Bargaining Theory

Bargaining Theory and Nash Bargaining over Drug Prices, Hospital Contracts and the Threshold

Many health care prices are agreed between two parties, such as a national payer and the patent holder of a new medicine, or a hospital system and an insurer. Bargaining theory predicts where such a deal settles within the range both sides would accept, and traces the answer to what each side gets if talks fail and how much bargaining power each holds. The central tools are the Nash bargaining solution and the alternating-offers model behind it. Under value-based pricing a cost-effectiveness threshold fixes the payer's walk-away price, as an illustrative price negotiation shows, and published applications cover hospitals, medical devices and medicines.

Negotiated prices as a bargaining problem

Bargaining theory starts where agreement creates value that neither party can capture alone. In health care, a payer covering most of a national population faces a manufacturer holding a patent, and a large hospital system faces an insurer whose members want access to it. Nash's 1950 paper treated this bilateral monopoly problem as a two-person game.

A two-party bargaining problem is described by a set of feasible agreements and a disagreement point, the payoff each party gets if no agreement is reached. In a negotiation over a medicine, the manufacturer will not accept less than its reservation price, and the payer will not pay more than the price at which the product stops being worth funding. The gap between the two is a surplus, and bargaining theory predicts how it is split.

The Nash bargaining solution and its weighted form

Nash proposed a rule that selects the agreement maximising the product of the two parties' gains over their disagreement payoffs. Applied work usually adds weights so that the parties can differ in bargaining power. The generalised Nash bargaining solution is as follows.

$$x^{*} = \arg\max_{x \in F} \left[ U_M(x) - d_M \right]^{\beta} \left[ U_P(x) - d_P \right]^{1-\beta}$$

where $x^{*}$ is the agreed outcome, $F$ is the set of feasible agreements, $U_M(x)$ and $U_P(x)$ are the payoffs to the manufacturer and the payer from agreement $x$, $d_M$ and $d_P$ are their disagreement payoffs, and $\beta$, between 0 and 1, is the manufacturer's bargaining weight, with $(1 - \beta)$ the payer's. Setting $\beta = 0.5$ gives Nash's original symmetric solution.

The symmetric solution is the only rule that satisfies four conditions together, namely Pareto efficiency, symmetry between parties in identical positions, invariance to a positive affine rescaling of either party's utility, and independence of irrelevant alternatives. The weighted form drops symmetry. Because the solution depends on payoffs only through gains over the disagreement point, a higher disagreement payoff raises a party's share. Strategic models show that this disagreement payoff is what each party receives while talks are stalled, its inside option, such as the benefit of current care the payer keeps during a delay. An outside option, available only by walking away for good, matters only when it binds, as the next section explains.

Prices when value is measured in money

When both payoffs are in money and move one for one with the price, the solution has a simple closed form. The manufacturer gains the price minus its reservation price, and the payer gains its maximum price minus the price paid. Maximising the weighted product of the two gains gives the following.

$$p^{*} = c + \beta \left( \bar{p} - c \right)$$

where $p^{*}$ is the negotiated price, $c$ is the manufacturer's reservation price (in the short run, its marginal cost of supply), $\bar{p}$ is the payer's maximum price and $\beta$ is the manufacturer's bargaining weight. The manufacturer receives a share $\beta$ of the surplus and the payer the rest, so the price runs from the reservation price to the payer's maximum as $\beta$ rises from 0 to 1.

Where bargaining power comes from

Strategic models explain the weight $\beta$ by describing the negotiation as a sequence of offers. In Rubinstein's alternating-offers model two parties take turns proposing how to divide a surplus of size one, and each discounts later agreement. The unique perfect equilibrium gives the party that proposes first the following share.

$$s_1 = \frac{1 - \delta_2}{1 - \delta_1 \delta_2}$$

where $s_1$ is the first proposer's share of the surplus, $\delta_1$ is the first proposer's discount factor per round and $\delta_2$ is the responder's, both between 0 and 1. A discount factor closer to one means a more patient party, and the more patient party obtains more of the surplus.

