Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Nash bargaining surplus-division function for negotiated health care prices

x* = argmax_(x in F) (U_M(x) - d_M)^beta * (U_P(x) - d_P)^(1 - beta)

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).

  • Nash bargaining price between a reservation price and a payer's maximum price

    p_star = c + beta * (p_bar - c)

    Gives the negotiated price when both parties' payoffs are in money and move one for one with the price: the manufacturer gains the price minus its reservation price c, and the payer gains its maximum price p_bar minus the price paid. Maximising the weighted Nash product of the two gains gives a weighted average of the two limits, so the manufacturer receives the share beta of the surplus p_bar minus c and the payer the share 1 minus beta. Berdud, Ferraro and Towse state the same weighted average in cost-effectiveness ratio units, from the developer's reserve ratio to the payer's maximum.

  • Payer's maximum price per patient from a cost-effectiveness threshold

    p_bar = k * Delta_E - Delta_C_o

    Gives the highest price per patient at which a payer that funds technologies with an incremental cost-effectiveness ratio at or below the threshold k still funds the product. At this price the ratio equals k and the incremental net monetary benefit (HE-FM-NMB-002) is zero. In bargaining terms it is the payer's walk-away price, the top of the range over which the price is negotiated.

  • Rubinstein first proposer's share of the surplus under alternating offers

    s_1 = (1 - delta_2) / (1 - delta_1 * delta_2)

    Gives the share of a surplus of size one obtained by the party that makes the first offer in the unique perfect equilibrium of Rubinstein's alternating-offers model, in which each party discounts agreement by a fixed factor per round of delay. The more patient a party, the larger its share, and the responder receives 1 minus s_1. Multiplied by the surplus p_bar minus c it gives the first proposer's gain, so s_1 can serve as the weight beta in HE-FM-BARG-001 for the party that opens.

  • Negotiated price when the payer holds a binding outside option

    p_neg = min(p_star, p_bar - v_P)

    Applies the outside option principle to the payer. An outside option, the net monetary benefit per patient v_P available by walking away for good, for example to a competing therapy, changes the price only if it is worth more than the payer's gain from the bargain without it, p_bar minus p_star. In that case the price falls until the payer's gain just equals the outside option; otherwise the threat is empty and the Nash price stands. The function min returns the smaller of its arguments.