Incentive Compatible Mechanism

Incentive Compatible Mechanism and Truthful Reporting in Health Economics

An incentive compatible mechanism is a set of decision and payment rules, of the kind studied in mechanism design, in which reporting private information truthfully is the best available strategy for every participant. In health economics the private information might be how much a resident would pay for a local public health programme, or which incompatible patient and donor pairs a transplant centre has registered. The 2007 scientific background of the Royal Swedish Academy of Sciences dates the notion to Leonid Hurwicz in 1972 and credits it with making mechanism design relevant to a wide variety of applications. The property matters to health economists because welfarist cost-benefit analysis rests on reported willingness to pay, and a valuation that respondents can profitably distort gives a biased basis for spending decisions. Truthful reporting has a price, since in the dominant-strategy Clarke-Groves rule the transfers that secure it generally fail to balance the budget, and the outcome is only as defensible as the decision rule applied to the reports.

Private information and the truth-telling condition

Hurwicz described a mechanism as a communication system in which participants send messages and a fixed rule turns the messages into an outcome. The messages may carry true or pretended information, such as a person's willingness to pay for a public good, and a participant may falsify them in the hope of paying less. A direct mechanism simply asks each participant to report a type, the private information that matters for the decision.

Osborne and Rubinstein give a compact version for a yes or no public decision. Each person $i$ holds a valuation known only to that person, and the mechanism maps everyone's reports to a decision and to money transfers. The mechanism is strategy-proof, the dominant-strategy form of incentive compatibility, when the condition below holds for every person, every true valuation, every alternative report and every set of reports by the others.

$$d(v_i, x_{-i})\,v_i + t_i(v_i, x_{-i}) \ge d(z_i, x_{-i})\,v_i + t_i(z_i, x_{-i})$$

where $d$ equals 1 when the project goes ahead and 0 when it does not, $v_i$ is person $i$'s true valuation in money, $z_i$ is any other report person $i$ could make, $x_{-i}$ stands for the reports of everyone else, and $t_i$ is the money transferred to person $i$, negative when person $i$ pays. The left side is the utility from telling the truth and the right side the utility from a misreport.

In a Bayesian mechanism the inequality need only hold on average over the others' private information, which the 2007 background notes makes the constraints easier to satisfy. A participation constraint is usually added, so that no one is made worse off by taking part. By the revelation principle any equilibrium outcome of any mechanism can be replicated by an incentive compatible direct mechanism, so the condition above is the constraint a designer works with.

Funding a local public health programme with the Clarke-Groves rule

For public goods the 2007 background notes that people may pretend to be relatively uninterested so as to reduce their share of the cost, and that before 1970 economists generally believed efficient provision was out of reach. Edward Clarke in 1971 and Theodore Groves in 1973 showed that, when utility is quasi-linear so that there are no income effects, truthful reporting can be a dominant strategy while the chosen level of the public good maximises social surplus.

Osborne and Rubinstein set out the binary form, the Vickrey-Clarke-Groves (VCG) mechanism. The project goes ahead if and only if the sum of reported valuations is positive. A person pays only when decisive, meaning that the report reverses the decision the others' reports point to. A decisive supporter pays the shortfall in the others' reports and a decisive opponent pays the others' net support, so no person can lower a payment by shading the report.

Worked example with five neighbourhoods and a mobile health checks service

All figures are illustrative. Five neighbourhood groups decide whether to fund a mobile health checks service from a local levy. Each valuation is net of an equal share of the cost, which reduces the problem to Osborne and Rubinstein's zero-cost case. The true net valuations are GBP 30 and GBP 20 for groups 1 and 2, and minus GBP 10, minus GBP 15 and minus GBP 15 for groups 3, 4 and 5.

1. The decision. The reports sum to a positive amount, so the service goes ahead.

$$\sum_{j=1}^{5} x_j = 30 + 20 - 10 - 15 - 15 = 10$$

where $x_j$ is the report of group $j$ in GBP, equal here to its true net valuation.

2. Who is decisive. Without group 1 the others sum to 20 − 10 − 15 − 15 = −20, and without group 2 they sum to 30 − 10 − 15 − 15 = −10, so both supporters are decisive. Without group 3 the others sum to 20, and without group 4 or 5 they sum to 25, so the service goes ahead anyway and no opponent pays.

3. Payments. Group 1 pays GBP 20 and group 2 pays GBP 10. Net payoffs are 30 − 20 = 10 for group 1, 20 − 10 = 10 for group 2, and −10, −15 and −15 for the rest.

4. Testing misreports. Group 1 gets the same result from any report above 20, and any report of 20 or less cancels the service, a payoff of 0 instead of 10. If group 3 reports minus GBP 25, the total is 20 − 25 = −5, the service is cancelled and group 3 pays the others' net support of 20, a payoff of −20 against −10. Blocking would cost group 4 GBP 25, a payoff of −25 against −15. No group gains by lying.

5. The alternatives. Summing reports with no payments lets group 3 cancel the service at no cost by reporting minus GBP 25, a payoff of 0 against −10. Majority rule is strategy-proof, but with two supporters and three opponents the service would be refused despite a positive net valuation.

6. The price of truth. The mechanism collects GBP 30, three times the net gain of GBP 10. If the money is withheld, the groups together end at 10 + 10 − 10 − 15 − 15 = −20, worse than without the service, and returning the money would destroy the truth-telling incentive. This is the budget balance problem the 2007 background describes.

Truthful answers in willingness to pay surveys

Willingness to pay in health is usually measured by survey, and the same logic applies to survey questions. Johnston and colleagues, whose guidance on stated preference studies focuses on public goods in environmental and human health settings, define an incentive compatible mechanism as one in which the respondent theoretically has the incentive to reveal private information truthfully, or in which truthful revelation is the dominant strategy. Their Recommendation 8 prefers incentive compatible formats for public goods and names a single binary choice per respondent, generally the status quo against the proposed change, as the most straightforward route.

