Mechanism Design

Mechanism Design and Writing Rules So Strategic Choices Serve Health Goals

Mechanism design starts from the outcome a health system wants, such as the cheapest reliable supplier of a medicine or the largest number of compatible kidney transplants, and works backwards to rules under which self-interested participants choose to produce it. Its central test is whether each participant does best by reporting private information truthfully and by taking part at all. It matters because purchasers, transplant programmes and funders rarely know suppliers' costs, which patient-donor pairs hospitals hold back, or firms' prospects, yet must still allocate scarce resources.

From predicting play to choosing the rules

Game theory in health usually takes the rules as given and predicts what players will do. Mechanism design reverses the direction. The designer fixes a goal and searches for rules that make that goal an equilibrium. The 2007 Nobel background traces the field to Leonid Hurwicz, who defined a mechanism as a communication system in which a pre-specified rule assigns an outcome to every set of messages.

Three ideas carry the theory. Hurwicz introduced incentive compatibility in 1972, so that the analysis could allow for self-interested participants with private information. The revelation principle, first formulated by Gibbard and developed in its greatest generality by Roger Myerson, states that any equilibrium outcome of any mechanism can be replicated by a direct mechanism in which participants report their private information and truth-telling is optimal. Eric Maskin's implementation theory addressed a gap it leaves open. A mechanism can have several equilibria, some poor, and Maskin characterised when every equilibrium delivers the goal. Hurwicz, Maskin and Myerson shared the 2007 prize.

The designer's problem of truthful reports and willing participants

The revelation principle turns a search over all institutions into a constrained optimisation over direct mechanisms. The designer chooses an allocation rule and a payment rule, both functions of what participants report.

$$\max_{x(\cdot),\,t(\cdot)} \; E_{\theta}\Big[ W\big(x(\theta), t(\theta)\big) \Big] \quad \text{subject to} \quad U_i(\theta_i \mid \theta_i) \ge U_i(\hat{\theta}_i \mid \theta_i), \qquad U_i(\theta_i \mid \theta_i) \ge \bar{u}_i$$

where $\theta = (\theta_1, \ldots, \theta_n)$ is the profile of participants' private information or types (a supplier's unit cost, a hospital's set of patient-donor pairs), $x(\theta)$ is the allocation rule (which supplier wins, which pairs exchange kidneys), $t(\theta)$ is the payment rule, $E_{\theta}$ is the expectation over the possible type profiles, $W$ is the designer's objective (such as health gain net of cost), $U_i(\hat{\theta}_i \mid \theta_i)$ is the expected payoff to participant $i$ of true type $\theta_i$ who reports $\hat{\theta}_i$ while others report truthfully, and $\bar{u}_i$ is what participant $i$ obtains by staying out. The first constraint is incentive compatibility (IC), required for every true and reported type; the second is individual rationality (IR), the participation constraint.

In a dominant-strategy mechanism, truth-telling must be best whatever others report; in a Bayesian mechanism it need only be best on average over others' types. The Nobel background notes that dominant-strategy mechanisms achieve little outside quasi-linear settings, where payoffs are linear in money, which is why much of the theory is Bayesian. The constraints also cost efficiency. Hurwicz showed that in a standard exchange economy no incentive-compatible mechanism satisfying participation can produce Pareto-optimal outcomes, and Myerson and Satterthwaite showed that, with voluntary participation, no mechanism realises all gains from bilateral trade when both sides hold private valuations. Institutions are therefore judged against incentive efficiency, the best attainable once incentives are respected, rather than Pareto efficiency under full information.

Matching doctors and kidneys without prices

The 2012 Nobel background on Roth and Shapley's work on stable allocations and market design notes that in most countries it is illegal to exchange organs for money, so organs must be assigned by other rules. Two properties then guide design. These are stability, which encourages voluntary participation, and incentive compatibility, which discourages manipulation.

