Contract Theory

Contract Theory for Incentives, Information and Payment Design in Health Care

Health care runs on delegation. Patients rely on clinicians, insurers and health systems pay hospitals and practices, and payers buy medicines whose performance in routine use is uncertain at launch. In each link one party acts for another that cannot fully observe, verify or enforce what is done, so the terms of the agreement shape the care delivered. The core framework (participation and incentive constraints, hidden action, hidden information and incomplete contracts) applies to provider payment and to outcome-based agreements between payers and manufacturers, and an illustrative blended payment and review evidence show how it works in practice.

Delegation in medical care, from Arrow onwards

Arrow's 1963 paper explained the special features of medical care as adaptations to uncertainty about the incidence of disease and the efficacy of treatment. Because the patient knows less than the physician, the patient cannot completely enforce standards of care, which gives rise to a relationship of trust. Arrow also noted that physicians act as a controlling agent for insurers by certifying the need for treatment, but called this a far from perfect check.

Contract theory gives these relationships one structure, in which patients, insurers and purchasers are principals, and clinicians, hospitals and manufacturers are agents. Ellis and McGuire modelled the hospital physician as an agent for two principals at once, the patient and the hospital. The general difficulty is the principal-agent problem, and contract theory concerns the agreements designed around it.

Hidden action, hidden information and unverifiable quality

Laffont and Martimort separate three information problems, in which the agent takes an action the principal does not observe, the agent holds private knowledge before contracting, or both parties observe something that no court can verify. They note that, in general, these problems prevent the allocation that full information would allow.

Hidden action is moral hazard; on the supply side it covers a provider's effort on diagnosis, quality and cost control. Hidden information is adverse selection in the contract-theory sense, where a provider may know more than the payer about a patient's expected cost, or about its own concern for patients. Both are forms of information asymmetry. Nonverifiability arises when both parties know care was poor but cannot write that judgement into an enforceable payment term.

The payer's problem of participation and incentive compatibility

The canonical model has the payer choose a payment rule that the provider accepts and then responds to in its own interest. The payer cannot pay for the action itself, only for verifiable signals of it. In a standard form it is written as follows.

$$\max_{t(\cdot),\,a} \; v(a) - E[t(y)] \quad \text{subject to} \quad E[u(t(y)) \mid a] - c(a) \ge \bar{u}, \qquad a \in \arg\max_{a'} \big( E[u(t(y)) \mid a'] - c(a') \big)$$

where $a$ is the agent's action (such as effort on quality or cost control), $v(a)$ is its expected value to the payer, $y$ is a verifiable signal (activity, coded diagnoses, a quality indicator), $t(y)$ is the payment rule, $u$ is the agent's utility from payment, $c(a)$ is the agent's private cost of the action and $\bar{u}$ is what the agent could obtain elsewhere. The first constraint is participation; the second, incentive compatibility, makes the action the payer wants the one the agent prefers.

Two results carry over to health care. When outcomes are noisy and the agent is risk averse, stronger incentives impose more risk, so the contract trades incentive strength against risk sharing. Holmström's informativeness principle adds that, in the simplest case, where effort is one-dimensional, payment should depend on any signal informative about the action, including signals that help filter out luck.

Payment rules as contracts and who bears the marginal cost

Ma and Mak frame provider payment through an internalisation principle, under which a payment system is efficient if and only if it makes the provider internalise the social benefit and cost of its choices. Each method allocates the cost of one more unit of care differently. The mechanics are on the provider payment page; the table summarises the predictions.

Case-based payment prices each episode as a base rate scaled by a case weight. Ellis and McGuire showed that cost-based reimbursement leads to too many services, and prospective payment to too few when the decision-maker weights hospital profit above patient benefit. Ma and Mak add that cost differences between patients invite dumping costly patients, creaming (over-providing quality to low-cost patients) and skimping (under-providing to high-cost ones), and that upcoding, misreporting cases into better-paid groups, is a separate response. Fee-for-service and capitation sit at the two ends of the volume incentive; selection is treated on the cream skimming and risk adjustment pages.

