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:

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 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.

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 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