Signature
p_neg = min(p_star, p_bar - v_P)
| Inputs | Definition | Unit |
|---|---|---|
p_star | Nash bargaining price per patient without the outside option, from HE-FM-BARG-001 | currency per patient |
p_bar | Payer's maximum price per patient, from HE-FM-BARG-002 | currency per patient |
v_P | Net monetary benefit per patient, compared with current care, that the payer obtains by walking away for good, for example by funding a competing therapy instead | currency per patient |
p_neg | Price per patient agreed once the payer's outside option is taken into account | currency per patient |
|---|
Function
Nash bargaining surplus-division function for negotiated health care prices
Maps the payoffs two negotiating parties obtain from each feasible agreement, their disagreement payoffs and their relative bargaining power to the agreement that maximises the weighted product of their gains over disagreement. In a price negotiation between a payer and a manufacturer the feasible agreements are prices from the manufacturer's reservation price up to the payer's maximum price, and the function returns the price that splits the surplus between them in proportion to their bargaining weights. A weight of one half gives Nash's original symmetric solution. The payer's maximum price can come from a cost-effectiveness threshold (HE-FM-BARG-002), the manufacturer's weight from the parties' patience in an alternating-offers model (HE-FM-BARG-003), and an outside option changes the result only when it binds (HE-FM-BARG-004).
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Implementations
Excel
Price with the payer's outside option in one cell
Excel takes the smaller of the Nash price and the payer's maximum price less the value of its outside option, using named cells.
=MIN(NashPrice,MaxPrice-PayerOutsideOption)
Assumptions
Outside option available only by leaving the negotiation
Taking the outside option ends the negotiation for good, and the payoffs while talks are stalled stay as in HE-FM-BARG-001. A payoff available during a delay is an inside option and enters through the disagreement point instead.
Outside option that still leaves a deal the manufacturer accepts
The value v_P is no larger than p_bar minus c, so the resulting price still covers the manufacturer's reservation price. A larger outside option leaves no price acceptable to both sides and the payer takes the alternative. The manufacturer has no outside option that binds.
Worked examples
Competing therapy worth £4,000 per patient to the payer
Under symmetric weights the Nash price is £4,500, leaving the payer £3,500 per patient. A competing therapy worth £4,000 beats that, so the price falls to £4,000 per patient, or £12,000 per QALY, as in the article.
p_star = 4500; p_bar = 8000; v_P = 4000; p_neg = 4000
Competing therapy worth £3,000 per patient as an empty threat
An outside option worth £3,000 is less than the £3,500 the payer already gains from the symmetric bargain, so the price stays at £4,500 per patient.
p_star = 4500; p_bar = 8000; v_P = 3000; p_neg = 4500
Common errors
Treating the outside option as the payer's disagreement payoff
Subtracting v_P from the payer's gain inside the Nash product and then splitting what is left overstates the payer's power. Under symmetric weights a £4,000 alternative would give a price of £2,500 rather than £4,000, and a £3,000 alternative, which should leave the price at £4,500, would cut it to £3,000.
Applying the rule when the outside option leaves no bargaining range
A competing therapy worth £7,500 per patient gives a formula price of £500, below the £1,000 reservation price. The manufacturer would not sell at that price, so no agreement is reached and the payer funds the alternative.
Sources
Muthoo on the outside option principle
Muthoo A. A non-technical introduction to bargaining theory. World Economics. 2000;1(2):145-166. Section 4, which shows that the agreed price is unchanged by an outside option worth less than what the party obtains without it and equals the outside option when it is worth more, the outside option principle; section 1.2 distinguishes outside options from inside options.
Canonical Identity
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