Game Theory problem using 2Xn Games Method & Example-1

1. Method & Example-1

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8. 2Xn Games example ( Enter your problem )

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Method & Example-1
Example-2

Other related methods

7. Matrix method
(Previous method)

2. Example-2
(Next example)

Player `B`

Player `A`

Method

2Xn Games Steps (Rule)
Step-1:	If this game does not have a saddle point and not reducible by the dominance method, then first write all 2`xx`2 sub-games and find the value of each 2`xx`2 sub-games.
Step-2:	Find 2`xx`2 sub-game which has the lowest value and this sub-game provides the solution to the larger game. Solve this sub-game using algebraic method.

Example-1
Find Solution of game theory problem using 2Xn Games
_{Player A}\^{Player B} B1 B2
A1 -3 4
A2 -1 1
A3 7 -2

Solution:
1. Saddle point testing
Players

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

		Player `B`
		`B_1`	`B_2`	Row Minimum
Player `A`	`A_1`	-3	(4)	`-3`
	`A_2`	[-1]	1	`[-1]`
	`A_3`	7	-2	`-2`
	Column Maximum	`7`	`(4)`

Select minimum from the maximum of columns
Column MiniMax = (4)

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

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

Also, no course of action dominates the other

So, we consider each 2`xx`2 sub-game and obtain their values

Players

2`xx`2 sub-game : (1) Row 1, 2

		Player `B`
		`B_1`	`B_2`
Player `A`	`A_1`	-3	4
Player `A`	`A_2`	-1	1

This game has saddle point and value of the game, `V_1` is -1

2`xx`2 sub-game : (2) Row 1, 3

		Player `B`
		`B_1`	`B_2`
Player `A`	`A_1`	-3	4
Player `A`	`A_3`	7	-2

This game has no saddle point, so we use the algebraic method.
Value of game, `V_2=(a * d - b * c)/((a + d) - (b + c))=((-3 xx -2) - (4 xx 7))/((-3 -2) - (4 +7))=(6 -28)/(-5 -11)=11/8`

2`xx`2 sub-game : (3) Row 2, 3

		Player `B`
		`B_1`	`B_2`
Player `A`	`A_2`	-1	1
Player `A`	`A_3`	7	-2

This game has no saddle point, so we use the algebraic method.
Value of game, `V_3=(a * d - b * c)/((a + d) - (b + c))=((-1 xx -2) - (1 xx 7))/((-1 -2) - (1 +7))=(2 -7)/(-3 -8)=5/11`

The 2`xx`2 sub-game with the lowest value is `-1` in (1) and hence the solution to this game provides the solution to the larger game.

Using algebraic method
1. Saddle point testing
Players

		Player `B`
		`B_1`	`B_2`
Player `A`	`A_1`	-3	4
Player `A`	`A_2`	-1	1