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6. Matrix method example ( Enter your problem )
 Method & Example-1Example-2 Other related methods

1. Method & Example-1

Method
 Step-1: Player A's optimal strategies =([[1,1]] xx P_(Adj))/([[1,1]] xx P_(Adj) xx [[1],[1]]) = [[p_1,p_2]] Step-2: Player B's optimal strategies =([[1,1]] xx P_(Cof))/([[1,1]] xx P_(Adj) xx [[1],[1]]) = [[q_1,q_2]] Step-3: Value of the game = (Player A's optimal strategies) xx (Payoff matrix P) xx (Player B's optimal strategies) V = [[p_1,p_2]] xx P xx [[q_1],[q_2]]

Example-1
1. Find Solution of game theory problem using matrix 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 RowMinimum Player A A_1 1 7 2 1 A_2 (6) [2] 7 [2] A_3 5 1 6 1 ColumnMaximum (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.

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

reduced matrix
 Player B B_1 B_2 Player A A_1 1 7 A_2 6 2

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

P_(Adj) = [[2,-7],[-6,1]]

and P_(Cof) = [[2,-6],[-7,1]]

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

=([[1,1]][[2,-7],[-6,1]])/([[1,1]][[2,-7],[-6,1]][[1],[1]])

=([[-4,-6]])/(-10)

=[[2/5,3/5]]

p_1=2/5 and p_2=3/5, where p_1 and p_2 represent the probabilities of player A's, using his strategies A_1 and A_2 respectively.

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

=([[1,1]][[2,-6],[-7,1]])/([[1,1]][[2,-7],[-6,1]][[1],[1]])

=([[-5,-5]])/(-10)

=[[1/2,1/2]]

q_1=1/2 and q_2=1/2, where q_1 and q_2 represent the probabilities of player B's, using his strategies B_1 and B_2 respectively.

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

V=[[2/5,3/5]][[1,7],[6,2]][[1/2],[1/2]]=4

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