Nash Product

Another solution concept is the maximization of the Nash product N.

N = (x₁-d₁)*(x₂-d₂)

d_i: Disagreement points

Graphically, this is the area under the pay-off points.

Numerical Example 1

Let’s assume we want to split 1 EUR. If the negotiation participants cannot come to an agreement, they receive 0 (d_i = 0).

The Nash product is:

N = (x₁-0)*(x₂-0)

It is maximal for x₁ = x₂ = ½

Now, we solve the equation with Lagrange.

N = a*b

N: Nash product

Side condition:

a+b = 1

1-a-b = 0

L = a*b+u(1-a-b)

dL/du = 1-a-b = 0

dL/da = b-u = 0

dL/db = a-u = 0

a = b = ½

Numerical Example 2

Player A receives a payment of 0.4 in case of a failed negotiation. This means the disagreement point for A is 0.4.

N = (x₁-0.4) * (x₂-0)

N = (x₁-0.4) * (1-x₁-0)

x₁ = 0.7

Negotiation Power

a is now a parameter for negotiation skills

N = (x₁-d₁)a*(x₂-d₂)(1-a)

• a = 0.5: both participants have the same power
• a > 0.5: participant 1 has more negotiation power than participant 2
• a < 0.5 participant 1 has less negotiation power than participant 2