The US market for new doctors is the classic case. Roth showed that its centralised clearinghouse, the NRMP, had improved a failing market with a procedure in effect equivalent to Gale and Shapley's deferred-acceptance algorithm. The employer-proposing version is not incentive compatible for applicants, while the applicant-proposing version is, and Roth proved that no stable matching mechanism makes truth-telling a dominant strategy for everyone. The NRMP adopted the Roth-Peranson applicant-proposing algorithm, adapted for couples, in 1997.

Roth, Sönmez and Ünver adapted the top trading cycle algorithm to patients whose willing donor is incompatible and set out how such exchanges can be arranged efficiently and incentive compatibly; their 2005 paper showed that when only pairwise exchanges are feasible and any compatible healthy kidney is equally acceptable, a wide class of constrained-efficient mechanisms is strategy-proof. Participation also needs design. Ashlagi and Roth report that hospitals can enrol only their hard-to-match pairs while arranging easy exchanges internally, and identify a mechanism that gives hospitals incentives to reveal all pairs. In the UK, NHS Blood and Transplant runs the UK Living Kidney Sharing Scheme, which forms two- and three-pair cycles and chains started by non-directed altruistic donors, with matching runs four times a year.

Tenders, auctions and advance market commitments

In medicine procurement, suppliers know their own costs and the buyer does not. The 2020 Nobel background on auction theory records Vickrey's result that in a second-price sealed-bid auction each bidder has a dominant strategy of bidding its true value, whereas first-price auctions, a common format for public contracts, have no dominant strategy because optimal bids depend on rivals' strategies.

Medicine tenders raise further concerns. Németh and colleagues report that hospital purchasing of medicines in Europe relies mainly on single-winner tenders, increasingly criticised for securing low prices at the expense of long-term competition, security of supply and patients with specific therapeutic needs. Their multi-winner concept uses maximum market shares and supply and sales guarantees, and asks how truth-telling can be encouraged in quantity bids, a mechanism design question in its own right.

An advance market commitment (AMC) is a pull mechanism, in which a funder commits to a subsidy policy before firms invest in research and capacity. In the pneumococcal pilot reviewed by Kremer, Levin and Snyder, five countries and the Gates Foundation pledged USD 1.5 billion in 2007; under Gavi's supervision firms competed for ten-year supply contracts capping the price at USD 3.50 per dose, and a firm committing to supply a share of the projected 200 million annual doses received that share of the fund as a per-dose subsidy. Their earlier model gives a design lesson. A pure per-dose subsidy lets a monopoly supplier claim the whole fund with low capacity, so funds should be tied to capacity commitments.

Outcome-based agreements apply the same logic to payment for medicines; their design is covered under contract theory and managed entry agreements.

Worked example of a reverse second-price tender and a kidney cycle

The figures are illustrative. A purchasing consortium tenders for 100,000 packs a year of a generic injectable, with a reserve (maximum) price of GBP 6.00 per pack. Three suppliers have private unit costs of GBP 4.00 for A, GBP 4.60 for B and GBP 5.20 for C. The lowest bid wins and the winner is paid the second-lowest bid.

1. Truthful bids. A bids 4.00, B 4.60 and C 5.20. A wins and is paid GBP 4.60 per pack, which gives the profit below.

$$\pi_A = (p - c_A) \times Q = (4.60 - 4.00) \times 100000 = 60000$$

where $\pi_A$ is A's annual profit in GBP, $p$ is the price per pack (the second-lowest bid), $c_A$ is A's unit cost and $Q$ is the contracted volume. The purchaser pays GBP 460,000 against a resource cost of GBP 400,000, the lowest available.

2. Misreporting by A. Bidding 4.50 or 3.50 leaves A the winner at the same price, because A's own bid never sets its payment. Bidding 4.70 hands the contract to B and A earns nothing.

3. Misreporting by B. Bidding 3.90 wins at A's bid of 4.00, below B's cost, a loss of 0.60 per pack or GBP 60,000. Bidding above 4.60 still loses. Truthful bidding is a dominant strategy for each supplier, the procurement mirror image of Vickrey's result.

