Game Theory

Calculus Method

This method is almost similar to the previous method except that instead of equating the two expected values, the expected value for a given player is maximized.

Consider the zero sum two person game given below:

Formulas:

The solution of the game is:

A play’s (p, 1 - p)

where:

B play’s (q, 1 - q)

where:

Value of the game, V = apq + c(1 – p)q + bp(1 – q) + d(1 – p)(1 – q)

To illustrate this method, consider the same example discussed in the previous section.

Example 1

Consider the following game:

Solution.

This game has no saddle point.

1 – p = 3/5

1 – q = 3/5

V = 2 X 2/5 X 2/5 + (-1) X 3/5 X 2/5 + (-1) X 2/5 X 3/5 + 1 X 3/5 X 3/5 = 1/5

Example 2

Solve the game whose pay-off matrix is given below:

Solution.

This game has no saddle point.

1 – p = 2/5

1 – q = 4/5

V = 1 X 3/5 X 1/5 + 5 X 2/5 X 1/5 + 3 X 3/5 X 4/5 + 2 X 2/5 X 4/5 = 13/5