Signature
s_1 = (1 - delta_2) / (1 - delta_1 * delta_2)
| Inputs | Definition | Unit |
|---|---|---|
delta_2 | Factor by which the party responding to the first offer discounts agreement for each round of delay | dimensionless, per round, above zero and below one |
delta_1 | Factor by which the first proposer discounts agreement for each round of delay; a value closer to one means a more patient party | dimensionless, per round, above zero and below one |
s_1 | Share of the surplus obtained in equilibrium by the party that makes the first offer | dimensionless share of the surplus |
|---|
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
First proposer's alternating-offers share in one cell
Excel returns the first proposer's share from named cells holding the proposer's and the responder's discount factors.
=(1-ResponderDelta)/(1-ProposerDelta*ResponderDelta)
Assumptions
Alternating offers with fixed per-round discounting
The parties take turns proposing how to divide a surplus of size one, with no limit on the number of rounds, and each discounts later agreement by a fixed factor per round. Both factors are below one, so delay is costly to each party.
Complete information and no outside options in Rubinstein bargaining
Each party knows the other's discount factor, there is no risk that talks break down for outside reasons, and neither party can walk away to an outside option. A payer's outside option is added with HE-FM-BARG-004.
Worked examples
Manufacturer opens with discount factors of 0.9 and 0.8
A manufacturer that opens and discounts each round by 0.9, facing a payer under clinical pressure that discounts by 0.8, obtains about 0.714 of the surplus, exactly five sevenths. Of the article's £7,000 surplus per patient that is £5,000, a price of £6,000.
delta_1 = 0.9; delta_2 = 0.8; s_1 = 0.714286
Payer opens with discount factors of 0.8 and 0.9
If the payer opens instead, its share as first proposer is about 0.357, exactly five fourteenths. The manufacturer's share falls to nine fourteenths, about 0.643, and the price to £5,500 per patient.
delta_1 = 0.8; delta_2 = 0.9; s_1 = 0.357143
Common errors
Swapping the proposer's and responder's discount factors
Putting the first proposer's own discount factor in the numerator gives the share the responder would obtain if it opened. With the manufacturer opening at 0.9 against 0.8 the swap gives about 0.357 instead of 0.714, and a price of £3,500 rather than £6,000 per patient.
Ignoring which party makes the first offer
The share s_1 includes the advantage of moving first. Using the manufacturer's opening share of five sevenths when the payer actually opens overstates the manufacturer's weight, which is nine fourteenths, and raises the price from £5,500 to £6,000 per patient.
Sources
Rubinstein on the perfect equilibrium partition with fixed discount factors
Rubinstein A. Perfect equilibrium in a bargaining model. Econometrica. 1982;50(1):97-109. Abstract, which gives the only perfect equilibrium partition when each player has a fixed discount factor as 1 minus delta_2, divided by 1 minus delta_1 times delta_2, for the player who proposes first.
Binmore, Rubinstein and Wolinsky on alternating offers and the Nash solution
Binmore K, Rubinstein A, Wolinsky A. The Nash bargaining solution in economic modelling. RAND Journal of Economics. 1986;17(2):176-188. Abstract, which shows that as the motivation to reach agreement becomes negligible the alternating-offers equilibrium approaches the Nash bargaining solution.
Muthoo on patience as a source of bargaining power
Muthoo A. A non-technical introduction to bargaining theory. World Economics. 2000;1(2):145-166. Section 2, which explains that a player's share of the surplus is greater the more patient that player is relative to the other negotiator.
Canonical Identity
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