Method

calculus method Steps (Rule)
Step-1:	A play's (`p_1`, `p_2`) `p_1=(d - c)/((a + d) - (b + c))` and `p_2 = 1 - p_1`
Step-2:	B play's (`q_1`, `q_2`) `q_1=(d - b)/((a + d) - (b + c))` and `q_2 = 1 - q_1`
Step-3:	Value of the game `V` `V=a*p_1*q_1+b*p_1*q_2+c*p_2*q_1+d*p_2*q_2`

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Example-1
1. Find Solution of game theory problem using calculus method

Player A\Player B	B1	B2
A1	1	3
A2	5	2

Solution:
1. Saddle point testing
Players

			Player `B`
			`B_1`	`B_2`
Player `A`	`A_1`		1	3
	`A_2`		5	2

We apply the maximin (minimax) principle to analyze the game.

			Player `B`
			`B_1`	`B_2`		Row Minimum
Player `A`	`A_1`		1	(3)		`1`
	`A_2`		5	[2]		`[2]`
	Column Maximum		`5`	`(3)`

Select minimum from the maximum of columns
Column MiniMax = (3)

Select maximum from the minimum of rows
Row MaxiMin = [2]

Here, Column MiniMax `!=` Row MaxiMin

`:.` This game has no saddle point.

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Matrix size is 2`xx`2, so dominance rule is not required.

Solution using calculus method
Here `a=1,b=3,c=5,d=2`

`p_1=(d - c)/((a + d) - (b + c))=(2 -5)/((1 +2) - (3 +5))=(-3)/(3 -8)=3/5`

`p_2=1-p_1=1-3/5=2/5`

`q_1=(d - b)/((a + d) - (b + c))=(2 -3)/((1 +2) - (3 +5))=(-1)/(3 -8)=1/5`

`q_2=1-q_1=1-1/5=4/5`

`V=a*p_1*q_1+b*p_1*q_2+c*p_2*q_1+d*p_2*q_2`

`=1*3/5*1/5 +3*3/5*4/5 +5*2/5*1/5 +2*2/5*4/5`

`=3/25 +36/25 +2/5 +16/25`

`=13/5`

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