Concept Architecture
Concept
Theoretically, Bargaining Theory analyses how two or more parties divide the gains from agreement when each party has alternative options, strategic power and preferences over possible outcomes. It is grounded in game theory and cooperative and non-cooperative models of negotiation. In health economics, bargaining theory explains negotiations between purchasers, providers, pharmaceutical manufacturers, insurers, clinicians and patients over prices, reimbursement, wages, contracts and access conditions.
Mathematically, bargaining theory represents negotiated outcomes through feasible payoff sets, disagreement payoffs and bargaining power parameters. The Nash bargaining solution selects the agreement that maximises the weighted product of each party?s gain relative to its disagreement position. Alternative models include Rubinstein alternating-offer bargaining and bilateral monopoly models, which incorporate timing, discounting and strategic offers.
In practice, bargaining models are estimated or calibrated using observed prices, reimbursement agreements, market shares, outside options and measures of negotiating power. Structural econometric models may recover bargaining parameters from hospital-insurer contracts, pharmaceutical prices or provider payment data. Scenario analysis is then used to assess how mergers, market concentration or changes in outside options affect negotiated prices and healthcare expenditure.
Purpose
Used to analyse negotiated healthcare prices, contracts and reimbursement outcomes by modelling the distribution of gains between parties with differing bargaining power and alternatives to agreement.
Mathematical Formulae
Primary Formula
Nash bargaining solution:
x* = argmax??F (U?(x) ? d?)? ? (U?(x) ? d?)???
where:
- F = feasible set of agreements
- U?(x), U?(x) = party payoffs under agreement x
- d?, d? = disagreement payoffs
- � = bargaining power of party 1
Supporting Formulae
Logarithmic representation:
x* = argmax??F [�ln(U?(x) ? d?) + (1 ? �)ln(U?(x) ? d?)]
Symmetric Nash bargaining solution:
x* = argmax??F (U?(x) ? d?)(U?(x) ? d?)
Discounted payoff:
PV = V / (1 + r)?
Related Mathematical Methods
- Nash Bargaining Solution
- Rubinstein Bargaining Model
- Cooperative Game Theory
- Non-Cooperative Game Theory
- Bilateral Monopoly
- Structural Econometric Modelling
Example
A hospital and an insurer negotiate the reimbursement price for a surgical episode. Agreement generates a total surplus of �2,000 per episode above their respective disagreement payoffs. The hospital has bargaining power � = 0.60.
Hospital share:
0.60 ? �2,000 = �1,200
Insurer share:
0.40 ? �2,000 = �800
If the hospital?s disagreement payoff corresponds to �7,000 per episode, the negotiated reimbursement price is:
�7,000 + �1,200 = �8,200
The result demonstrates how stronger hospital bargaining power increases the negotiated price.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| PRODUCT | =(B2-D2)^$F$1*(C2-E2)^(1-$F$1) | Calculates the weighted Nash product for a proposed agreement. |
| LN | =$F$1*LN(B2-D2)+(1-$F$1)*LN(C2-E2) | Calculates the logarithmic Nash objective. |
| MAX | =MAX(ObjectiveRange) | Identifies the agreement producing the largest bargaining objective. |
| INDEX | =INDEX(PriceRange,MATCH(MAX(ObjectiveRange),ObjectiveRange,0)) | Retrieves the negotiated price associated with the maximum Nash product. |
| LET | =LET(Surplus,B2,Power,C2,Surplus*Power) | Allocates negotiated surplus according to bargaining power. |
VBA (Optional)
VBA can automate negotiation simulations across alternative disagreement payoffs, bargaining powers and feasible reimbursement prices.
Sources
- Nash JF. The bargaining problem. Econometrica.
- Rubinstein A. Perfect equilibrium in a bargaining model. Econometrica.
- Muthoo A. Bargaining Theory with Applications. Cambridge University Press.
- Grennan M. Price discrimination and bargaining: empirical evidence from medical devices. American Economic Review.
- Gowrisankaran G, Nevo A, Town R. Mergers when prices are negotiated: evidence from the hospital industry. American Economic Review.
Related Concepts (2)
Frequently Asked Questions (6)
What is bargaining theory?
A branch of economic theory analysing how parties negotiate to reach a mutually acceptable agreement when their interests partly conflict.
Source: Nash 1950
How does bargaining theory differ from perfect competition?
Under perfect competition no single buyer or seller can influence the price, which is set impersonally by the market, so there is nothing to negotiate. Bargaining theory addresses the opposite situation, where a small number of parties must agree terms between themselves and each has some power to hold out for a better deal. The outcome then depends on their relative patience, their alternatives if talks fail, and the rules of negotiation rather than on a market-clearing price. Muthoo (1999) sets out this contrast.
Source: Muthoo 1999
What determines the outcome in bargaining theory?
The outcome depends on the gain available from agreement and on each party's position if no agreement is reached, the disagreement or threat point. A party that loses little from failing to agree is in a stronger position and can secure more favourable terms, while a party with much to lose is weaker. Nash's solution selects the division satisfying certain reasonable conditions, and it shows the settlement shifting toward the party with the better outside option.
Source: Nash 1950
What is the Nash bargaining solution?
The Nash bargaining solution is a proposed outcome for a bargaining problem, derived by Nash from a small set of reasonable axioms such as efficiency and symmetry. It selects the agreement that maximises the product of the parties' gains over their disagreement points, giving a unique division of the surplus. The solution provides a benchmark for how rational parties might split the gain from agreement, with the split depending on their preferences and their positions should bargaining break down.
Source: Nash 1950
What is the threat point in bargaining?
The threat point, or disagreement point, is the outcome each party receives if no agreement is reached, and it anchors the bargain. The terms of any settlement must leave each party at least as well off as at its threat point, or that party would refuse to agree. A better outside option raises a party's threat point and strengthens its bargaining position, shifting the agreed terms in its favour. The threat point therefore determines much of the balance of power in a negotiation.
Source: Nash 1950
How does bargaining theory apply to health care?
Bargaining theory applies to health care wherever terms are negotiated rather than set by open competition, such as between insurers and hospitals over prices, between payers and drug manufacturers over reimbursement, or between employers and health plans. It explains how the balance of the settlement depends on each side's alternatives: a hospital that a payer's network cannot do without commands better terms, while a payer with other options is stronger. The framework clarifies why negotiated health care prices vary with bargaining power.
Source: Nash 1950
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 9 Sep 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/bargaining-theory
- Term code
- HE-EE-ME-005
Stable URI · Machine-readable · Resolvable · CC BY 4.0