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 |

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

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