Binmore, Rubinstein and Wolinsky showed that as the incentive to settle quickly becomes negligible, the equilibrium of alternating-offers models approaches the Nash bargaining solution, provided utilities and the disagreement point are chosen to reflect that incentive. Their results provide a guide for applying the Nash solution in economic modelling. In pharmaceutical markets, Berdud, Ferraro and Towse attribute the payer's impatience to clinical need and the developer's to the sunk nature of research and development costs.

Muthoo's outside option principle makes this precise. An outside option, the best deal available by walking away for good, raises a party's bargaining power only if it is worth more than what that party would get from the negotiation without it; otherwise it is an empty threat. A payer therefore gains from a competing therapy only when the competitor's offer beats the payer's share of the bargain.

The threshold as the payer's walk-away price

Under value-based pricing the payer's maximum price comes from an economic evaluation. A health system that funds technologies whose incremental cost-effectiveness ratio is at or below a cost-effectiveness threshold will pay at most the price at which the two are equal. Using net monetary benefit, and assuming one course per patient, that maximum price is as follows.

$$\bar{p} = k \, \Delta E - \Delta C_{o}$$

where $\bar{p}$ is the payer's maximum price per patient, $k$ is the threshold in money per quality-adjusted life year, $\Delta E$ is the incremental QALYs per patient, and $\Delta C_{o}$ is the incremental cost per patient other than the product's price, such as administration costs.

Claxton and colleagues pointed out in 2008 that a price at which the ratio just equals the threshold leaves the NHS with zero net health benefit, so all of the benefit of the innovation goes to the manufacturer as revenue. In bargaining terms, pricing at the threshold is the outcome predicted when the manufacturer holds all the bargaining power. Jena and Philipson noted that improved cost-effectiveness, which raises consumer surplus, often clashes with incentives for producer surplus such as patents; in their illustrative sample the median technology's producer appropriation was about 15 per cent.

Berdud, Ferraro and Towse placed Nash bargaining inside this framework. After the payer commits to a threshold, the two sides bargain over the effective ratio, and the Nash solution is the same weighted average in cost-effectiveness ratio units, between the developer's reserve ratio and the payer's maximum. They found that, depending on research and development costs and when the payer has enough bargaining power, a decision threshold above the supply-side threshold, the one that reflects the health displaced by extra spending, can be socially efficient, because negotiated prices then settle below the threshold price. An earlier version was published as a consulting report commissioned by a manufacturer.

Worked example of a payer and a manufacturer negotiating a price per course

The figures are illustrative. Each patient receives one course of a new treatment, so all amounts below are per patient. The treatment gives 0.5 incremental QALYs and adds £2,000 in other costs. The payer uses a threshold of £20,000 per QALY, and the manufacturer's reservation price, its short-run cost of supplying the course, is £1,000.

1. Find the payer's maximum price. The threshold condition gives the following.

$$\bar{p} = 20000 \times 0.5 - 2000 = 8000$$

where $\bar{p}$ is the maximum price in pounds per patient. The surplus to divide is £8,000 minus £1,000, or £7,000 per patient.

2. Apply the symmetric Nash solution. With equal weights the price is as follows.

$$p^{*} = 1000 + 0.5 \times 7000 = 4500$$

where $p^{*}$ is the negotiated price in pounds per patient. The cost per QALY at that price is as follows.

$$ICER = \frac{4500 + 2000}{0.5} = 13000$$

where $ICER$ is the incremental cost per QALY gained. Each side gains £3,500 per patient, the payer as net monetary benefit and the manufacturer above its reservation price.

