Example 1. Nash arbitration with the golden section search method
Assume we have the following partial conflict game between two players:
Player II Strategies C1 C2
Player I R1 (2, 6) (5,4) R2 (10,5) (4,8)
Assume in this example we have already found the Nash equilibrium, (4.666, 5.6). We plot the points and obtain the convex polygon shown in figure 3.
Figure 3. Payoff polygon and Pareto optimal line segment
We find the Nash equilibrium is not Pareto optimal. To be Pareto optimal, the Nash equilibrium must line on the Pareto optimal line segment. For sake of illustration, we assume we have tried all other strategic moves’ methods to improve our outcomes and we move on to arbitration.
We use the end points (4, 8) and (10, 5) to obtain the equation of the Pareto Optimal line shown in figure 3 using the point-slope formula of a line: .
We find the equation as (y-8) =-3/6 (x-4) which simplifies to y= -1/2 x +10.
Now, we can return to the Nash arbitration scheme that says in particular find the values of (x, y) along the line y=-1/2 x + 10 where x > 4.6666, y > 5.6 that maximizes the product (x-4.6666) (y-5.6).
To use golden section in one dimension, we simply create a function of one variable by substituting -1/2 x + 10 for y in the function to be maximized, (x-4.6666) (-1/2 x + 10-5.6) or (x-4.6666) (-1/2 x + 4.4). We can simplify this through multiplication, if desired, to obtain -1/2 x2 + 6.73333 x -20.53333. The next crucial element is to determine the interval. The value of x must be greater than 4.6666 and less than the x coordinate of right most end of point of the Pareto optimal line, which in this case is 10.
Therefore, we will use golden section to find the value of x that maximizes the function, -1/2 x2 + 6.73333 x -20.53333, within the interval [4.6666, 10].
We have built a template, available upon request, to perform the calculations. We find the value of x that maximizes this function within the interval to be x= 6.733241. So how do we find the y coordinate? We substitute x= 6.733241 into the equation y = -1/2 x + 10 and find y =6.63338. Our Nash arbitration