The conditions are demanding. The choice must be perceived as consequential, with a non-zero chance that answers influence the decision and that respondents would have to pay, and the payment vehicle must be binding. Johnston and colleagues treat consequentiality as necessary but not sufficient. Classic open-ended questions are not incentive compatible and attract many unrealistically high or zero answers, double-bounded binary formats generally violate incentive compatibility, and voluntary payment vehicles invite free riding.

For a contingent valuation study of a community vaccination campaign, the guidance points to one referendum question tied to a credible local levy. In a discrete choice experiment, binary choices give the clearest incentive properties, and multinomial formats are incentive compatible only under more restrictive conditions. Social choice theory gives the same pattern. With two options majority rule is strategy-proof, while the Gibbard-Satterthwaite theorem shows that with three or more options every unanimous strategy-proof rule is a dictatorship.

Kidney exchange, regulation and price negotiation

Organ allocation runs without prices, so the allocation rule alone must secure truthful reports. Roth, Sönmez and Ünver build on Roth's 1982 result that the top trading cycles mechanism makes stating true preferences a dominant strategy, and show that their top trading cycles and chains mechanism for kidney exchange is strategy-proof under several chain selection rules. Strategy-proofness is lost under a rule that chooses among the longest chains, so a rule that looks attractive for transplant numbers can invite patients to misstate their preferences.

Baron and Myerson modelled a regulator that cannot observe a monopolist's true costs, with an optimal scheme trading off rent extraction against efficient output while keeping the firm in the market. A health payer pricing the services of providers whose costs it cannot see faces the same structure, treated under contract theory and the principal-agent problem.

Bilateral bargaining carries a starker limit. Myerson and Satterthwaite showed that when buyer and seller each hold private valuations, no incentive compatible mechanism that respects voluntary participation realises all gains from trade. For a payer and a manufacturer negotiating a price after a health technology assessment, the result implies that some agreements both sides would value can fail under any negotiating protocol, which bargaining theory treats from the strategic side.

Incentive compatibility in welfarist and extra-welfarist evaluation

The VCG rule asks whether the money valuations of gainers exceed those of losers, the potential Pareto test of welfarist welfare economics. Osborne and Rubinstein warn that this criterion is not necessarily desirable where wealth differs widely. In their example two wealthy people value a project at 100 each and 99 poor people value it at minus 1, so the project goes ahead and the wealthy pay nothing.

Brouwer and colleagues explain why this matters for health. Applied cost-benefit analysis expresses net utility through willingness and ability to pay, whereas the extra-welfarist use of need, seen archetypically in cost-utility analysis, is unrelated to ability to pay and insulates welfare judgements from the prevailing income distribution, which matters because income is correlated with the incidence and prevalence of disease. A truthful report of willingness to pay leaves open whether summed willingness to pay should settle a health decision, a value judgement that a social welfare function makes explicit. The incentive constraints still bind an extra-welfarist decision maker who relies on providers, manufacturers and transplant centres for information they hold privately.

Incentive compatibility beside nearby ideas

A Nash equilibrium describes behaviour once the rules are fixed, while incentive compatibility is a property of the rules that makes truth-telling an equilibrium or a dominant strategy. The Vickrey second-price auction, which Osborne and Rubinstein treat as a strategy-proof direct mechanism with transfers, is the standard case in auction theory. Screening menus of insurance plans or payment schedules satisfy incentive constraints of the same kind, made necessary by information asymmetry and adverse selection. The Lindahl equilibrium prices a public good by each person's marginal valuation, which people have an incentive to understate.

Limits and common misreadings

The guarantees are narrower than the name suggests. Five errors recur in health applications.

1. Treating truth-telling as the predicted outcome. The 2007 background states that incentive compatibility guarantees truth-telling is an equilibrium, not the only one.
2. Equating incentive compatibility with efficiency. With voluntary participation and unanimity, Mailath and Postlewaite showed that the probability of funding a public project tends to zero as numbers grow.
3. Assuming survey formats are incentive compatible by design. Johnston and colleagues criticise guidance implying that some question types are incentive compatible regardless of the good, the payment vehicle or perceived consequentiality.
4. Ignoring transparency and acceptability. Osborne and Rubinstein note that the VCG mechanism is not very transparent and that people may find paying unacceptable when the project is not carried out.
5. Reading strategy-proofness too widely. Roth, Sönmez and Ünver note that a kidney has an option value for its intended recipient, since a better trade may come later, so the result covers stated preferences within one matching run.

Sources

• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. https://doi.org/10.1016/j.jhealeco.2007.07.003
• Johnston RJ, Boyle KJ, Adamowicz W, Bennett J, Brouwer R, Cameron TA, Hanemann WM, Hanley N, Ryan M, Scarpa R, Tourangeau R, Vossler CA. Contemporary guidance for stated preference studies. Journal of the Association of Environmental and Resource Economists. 2017;4(2):319-405. https://doi.org/10.1086/691697
• Osborne MJ, Rubinstein A. Models in Microeconomic Theory. Expanded 2nd ed. Open Book Publishers; 2023. Chapters 17 and 20. https://doi.org/10.11647/OBP.0362
• Roth AE, Sönmez T, Ünver MU. Kidney exchange. NBER Working Paper 10002; 2003. https://www.nber.org/papers/w10002
• Royal Swedish Academy of Sciences. Mechanism design theory. Scientific background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2007. 2007. https://www.nobelprize.org/uploads/2018/06/advanced-economicsciences2007.pdf