4. The price of truth-telling. The GBP 60,000 above A's cost is the rent left to the low-cost supplier to make honesty and participation worthwhile. If A bid alone, the rule would pay the reserve price of GBP 6.00, or GBP 200,000 a year above A's cost, so the reserve price caps that exposure.

5. A kidney cycle. Three incompatible pairs are registered. Donor 1 suits patient 2, donor 2 suits patient 3 and donor 3 suits patient 1. No two pairs can swap, so a pairwise-only rule yields no transplants, while a three-way cycle yields three. An altruistic donor who suits patient 1 adds a fourth transplant either way, by giving to the waiting list alongside the cycle or by starting a chain (to patient 1, then donors 1, 2 and 3 to patient 2, patient 3 and the waiting list) whose operations need not be simultaneous, though it is longer than the UK scheme's chains of one or two pairs.

The designer buys truthful information by paying for it, and richer exchange rules raise the number of feasible transplants.

Design choices that change the result

The guarantees hold only under their assumptions, and small changes in rules or setting can remove them. Four choices matter most in health care.

Using mechanism design in evaluation and HTA

Mechanism design produces no cost-effectiveness ratio, but it shapes the prices, volumes and allocations that economic evaluation takes as inputs. A tender rule fixes the acquisition cost used in cost-effectiveness analysis and budget impact analysis, and an outcome-based agreement changes the expected net price a health technology assessment considers. Evaluating a new procurement, matching or payment rule therefore means modelling how participants respond, not holding prices and supply fixed.

Neighbouring concepts

Nash equilibrium in health supplies the solution concepts; mechanism design chooses the rules under which equilibria arise. Contract theory covers bilateral agreements; its screening models are mechanism design problems with one principal. The principal-agent problem describes the delegation setting many health mechanisms address, and adverse selection and information asymmetry describe the private information that IC constraints must manage.

Limits and common misreadings

Incentive compatibility makes truth-telling an equilibrium, not the only one. Experiments cited in the 2020 background found that subjects reach their dominant strategy more easily in an English auction than in a second-price auction, so a design that is strategy-proof on paper may still need to be simple to use. A mechanism optimal for one goal can be poor for another, as when a lowest-price single-winner tender weakens future supply. Idealised results may not carry over. With couples in the doctors' market a stable matching may not exist, so market designers combine theory with simulation. The objective itself, weighing price against access, equity and security of supply, remains a policy judgement.

Sources

• Ashlagi I, Roth AE. Free riding and participation in large scale, multi-hospital kidney exchange. Theoretical Economics. 2014;9(3):817-863.
• Kremer M, Levin JD, Snyder CM. Advance market commitments: insights from theory and experience. NBER Working Paper 26775; 2020. Published in AEA Papers and Proceedings. 2020;110:269-273.
• Németh G, Mágó ML, Kaló Z, Lám J, Balogh T, Brodszky V. A concept for multi-winner tenders for medicinal products with balancing between efficient prices, long-term competition and sustainability of supply. Frontiers in Medicine. 2023;10:1282698.
• NHS Blood and Transplant. UK Living Kidney Sharing Scheme. ODT Clinical website; accessed 3 Oct 2026.
• Roth AE, Sönmez T, Ünver MU. Kidney exchange. Quarterly Journal of Economics. 2004;119(2):457-488.
• Roth AE, Sönmez T, Ünver MU. Pairwise kidney exchange. Journal of Economic Theory. 2005;125(2):151-188.
• 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.
• Royal Swedish Academy of Sciences. Press release: the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2007. 15 Oct 2007.
• Royal Swedish Academy of Sciences. Stable allocations and the practice of market design. Scientific background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2012. 2012.
• Royal Swedish Academy of Sciences. Improvements to auction theory and inventions of new auction formats. Scientific background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2020. 2020.