Calculus Method: Game Theory
The Calculus method is almost similar to the previous method (algebraic 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:
| Player B | |||
|---|---|---|---|
| Player A | I | II | |
| I | a | b | |
| II | c | d | |
Formulas: Calculus Method
The solution of the game is:
A play’s (p, 1 - p)
where:
| p = | d - c ------------------- (a + d) - (b + c) |
B play’s (q, 1 - q)
where:
| q = | d - b -------------------- (a + d) - (b + c) |
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 Calculus Method: Game Theory
Consider the following game:
| Player B | |||
|---|---|---|---|
| Player A | I | II | |
| I | 2 | -1 | |
| II | -1 | 1 | |
Solution.
This game has no saddle point.
| p = | 1 - (-1) ----------------------- (2 + 1) - (-1 - 1) |
= | 2 ---- 5 |
1 – p = 3/5
| q = | 1 - (-1) ----------------------- (2 + 1) - (-1 - 1) |
= | 2 ---- 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
Calculus Method Example 2: Game Theory
Solve the game whose pay-off matrix is given below:
| Player B | |||
|---|---|---|---|
| Player A | I | II | |
| I | 1 | 3 | |
| II | 5 | 2 | |
Solution.
This game has no saddle point.
| p = | 2 - 5 ----------------------- (1 + 2) - (3 + 5) |
= | 3 ---- 5 |
1 – p = 2/5
| q = | 2 - 3 ----------------------- (1 + 2) - (3 + 5) |
= | 1 ---- 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
