C - Appendix to Chapter 5: Application to Games

Summary

The Nash solution of the bargaining problem whose frontier is the cubic curve with Equation (5.1.11) is the point on the frontier where [(1 + A)x − y − a] (y − b) takes its maximum value. We use the Lagrange multiplier method to find the maximum. Set the Lagrangian equal to

Then

It follows from the condition ∂Λ/∂y = 0 that

Set a − b = α. Then

However, if

then

Substitute the expression for y into Equation (5.1.11) and clear the denominator. The result is the following equation in x.

To find the optima in Example 5.2.1., we computed the partial derivative of P, the payoff function, with respect to y. The result is a quadratic in x with a single solution S(d, x) that is in the interval [0, 1] when x ∈ [0, 1] and d ∈ (0.5, 1.5). The solution S(d, x) substituted for y in P. When the resulting function of x and d is optimized, values to be in chosen in [0, 1], the result is the expression given in the second section of Chapter 5. One can then compute the y coordinate ŷ for the optimal point. The expression for is the following.

Each of the entries F1, …, F7 is an expression in d. The expressions are the following.