Find Solution of game theory problem using matrix 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.

---

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

---

reduced matrix

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

For this reduced matrix, calculate `P_(Adj)` and `P_(Cof)`

`P_(Adj) = [[10,-15],[-14,12]]`

and `P_(Cof) = [[10,-14],[-15,12]]`

Player A's optimal strategies `=([[1,1]] xx P_(Adj))/([[1,1]] xx P_(Adj) xx [[1],[1]])`

`=([[1,1]][[10,-15],[-14,12]])/([[1,1]][[10,-15],[-14,12]][[1],[1]])`

`=([[-4,-3]])/(-7)`

`=[[4/7,3/7]]`

`p_1=4/7` and `p_2=3/7`, where `p_1` and `p_2` represent the probabilities of player A's, using his strategies `A_2` and `A_3` respectively.


Similarly,
Player B's optimal strategies `=([[1,1]] xx P_(Cof))/([[1,1]] xx P_(Adj) xx [[1],[1]])`

`=([[1,1]][[10,-14],[-15,12]])/([[1,1]][[10,-15],[-14,12]][[1],[1]])`

`=([[-5,-2]])/(-7)`

`=[[5/7,2/7]]`

`q_1=5/7` and `q_2=2/7`, where `q_1` and `q_2` represent the probabilities of player B's, using his strategies `B_2` and `B_3` respectively.


Hence, Value of the game `V` = (Player A's optimal strategies) `xx` (Payoff matrix `P_(ij)`) `xx` (Player B's optimal strategies)

`V=[[4/7,3/7]][[12,15],[14,10]][[5/7],[2/7]]=90/7`

---

---