Find Solution of game theory problem using arithmetic method

Player A\Player B	B1	B2	B3
A1	1	7	2
A2	6	2	7
A3	5	1	6

Solution:
1. Saddle point testing
Players

			Player `B`
			`B_1`	`B_2`	`B_3`
Player `A`	`A_1`		1	7	2
	`A_2`		6	2	7
	`A_3`		5	1	6

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

			Player `B`
			`B_1`	`B_2`	`B_3`		Row Minimum
Player `A`	`A_1`		1	7	2		`1`
	`A_2`		(6)	[2]	7		`[2]`
	`A_3`		5	1	6		`1`
	Column Maximum		`(6)`	`7`	`7`

Select minimum from the maximum of columns
Column MiniMax = (6)

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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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`		1	7	2
	`A_2`		6	2	7
	`A_3`		5	1	6

row-3 `<=` row-2, so remove row-3

			Player `B`
			`B_1`	`B_2`	`B_3`
Player `A`	`A_1`		1	7	2
	`A_2`		6	2	7

column-3 `>=` column-1, so remove column-3

			Player `B`
			`B_1`	`B_2`
Player `A`	`A_1`		1	7
	`A_2`		6	2

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reduced matrix

			Player `B`
			`B_1`	`B_2`
Player `A`	`A_1`		1	7
	`A_2`		6	2

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

			Player `B`
			`B_1`	`B_2`
Player `A`	`A_1`		1	7		`|6-2|=4` `:. p_1=(4)/(6+4)=2/5`
	`A_2`		6	2		`|1-7|=6` `:. p_2=(6)/(6+4)=3/5`
			`|7-2|=5` `:. q_1=(5)/(5+5)=1/2`	`|1-6|=5` `:. q_2=(5)/(5+5)=1/2`

1. Find absolute difference between the two values in the first row and put it against second row of the matrix
`|1-7|=6`

2. Find absolute difference between the two values in the second row and put it against first row of the matrix
`|6-2|=4`

`:. p_1=(4)/(6+4)=2/5`

`:. p_2=(6)/(6+4)=3/5`


3. Find absolute difference between the two values in the first column and put it against second column of the matrix
`|1-6|=5`

4. Find absolute difference between the two values in the second column and put it against first column of the matrix
`|7-2|=5`

`:. q_1=(5)/(5+5)=1/2`

`:. q_2=(5)/(5+5)=1/2`


Hence, firm `A` should adopt strategy `A_1` and `A_2` with 40% of time and 60% of time respectively.

Similarly, firm `B` should adopt strategy `B_1` and `B_2` with 50% of time and 50% of time respectively.


Expected gain of Firm A
`(1)` `1 xx 2/5+6 xx 3/5 = 4 ,` Firm `B` adopt `B_1`

`(2)` `7 xx 2/5+2 xx 3/5 = 4 ,` Firm `B` adopt `B_2`


Expected loss of Firm B
`(1)` `1 xx 1/2+7 xx 1/2 = 4 ,` Firm `A` adopt `A_1`

`(2)` `6 xx 1/2+2 xx 1/2 = 4 ,` Firm `A` adopt `A_2`

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