3. Derive the weight from patience. Suppose the manufacturer opens and discounts each round of delay by 0.9, while the payer, under clinical pressure to give patients access, discounts by 0.8. The manufacturer's share is as follows.

$$s_M = \frac{1 - 0.8}{1 - 0.9 \times 0.8} = \frac{0.2}{0.28} \approx 0.714$$

where $s_M$ is the manufacturer's share of the surplus, exactly five sevenths. This share includes the advantage of moving first; had the payer opened, the manufacturer's share would be nine fourteenths. Five sevenths of £7,000 is £5,000, so the price is £6,000, the ratio rises to £16,000 per QALY and the payer keeps £2,000 per patient.

4. Add an outside option for the payer. Under symmetric weights, suppose a competing therapy would give the payer a net monetary benefit of £4,000 per patient. This beats the £3,500 from the symmetric bargain, so under the outside option principle the price falls to £4,000 per patient, or £12,000 per QALY. A competing offer worth £3,000 would leave the price at £4,500.

Every price in the table is cost-effective at £20,000 per QALY, so the evaluation fixes only the ceiling; where the price settles below it decides how much of the health gain the system keeps.

Hospital-insurer, device and medicine negotiations in empirical work

Most empirical applications in health care come from industrial organisation, where many buyers negotiate with many sellers at once. The standard "Nash-in-Nash" model builds on Horn and Wolinsky's analysis of bilateral monopolies and was given a non-cooperative foundation by Collard-Wexler, Gowrisankaran and Lee. Each buyer-seller pair reaches its Nash bargaining solution, taking the other pairs' agreements as given.

Gowrisankaran, Nevo and Town estimated a bargaining model between hospitals and managed care organisations in the United States and found that insurer bargaining restrains hospital prices significantly. A tenfold increase in patient coinsurance would reduce prices by 16 per cent, and a challenged hospital acquisition in Northern Virginia would have raised prices significantly.

Grennan applied bilateral Nash bargaining between hospitals and coronary stent manufacturers and found that more uniform pricing works against hospitals by softening competition. For medicines, Dubois, Gandhi and Vasserman modelled the United States alongside reference countries such as Canada, where prices are negotiated between manufacturers and government. Simulating external reference pricing by the United States, with firms internalising the cross-country externalities the policy creates, they found much smaller price decreases in the United States than price increases in the reference countries.

Assumptions that change the negotiated price

A bargaining model's prediction depends heavily on how it is set up. The disagreement point is usually the net benefit of current care for the payer and the profit from not selling in that market for the manufacturer, which falls if a failed deal also damages prices elsewhere.

The payoff measures matter as much. Measuring the payer's gain at the threshold assumes the threshold reflects the health displaced by spending, and a marginal-cost reservation price ignores research and development spending, which is sunk once the product exists but still shapes future investment. The Nash solution also assumes each side knows the other's payoffs.

Bargaining theory beside related ideas

Bargaining theory is a branch of game theory with cooperative and non-cooperative sides. The Nash bargaining solution differs from a Nash equilibrium. The first divides a surplus between parties who can make a binding agreement, and the second is a set of strategies from which no player wants to deviate alone. Nash-in-Nash models combine the two.

Access negotiation is the practical process through which a supplier and a payer agree price and access conditions, and bargaining theory models its outcome. Contract theory asks how terms should be designed when one side knows more than the other, the central problem of outcome-based managed entry agreements. When a managed entry agreement takes the form of a simple or confidential discount, the discounted price is itself a bargained price. Auction theory applies when several suppliers compete in a tender, so the buyer's power comes from rivalry among bidders.

Where bargaining models mislead

A common misreading treats the Nash solution as a statement of fairness. It is a prediction derived from conditions on outcomes, and an equal split is a modelling assumption rather than an ethical default. A second error is to read the threshold as the expected price rather than the payer's maximum; which share of the surplus best supports long-run innovation remains contested.

Estimated weights also mislead when the data do not show the price actually paid, since confidential discounts mean list prices overstate the manufacturer's share. Treating every alternative as the payer's disagreement payoff overstates its power, because an alternative matters only when it beats the bargain on offer.

Sources

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