1. Find Solution of game theory problem using arithmetic method

Player A\Player B	B1	B2	B3
A1	10	5	-2
A2	13	12	15
A3	16	14	10

Solution:
1. Saddle point testing
Players

			Player `B`
			`B_1`	`B_2`	`B_3`
Player `A`	`A_1`		10	5	-2
	`A_2`		13	12	15
	`A_3`		16	14	10

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

			Player `B`
			`B_1`	`B_2`	`B_3`		Row Minimum
Player `A`	`A_1`		10	5	-2		`-2`
	`A_2`		13	[12]	15		`[12]`
	`A_3`		16	(14)	10		`10`
	Column Maximum		`16`	`(14)`	`15`

Select minimum from the maximum of columns
Column MiniMax = (14)

Select maximum from the minimum of rows
Row MaxiMin = [12]

Here, Column MiniMax `!=` Row MaxiMin

`:.` This game has no saddle point.

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2. Dominance rule to reduce the size of the payoff matrix
Using dominance property

			Player `B`
			`B_1`	`B_2`	`B_3`
Player `A`	`A_1`		10	5	-2
	`A_2`		13	12	15
	`A_3`		16	14	10

row-1 `<=` row-3, so remove row-1

			Player `B`
			`B_1`	`B_2`	`B_3`
Player `A`	`A_2`		13	12	15
	`A_3`		16	14	10

column-1 `>=` column-2, so remove column-1

			Player `B`
			`B_2`	`B_3`
Player `A`	`A_2`		12	15
	`A_3`		14	10

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			Player `B`
			`B_2`	`B_3`
Player `A`	`A_2`		12	15
	`A_3`		14	10

Using arithemetic method to get optimal mixed strategies for both the firms.

			Player `B`
			`B_2`	`B_3`
Player `A`	`A_2`		12	15		`|14-10|=4` `:. p_1=(4)/(3+4)=4/7`
	`A_3`		14	10		`|12-15|=3` `:. p_2=(3)/(3+4)=3/7`
			`|15-10|=5` `:. q_1=(5)/(2+5)=5/7`	`|12-14|=2` `:. q_2=(2)/(2+5)=2/7`

1. Find absolute difference between the two values in the first row and put it against second row of the matrix
`|12-15|=3`

2. Find absolute difference between the two values in the second row and put it against first row of the matrix
`|14-10|=4`

`:. p_1=(4)/(3+4)=4/7`

`:. p_2=(3)/(3+4)=3/7`


3. Find absolute difference between the two values in the first column and put it against second column of the matrix
`|12-14|=2`

4. Find absolute difference between the two values in the second column and put it against first column of the matrix
`|15-10|=5`

`:. q_1=(5)/(2+5)=5/7`

`:. q_2=(2)/(2+5)=2/7`


Hence, firm `A` should adopt strategy `A_2` and `A_3` with 57% of time and 43% of time respectively.

Similarly, firm `B` should adopt strategy `B_2` and `B_3` with 71% of time and 29% of time respectively.


Expected gain of Firm A
`(1)` `12 xx 4/7+14 xx 3/7 = 90/7 ,` Firm `B` adopt `B_2`

`(2)` `15 xx 4/7+10 xx 3/7 = 90/7 ,` Firm `B` adopt `B_3`


Expected loss of Firm B
`(1)` `12 xx 5/7+15 xx 2/7 = 90/7 ,` Firm `A` adopt `A_2`

`(2)` `14 xx 5/7+10 xx 2/7 = 90/7 ,` Firm `A` adopt `A_3`