Mixed payment and the index of agency

Ellis and McGuire proposed paying part of each case prospectively and reimbursing a fixed share of cost. The decision-maker weights patient benefit against hospital net revenue at a rate they call the index of agency. In a linear version of their model, revenue per case is as follows.

$$R(q) = P + r\,c\,q$$

where $R(q)$ is revenue for a case receiving $q$ units of hospital care, $P$ is the prospective amount, $r$ is the share of cost reimbursed and $c$ is the constant marginal cost of a unit of care. With $\alpha$ constant, as Ellis and McGuire assume when amounts per patient are small, the decision is equivalent to choosing $q$ to maximise $\alpha B(q) + R(q) - c\,q$, a simplification used here, which gives the following.

$$\alpha\, b(q) = (1 - r)\, c$$

where $B(q)$ is the patient's benefit in money terms, $b(q)$ is its marginal value and $\alpha$ is the index of agency, the weight on a pound of patient benefit relative to a pound of net revenue. Efficiency requires $b(q) = c$, which holds under the following condition.

$$r^{*} = 1 - \alpha$$

where $r^{*}$ is the efficient share of cost reimbursed. A perfect agent ($\alpha = 1$) needs pure prospective payment; an agent who gives profit more weight ($\alpha < 1$) under-supplies care under it, and partial cost reimbursement restores the efficient quantity. Ellis and McGuire showed in their model that the mixed system also reduces providers' risk and their incentives to admit or reclassify patients unnecessarily.

Worked example of a blended case payment and the provider's choice

The figures are illustrative. A patient's benefit, in pounds, from $q$ bed-days is $B(q) = 1200q - 50q^2$, so the marginal benefit of a day is $b(q) = 1200 - 100q$. Each day costs £400, and the treating team values a pound of patient benefit at half a pound of hospital net revenue, so $\alpha = 0.5$.

1. The efficient stay. Marginal benefit is set equal to marginal cost, as follows.

$$1200 - 100q = 400 \;\Rightarrow\; q = 8$$

where $q$ is the number of bed-days that maximises benefit net of cost.

2. Pure case payment, no cost reimbursed. The provider bears each extra day in full, as follows.

$$0.5 \times (1200 - 100q) = 400 \;\Rightarrow\; 1200 - 100q = 800 \;\Rightarrow\; q = 4$$

where the left side is the weighted marginal benefit and the right side the provider's marginal cost. The stay halves.

3. Full cost reimbursement. With $r = 1$ the provider bears no marginal cost, so care continues until marginal benefit falls to zero, at 12 days.

4. Blended payment with half of cost reimbursed. With $r = 1 - \alpha = 0.5$ the provider bears £200 per day, as follows.

$$0.5 \times (1200 - 100q) = 0.5 \times 400 \;\Rightarrow\; q = 8$$

where both sides are halved, restoring the efficient stay. Keeping the payment for an eight-day case at its cost of £3,200 sets the fixed part at $P = 3200 - 0.5 \times 400 \times 8 = 1600$, that is £1,600.

5. An unusually costly case. Two extra days cost 2 × £400, or £800. The pure case price leaves all £800 with the hospital; the blend leaves half of it, £400.

The blend gives the highest benefit net of cost and halves exposure to costly cases. Cutting the pure case price to £1,600 would remove the margin but not the under-treatment. The result rests on an $\alpha$ the payer cannot observe directly, and the model omits cost-reduction effort, which weakens under partial reimbursement.

Multitasking, altruism and the limits of pay for performance

Holmström and Milgrom showed that when one of several tasks is hard to measure, strong incentives on the measured task divert effort from the other, so the optimal contract may carry weak explicit incentives or none, as when a practice is paid for a recorded blood pressure but not for the quality of the consultation. Ma and Mak report that a lower provider share of cost can partly restore balance across qualities, and that a provider's concern for patients can substitute for a profit margin, although when that concern is private information the first-best outcome generally cannot be reached.

