condition

In the textbook insurance setting people differ only in their chance of a loss of fixed size. In the notation of the Nobel background, a contract is a premium and a payment received if the loss occurs, valued by expected utility.

$$V_i(a, b) = (1 - p_i)\, u(y - a) + p_i\, u(y - a - d + b)$$

where $V_i$ is the expected utility of a person of risk type $i$, $p_i$ is that type's probability of the loss, $y$ is income, $d$ is the size of the loss, $a$ is the premium, $b$ is the payment if the loss occurs, $u$ is a concave utility of income and $d - b$ is the deductible.

What makes sorting possible is that the two types value extra cover differently at every contract. Along an indifference curve, the extra premium a person will pay for one more pound of cover is as follows.

$$\frac{da}{db}\bigg|_{V_i} = \frac{p_i\, u'(y - a - d + b)}{(1 - p_i)\, u'(y - a) + p_i\, u'(y - a - d + b)}$$

where $u'$ is marginal utility in each state and the left side is the slope of type $i$'s indifference curve in premium and cover space. The slope rises with $p_i$, so a high risk's indifference curve is steeper than a low risk's at any contract and the two cross only once. The Nobel background traces conditions of this kind to the single-crossing condition of Mirrlees, often called the Mirrlees-Spence condition.

The design problem with participation and self-selection constraints

An insurer with market power chooses a menu to maximise expected profit. The menu must meet two sets of constraints, written here for two risk types.

$$\max_{(a_H, b_H),\,(a_L, b_L)} \; \sum_{i \in \{H, L\}} \lambda_i\, (a_i - p_i b_i) \quad \text{subject to} \quad V_i(a_i, b_i) \ge V_i(0, 0), \qquad V_i(a_i, b_i) \ge V_i(a_j, b_j)$$

where $H$ and $L$ are the high and low risk types, $\lambda_i$ is the share of type $i$ in the population, $a_i - p_i b_i$ is the expected profit on the contract meant for type $i$, $V_i(0, 0)$ is that type's expected utility without insurance, and $j$ is the other type. The first constraint is participation, so each type buys at all. The second is self-selection, or incentive compatibility, so each type prefers its own contract.

The 2014 Nobel background on Tirole sets out the general lessons of such problems. Types that could gain by mimicking another type earn information rents, surpluses the principal concedes to keep them truthful, and the principal accepts distorted contracts for other types in order to reduce those rents. The contract of the most able or most efficient type is left undistorted, which the background calls no distortion at the top. In the insurance version the undistorted contract is full cover for high risks, who keep the rent, while low risks receive partial cover. Stiglitz analysed the monopoly insurer's version of the problem in 1977.

Competitive screening and the Rothschild-Stiglitz result

Rothschild and Stiglitz asked what competing insurers can do when none can observe a customer's risk. In the Nobel summary the model has no pooling equilibrium, because a rival could profitably cream-skim with a contract better for low risks and worse for high risks. The only possible equilibrium separates the types. High risks buy full cover at a high premium and low risks buy partial cover at a lower premium, with a deductible that only just deters high risks from taking the cheaper contract.

$$a_H = p_H d, \qquad b_H = d, \qquad a_L = p_L b_L, \qquad V_H(a_H, b_H) = V_H(a_L, b_L)$$

where each contract breaks even at its own type's risk, the high risks are fully insured, and the last condition sets the low-risk cover $b_L$ at the largest level that leaves high risks no better off with it.

Rothschild and Stiglitz also found conditions under which no equilibrium in pure strategies exists. Stiglitz, Yun and Kosenko explain why. Separation leaves low risks with too little insurance compared with full information, and when that distortion is too large there is always a pooling contract that both types prefer, so the separating pair cannot be sustained and no competitive equilibrium exists. The result depends on exclusivity, meaning that an insurer can see and limit the other cover a customer buys. If undisclosed top-up contracts can be sold, they show, high risks would buy the low-risk contract plus secret extra cover and the standard contracts would no longer separate.

Worked example with two risk groups and a deductible menu

All figures in this example are illustrative. Each person earns GBP 40,000 and may face a treatment cost of GBP 20,000 in the year. High risks have a 30% chance of the cost and low risks a 10% chance. Utility is the natural logarithm of income, and insurers compete and break even.

1. High-risk contract. Full cover at the high-risk fair premium costs 0.30 × 20,000 = GBP 6,000, leaving GBP 34,000 in both states. Its expected utility is ln 34,000 = 10.43412.

2. Low-risk contract. A low-risk contract with cover $b$ costs 0.10 × $b$. The cover is raised until a high risk is just indifferent, which here gives cover of GBP 4,292 at a premium of GBP 429, a deductible of GBP 15,708. A high risk who took it would have GBP 39,571 if well and GBP 23,863 if ill, with logarithms 10.58585 and 10.08008. Weighting gives 0.70 × 10.58585 = 7.41009 and 0.30 × 10.08008 = 3.02402, which sum to 10.43412, the same as full cover, so the deductible only just keeps high risks away.

3. What separation costs low risks. For a low risk the same contract gives 0.90 × 10.58585 = 9.52726 and 0.10 × 10.08008 = 1.00801, which sum to 10.53527, a certainty equivalent of GBP 37,619. Uninsured, a low risk has 0.90 × 10.59663 + 0.10 × 9.90349 = 10.52732, or GBP 37,321. If insurers could observe risk, a low risk would buy full cover for 0.10 × 20,000 = GBP 2,000 and keep GBP 38,000. The menu is worth 37,619 minus 37,321 = GBP 298 to each low risk, but the hidden information costs each one 38,000 minus 37,619 = GBP 381. High risks are no worse off than under full information.

