Find Solution of game theory problem using calculus method

Player A\Player B	B1	B2
A1	2	-1
A2	-1	1

Solution:
1. Saddle point testing
Players

			Player `B`
			`B_1`	`B_2`
Player `A`	`A_1`		2	-1
	`A_2`		-1	1

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

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

Select minimum from the maximum of columns
Column MiniMax = (1)

Select maximum from the minimum of rows
Row MaxiMin = [-1]

Here, Column MiniMax `!=` Row MaxiMin

`:.` This game has no saddle point.

---

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

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

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

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

`q_2=1-q_1=1-2/5=3/5`

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

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

`=8/25 -6/25 -6/25 +9/25`

`=1/5`

---

---