Incomplete contracts and quality that cannot be written down

Some aspects of care cannot be specified in advance or verified afterwards. The incomplete-contracts approach of Grossman, Hart and Moore turns to residual control rights, the right to decide matters the contract leaves open, which can stand in for rewards that cannot be written down. Hart, Shleifer and Vishny applied this to privately run public services, where the incentive to cut cost can lower quality that is hard to specify, a concern the Nobel committee's review notes has been raised for hospitals among other services.

Outcome-based agreements between payers and manufacturers

Contracts for new medicines face the same design questions under uncertainty about real-world performance. The ISPOR task force describes performance-based risk-sharing arrangements as tracking a product's performance in a defined population over a set period, with reimbursement based on the health and cost outcomes achieved. On this site such arrangements fall under managed entry agreements.

Towse and Garrison argued that the party best able to manage or bear a risk should do so, favoured a value of information and real options framework, and named the difficulty of writing and policing agreements as a constraint. Barros argued that paying by results can lead to too many patients being treated and to prices raised in anticipation, so welfare effects are ambiguous. The ISPOR task force adds that negotiation, monitoring and evaluation costs can be substantial.

What reviews of payment incentives have found

The theory predicts the direction of responses; Cochrane reviews show that their size and persistence are often uncertain. Four reviews cover primary care, outpatient and hospital payment.

• Primary care payment (2000). Four studies with 640 physicians found more visits and diagnostic services under fee-for-service than capitation, but fewer hospital referrals and repeat prescriptions.
• Outpatient payment (2021). In a review of 27 studies, added pay for performance probably improved child immunisation (risk ratio 1.27, 95% confidence interval 1.19 to 1.36); in one trial the gain faded once incentives ended.
• Hospital pay for performance (2019). Across 27 studies of six programmes, effects on mortality and adverse events were very slight or absent, at very low certainty; penalties appeared slightly more effective than rewards.
• Primary care quality incentives (2011). Six of seven studies found modest effects on some measures; the evidence was judged insufficient to support or reject such schemes.

Ma and Mak's review adds that stays were about 25% longer under cost reimbursement in studies of the 1983 Medicare reform, and that hospitals upcoded when Medicare changed its group definitions.

Contract terms inside economic evaluation and HTA

The payment rule is part of the intervention being evaluated. A model of a new payment scheme should allow for predicted responses in volume, patient selection and coding rather than hold practice fixed, with sensitivity analysis over their size. In health technology assessment, an outcome-based agreement changes the expected net price and adds monitoring costs, and the task force judges its societal case by whether extra data collection is worth its cost in better resource allocation.

Neighbouring concepts and frequent misreadings

Mechanism design overlaps with the hidden-information side of contract theory, because screening contracts that offer a menu of terms so that providers with different private information choose different options are mechanism design problems. Game theory in health covers strategic interaction with or without a contract. Moral hazard has a demand-side meaning (the response to insurance cover) and a supply-side one (a provider's hidden action).

Theory predicts the direction of a response, not its size. A blended payment is not better by construction, because its value depends on a cost share matched to motives the payer cannot observe directly. Paying for a measured indicator is not paying for health, and a rule that cannot be enforced through verifiable coding or outcomes will not work as written.

Sources

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• Barros PP. The simple economics of risk-sharing agreements between the NHS and the pharmaceutical industry. Health Economics. 2011;20(4):461-470.
• Ellis RP, McGuire TG. Provider behavior under prospective reimbursement: cost sharing and supply. Journal of Health Economics. 1986;5(2):129-151.
• Garrison LP Jr, Towse A, Briggs A, et al. Performance-based risk-sharing arrangements: good practices for design, implementation, and evaluation. Report of the ISPOR Good Practices for Performance-Based Risk-Sharing Arrangements Task Force. Value in Health. 2013;16(5):703-719.
• Gosden T, Forland F, Kristiansen I, et al. Capitation, salary, fee-for-service and mixed systems of payment: effects on the behaviour of primary care physicians. Cochrane Database of Systematic Reviews. 2000, Issue 3. Art. No.: CD002215.
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