4. Testing whether the menu survives. Suppose 80% of buyers are low risk. A contract priced at the average risk of 0.8 × 0.10 + 0.2 × 0.30 = 0.14 would attract both types. The best such contract for a low risk has cover of GBP 7,641 at a premium of GBP 1,070, worth GBP 37,471 as a certainty equivalent. That is below GBP 37,619, so low risks stay with their own contract and the separating menu stands.

5. When low risks are the large majority. With 90% low risks the average risk is 0.9 × 0.10 + 0.1 × 0.30 = 0.12. The best pooled contract for a low risk now has cover of GBP 12,879 at a premium of GBP 1,545, leaving GBP 38,455 if well and GBP 31,333 if ill. Its value is 0.90 × 10.55723 + 0.10 × 10.35244 = 10.53675, or GBP 37,675, above GBP 37,619. A rival could offer a contract close to it and draw away both types, breaking the separating menu, yet no pooled contract can survive either, so no equilibrium exists. Here the change comes once low risks exceed about 88% of buyers.

Screening through health plan menus and benefit design

Employer plan menus are the most familiar screening devices in health care, because buyers choose among plans that differ in premium and deductible. How well the menu sorts depends on how people choose. Handel studied a large firm whose plan menu changed, found substantial inertia in employees' choices, and estimated that policies nudging them towards better choices would have made adverse selection substantially worse, reducing welfare by enough to double the existing welfare loss from adverse selection.

Insurers can also screen through the services they offer rather than through prices. Frank, Glazer and McGuire show that health plans paid by capitation have an incentive to distort the quality of services so as to attract profitable enrolees and deter unprofitable ones, and model this rationing as a shadow price on access to each area of care. Their index, which draws partly on people's forecasts of their own costs and on the risk adjustment used for payment, identifies the services most exposed to such distortion. Here screening meets cream skimming, and risk adjustment of plan payments is the usual response.

Menus of payment contracts for providers

The same logic applies when a provider knows its own costs and the payer does not. The 2014 Nobel background describes Laffont and Tirole's optimal procurement mechanism as a menu of contracts, each combining a lump-sum payment with a share of any cost overrun, chosen by the firm according to its expected cost. The most efficient firm receives a fixed-price contract, every other type has part of its costs reimbursed and so makes less than first-best effort to cut costs, and all types except the least efficient earn information rents.

The background also reports Rogerson's result that a menu of only a cost-reimbursement contract and a fixed-price contract captures at least three quarters of the surplus of the optimal mechanism, under quadratic utility and uniformly distributed types. In health care this casts a choice between cost-based and fixed hospital payment as a screening problem, in which a payer weighs stronger cost incentives against rents to low-cost providers. Payment rules themselves are covered under contract theory and provider payment.

What screening implies for evaluation and policy

The Nobel background notes two difficulties in testing for asymmetric information. Adverse selection is hard to tell apart from moral hazard, and screening and signalling partly remove the effects of the asymmetry. A study that finds no link between cover and claims, as Chiappori and Salanié reported for car insurance, has therefore not shown that private information is absent.

In an economic evaluation of cost sharing or benefit design, a new deductible or plan option changes who buys which plan as well as how much care each person uses, so holding the mix of enrolees fixed misstates costs and welfare. On welfare, Stiglitz, Yun and Kosenko state that the Rothschild-Stiglitz equilibrium, when it exists, is Pareto efficient given exclusivity, and that when it does not exist the Pareto efficient contracts involve a subsidy paid by low risks to high risks. Compulsory pooled schemes such as social health insurance achieve that kind of subsidy directly through risk pooling.

Screening beside signalling, mechanism design and adverse selection

In signalling the informed party takes a costly action to reveal its type, while in screening the uninformed party offers the options. Mechanism design is the general theory of choosing rules when participants hold private information, and a screening menu is a mechanism with one principal. Adverse selection is the problem that screening addresses, information asymmetry the condition behind it, and moral hazard a different problem of hidden action after the contract is signed. The term is unrelated to screening for disease, whose evaluation is covered under screening economics.

Limits and common misreadings

Sorting by plan is often taken as proof of a working screen, but plan choice also reflects income and habit, and Handel's results show that choice frictions shape the outcome. Separation is often presented as costless, yet in the competitive model all of its cost falls on the low risks through under-insurance. The standard results also rest on exclusivity and on the equilibrium concept used. The Nobel background notes that Wilson's looser definition of equilibrium makes existence more likely, and Stiglitz, Yun and Kosenko cite work showing the results are sensitive to the game-theoretic formulation. Two-type models also leave out the many risk levels and the regulation that real menus face.

Sources

• Frank RG, Glazer J, McGuire TG. Measuring adverse selection in managed health care. Journal of Health Economics. 2000;19(6):829-854. https://doi.org/10.1016/S0167-6296(00)00059-X
• Handel BR. Adverse selection and inertia in health insurance markets: when nudging hurts. American Economic Review. 2013;103(7):2643-2682. https://doi.org/10.1257/aer.103.7.2643
• Royal Swedish Academy of Sciences. Markets with asymmetric information. Scientific background on the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel 2001. 10 October 2001. https://www.nobelprize.org/uploads/2018/06/advanced-economicsciences2001.pdf
• Royal Swedish Academy of Sciences. Jean Tirole: market power and regulation. Scientific background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2014. 13 October 2014. https://www.nobelprize.org/uploads/2018/06/advanced-economicsciences2014.pdf
• Stiglitz JE, Yun J, Kosenko A. Equilibrium in a competitive insurance market under adverse selection with endogenous information. NBER Working Paper 23556; 2017. https://www.nber.org/papers/w23556