# Game Theory: 2 x n Games

Games where one player has only two courses of action while the other has more than two, are called 2 X n or n X 2 games.

If these games do not have a saddle point or are reducible by the dominance method, then before solving these games we write all 2 X 2 sub-games and determine the value of each 2 X 2 sub-game.

This method is illustrated by the following example.

## Example: 2 x n Games

Determine the solution of game for the pay-off matrix given below:

Player B
Player A   I II III
I -3 -1 7
II 4 1 -2

Solution.

Obviously, there is no saddle point and also no course of action dominates the other. Therefore, we consider each 2 X 2 sub-game and obtain their values.

(a)

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Player B
Player A   I II
I -3 -1
II 4 1

The saddle point is 1. So the value of game, V1 is 1.

(b)

Player B
Player A   I II
I -3 7
II 4 -2

This game has no saddle point, so we use the algebraic method.

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 Value of game, V2 = (-3) X (-2) - (7 X 4) ------------------------- (-3 - 2) - (7 + 4) = 11 --- 8

(c)

Player B
Player A   II III
I -1 7
II 1 -2

This game has no saddle point, so we use the algebraic method.

 Value of game, V3 = (-1) X (-2) - (7 X 1) ----------------------- (-1 - 2) - (7 + 1) = 5 --- 11

The 2 X 2 sub-game with the lowest value is (c) and hence the solution to this game provides the solution to the larger game.

Using algebraic method:

A plays ( 3/11, 8/11)
B plays (0, 9/11, 2/11)
Value of game is 5/11.

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