Game Theory problem using Matrix method Example-2

2. Example-2

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Method & Example-1
Example-2

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8. 2Xn Games
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Player `B`

Player `A`

Player `A`

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
Player `A`	`A_3`	16	14	10

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

reduced matrix

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`

This material is intended as a summary. Use your textbook for detail explanation.
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